Python Programming

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Slide 1: Python Programming: An Introduction to Computer Science John M. Zelle, Ph.D. Version 1.0rc2 Fall 2002 Slide 2: Copyright c 2002 by John M. Zelle All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior written permission of the author. A This document was prepared with L TEX 2ε and reproduced by Wartburg College Printing Services. Slide 3: Contents 1 Computers and Programs 1.1 The Universal Machine . . . 1.2 Program Power . . . . . . . 1.3 What is Computer Science? . 1.4 Hardware Basics . . . . . . 1.5 Programming Languages . . 1.6 The Magic of Python . . . . 1.7 Inside a Python Program . . 1.8 Chaos and Computers . . . . 1.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 2 2 3 4 5 8 10 11 13 13 13 15 15 15 16 17 17 18 19 20 22 24 25 25 27 28 31 32 34 35 39 39 41 43 43 44 45 48 2 Writing Simple Programs 2.1 The Software Development Process . . . 2.2 Example Program: Temperature Converter 2.3 Elements of Programs . . . . . . . . . . . 2.3.1 Names . . . . . . . . . . . . . . 2.3.2 Expressions . . . . . . . . . . . . 2.4 Output Statements . . . . . . . . . . . . . 2.5 Assignment Statements . . . . . . . . . . 2.5.1 Simple Assignment . . . . . . . . 2.5.2 Assigning Input . . . . . . . . . . 2.5.3 Simultaneous Assignment . . . . 2.6 Deﬁnite Loops . . . . . . . . . . . . . . 2.7 Example Program: Future Value . . . . . 2.8 Exercises . . . . . . . . . . . . . . . . . 3 Computing with Numbers 3.1 Numeric Data Types . . . . . . . . 3.2 Using the Math Library . . . . . . . 3.3 Accumulating Results: Factorial . . 3.4 The Limits of Int . . . . . . . . . . 3.5 Handling Large Numbers: Long Ints 3.6 Type Conversions . . . . . . . . . . 3.7 Exercises . . . . . . . . . . . . . . 4 Computing with Strings 4.1 The String Data Type . . . . . . . 4.2 Simple String Processing . . . . . 4.3 Strings and Secret Codes . . . . . 4.3.1 String Representation . . . 4.3.2 Programming an Encoder 4.3.3 Programming a Decoder . 4.3.4 Other String Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Slide 4: ii 4.3.5 From Encoding to Encryption . . . . Output as String Manipulation . . . . . . . . 4.4.1 Converting Numbers to Strings . . . . 4.4.2 String Formatting . . . . . . . . . . . 4.4.3 Better Change Counter . . . . . . . . File Processing . . . . . . . . . . . . . . . . 4.5.1 Multi-Line Strings . . . . . . . . . . 4.5.2 File Processing . . . . . . . . . . . . 4.5.3 Example Program: Batch Usernames 4.5.4 Coming Attraction: Objects . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 49 49 50 51 52 52 53 55 56 57 61 61 62 64 68 73 75 75 76 79 79 79 81 81 81 82 85 85 86 89 90 93 95 97 101 101 101 103 104 105 107 109 112 112 113 114 116 116 116 4.4 4.5 4.6 5 Objects and Graphics 5.1 The Object of Objects . . . . . 5.2 Graphics Programming . . . . 5.3 Using Graphical Objects . . . 5.4 Graphing Future Value . . . . 5.5 Choosing Coordinates . . . . . 5.6 Interactive Graphics . . . . . . 5.6.1 Getting Mouse Clicks 5.6.2 Handling Textual Input 5.7 Graphics Module Reference . 5.7.1 GraphWin Objects . . 5.7.2 Graphics Objects . . . 5.7.3 Entry Objects . . . . . 5.7.4 Displaying Images . . 5.7.5 Generating Colors . . 5.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Deﬁning Functions 6.1 The Function of Functions . . . . . . . . . 6.2 Functions, Informally . . . . . . . . . . . . 6.3 Future Value with a Function . . . . . . . . 6.4 Functions and Parameters: The Gory Details 6.5 Functions that Return Values . . . . . . . . 6.6 Functions and Program Structure . . . . . . 6.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Control Structures, Part 1 7.1 Simple Decisions . . . . . . . . . . . . . . . . . 7.1.1 Example: Temperature Warnings . . . . . 7.1.2 Forming Simple Conditions . . . . . . . 7.1.3 Example: Conditional Program Execution 7.2 Two-Way Decisions . . . . . . . . . . . . . . . . 7.3 Multi-Way Decisions . . . . . . . . . . . . . . . 7.4 Exception Handling . . . . . . . . . . . . . . . . 7.5 Study in Design: Max of Three . . . . . . . . . . 7.5.1 Strategy 1: Compare Each to All . . . . 7.5.2 Strategy 2: Decision Tree . . . . . . . . 7.5.3 Strategy 3: Sequential Processing . . . . 7.5.4 Strategy 4: Use Python . . . . . . . . . . 7.5.5 Some Lessons . . . . . . . . . . . . . . 7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Slide 5: CONTENTS 8 Control Structures, Part 2 8.1 For Loops: A Quick Review . . . . . . . 8.2 Indeﬁnite Loops . . . . . . . . . . . . . . 8.3 Common Loop Patterns . . . . . . . . . . 8.3.1 Interactive Loops . . . . . . . . . 8.3.2 Sentinel Loops . . . . . . . . . . 8.3.3 File Loops . . . . . . . . . . . . 8.3.4 Nested Loops . . . . . . . . . . . 8.4 Computing with Booleans . . . . . . . . 8.4.1 Boolean Operators . . . . . . . . 8.4.2 Boolean Algebra . . . . . . . . . 8.5 Other Common Structures . . . . . . . . 8.5.1 Post-Test Loop . . . . . . . . . . 8.5.2 Loop and a Half . . . . . . . . . 8.5.3 Boolean Expressions as Decisions 8.6 Exercises . . . . . . . . . . . . . . . . . iii 119 119 120 121 121 123 125 126 127 127 129 130 130 132 132 134 137 137 137 138 138 140 140 141 142 143 144 146 148 148 148 149 150 150 151 152 155 155 156 156 156 159 159 160 162 164 164 164 166 166 166 169 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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. . . . . 9.3.2 Separation of Concerns . . . . . . . . 9.3.3 Second-Level Design . . . . . . . . . 9.3.4 Designing simNGames . . . . . . . . 9.3.5 Third-Level Design . . . . . . . . . . 9.3.6 Finishing Up . . . . . . . . . . . . . 9.3.7 Summary of the Design Process . . . 9.4 Bottom-Up Implementation . . . . . . . . . . 9.4.1 Unit Testing . . . . . . . . . . . . . . 9.4.2 Simulation Results . . . . . . . . . . 9.5 Other Design Techniques . . . . . . . . . . . 9.5.1 Prototyping and Spiral Development . 9.5.2 The Art of Design . . . . . . . . . . 9.6 Exercises . . . . . . . . . . . . . . . . . . . 10 Deﬁning Classes 10.1 Quick Review of Objects . . . . . . . . . 10.2 Example Program: Cannonball . . . . . . 10.2.1 Program Speciﬁcation . . . . . . 10.2.2 Designing the Program . . . . . . 10.2.3 Modularizing the Program . . . . 10.3 Deﬁning New Classes . . . . . . . . . . . 10.3.1 Example: Multi-Sided Dice . . . 10.3.2 Example: The Projectile Class . . 10.4 Objects and Encapsulation . . . . . . . . 10.4.1 Encapsulating Useful Abstractions 10.4.2 Putting Classes in Modules . . . . 10.5 Widget Objects . . . . . . . . . . . . . . 10.5.1 Example Program: Dice Roller . . 10.5.2 Building Buttons . . . . . . . . . 10.5.3 Building Dice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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Slide 6: iv CONTENTS 10.5.4 The Main Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 10.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 11 Data Collections 11.1 Example Problem: Simple Statistics . . . . 11.2 Applying Lists . . . . . . . . . . . . . . . 11.2.1 Lists are Sequences . . . . . . . . . 11.2.2 Lists vs. Strings . . . . . . . . . . . 11.2.3 List Operations . . . . . . . . . . . 11.3 Statistics with Lists . . . . . . . . . . . . . 11.4 Combining Lists and Classes . . . . . . . . 11.5 Case Study: Python Calculator . . . . . . . 11.5.1 A Calculator as an Object . . . . . 11.5.2 Constructing the Interface . . . . . 11.5.3 Processing Buttons . . . . . . . . . 11.6 Non-Sequential Collections . . . . . . . . . 11.6.1 Dictionary Basics . . . . . . . . . . 11.6.2 Dictionary Operations . . . . . . . 11.6.3 Example Program: Word Frequency 11.7 Exercises . . . . . . . . . . . . . . . . . . 12 Object-Oriented Design 12.1 The Process of OOD . . . . . . . . . . 12.2 Case Study: Racquetball Simulation . . 12.2.1 Candidate Objects and Methods 12.2.2 Implementing SimStats . . . . . 12.2.3 Implementing RBallGame . . . 12.2.4 Implementing Player . . . . . . 12.2.5 The Complete Program . . . . . 12.3 Case Study: Dice Poker . . . . . . . . . 12.3.1 Program Speciﬁcation . . . . . 12.3.2 Identifying Candidate Objects . 12.3.3 Implementing the Model . . . . 12.3.4 A Text-Based UI . . . . . . . . 12.3.5 Developing a GUI . . . . . . . 12.4 OO Concepts . . . . . . . . . . . . . . 12.4.1 Encapsulation . . . . . . . . . . 12.4.2 Polymorphism . . . . . . . . . 12.4.3 Inheritance . . . . . . . . . . . 12.5 Exercises . . . . . . . . . . . . . . . . 13 Algorithm Analysis and Design 13.1 Searching . . . . . . . . . . . . . . . . 13.1.1 A Simple Searching Problem . . 13.1.2 Strategy 1: Linear Search . . . . 13.1.3 Strategy 2: Binary Search . . . 13.1.4 Comparing Algorithms . . . . . 13.2 Recursive Problem-Solving . . . . . . . 13.2.1 Recursive Deﬁnitions . . . . . . 13.2.2 Recursive Functions . . . . . . 13.2.3 Recursive Search . . . . . . . . 13.3 Sorting Algorithms . . . . . . . . . . . 13.3.1 Naive Sorting: Selection Sort . 13.3.2 Divide and Conquer: Merge Sort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 177 178 178 179 180 181 184 188 188 188 190 193 193 194 194 198 201 201 202 203 203 205 207 207 210 210 210 211 214 216 221 221 222 222 223 225 225 225 226 226 227 228 229 230 230 231 231 232 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Slide 7: CONTENTS 13.3.3 Comparing Sorts . . 13.4 Hard Problems . . . . . . . 13.4.1 Towers of Hanoi . . 13.4.2 The Halting Problem 13.4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v 234 235 236 239 241 Slide 8: vi CONTENTS Slide 9: Chapter 1 Computers and Programs Almost everyone has used a computer at one time or another. Perhaps you have played computer games or used a computer to write a paper or balance your checkbook. Computers are used to predict the weather, design airplanes, make movies, run businesses, perform ﬁnancial transactions, and control factories. Have you ever stopped to wonder what exactly a computer is? How can one device perform so many different tasks? These basic questions are the starting point for learning about computers and computer programming. 1.1 The Universal Machine A modern computer might be deﬁned as “a machine that stores and manipulates information under the control of a changeable program.” There are two key elements to this deﬁnition. The ﬁrst is that computers are devices for manipulating information. This means that we can put information into a computer, and it can transform the information into new, useful forms, and then output or display the information for our interpretation. Computers are not the only machines that manipulate information. When you use a simple calculator to add up a column of numbers, you are entering information (the numbers) and the calculator is processing the information to compute a running sum which is then displayed. Another simple example is a gas pump. As you ﬁll your tank, the pump uses certain inputs: the current price of gas per gallon and signals from a sensor that reads the rate of gas ﬂowing into your car. The pump transforms this input into information about how much gas you took and how much money you owe. We would not consider either the calculator or the gas pump as full-ﬂedged computers, although modern versions of these devices may actually contain embedded computers. They are different from computers in that they are built to perform a single, speciﬁc task. This is where the second part of our deﬁnition comes into the picture: computers operate under the control of a changeable program. What exactly does this mean? A computer program is a detailed, step-by-step set of instructions telling a computer exactly what to do. If we change the program, then the computer performs a different sequence of actions, and hence, performs a different task. It is this ﬂexibility that allows your PC to be at one moment a word processor, at the next moment a ﬁnancial planner, and later on, an arcade game. The machine stays the same, but the program controlling the machine changes. Every computer is just a machine for executing (carrying out) programs. There are many different kinds of computers. You might be familiar with Macintoshes and PCs, but there are literally thousands of other kinds of computers both real and theoretical. One of the remarkable discoveries of computer science is the realization that all of these different computers have the same power; with suitable programming, each computer can basically do all the things that any other computer can do. In this sense, the PC that you might have sitting on your desk is really a universal machine. It can do anything you want it to, provided you can describe the task to be accomplished in sufﬁcient detail. Now that’s a powerful machine! 1 Slide 10: 2 CHAPTER 1. COMPUTERS AND PROGRAMS 1.2 Program Power You have already learned an important lesson of computing: Software (programs) rules the hardware (the physical machine). It is the software that determines what any computer can do. Without programs, computers would just be expensive paperweights. The process of creating software is called programming, and that is the main focus of this book. Computer programming is a challenging activity. Good programming requires an ability to see the big picture while paying attention to minute detail. Not everyone has the talent to become a ﬁrst-class programmer, just as not everyone has the skills to be a professional athlete. However, virtually anyone can learn how to program computers. With some patience and effort on your part, this book will help you to become a programmer. There are lots of good reasons to learn programming. Programming is a fundamental part of computer science and is, therefore, important to anyone interested in becoming a computer professional. But others can also beneﬁt from the experience. Computers have become a commonplace tool in our society. Understanding the strengths and limitations of this tool requires an understanding of programming. Non-programmers often feel they are slaves of their computers. Programmers, however, are truly the masters. If you want to become a more intelligent user of computers, then this book is for you. Programming can also be loads of fun. It is an intellectually engaging activity that allows people to express themselves through useful and sometimes remarkably beautiful creations. Believe it or not, many people actually write computer programs as a hobby. Programming also develops valuable problem-solving skills, especially the ability to analyze complex systems by reducing them to interactions of understandable subsystems. As you probably know, programmers are in great demand. More than a few liberal arts majors have turned a couple computer programming classes into a lucrative career option. Computers are so commonplace in the business world today that the ability to understand and program computers might just give you the edge over your competition, regardless of your occupation. 1.3 What is Computer Science? You might be surprised to learn that computer science is not the study of computers. A famous computer scientist named Edsgar Dijkstra once quipped that computers are to computer science what telescopes are to astronomy. The computer is an important tool in computer science, but it is not itself the object of study. Since a computer can carry out any process that we can describe, the real question is What processes can we describe? Put another way, the fundamental question of computer science is simply What can be computed? Computer scientists use numerous techniques of investigation to answer this question. The three main ones are design, analysis, and experimentation. One way to demonstrate that a particular problem can be solved is to actually design a solution. That is, we develop a step-by-step process for achieving the desired result. Computer scientists call this an algorithm. That’s a fancy word that basically means “recipe.” The design of algorithms is one of the most important facets of computer science. In this book you will ﬁnd techniques for designing and implementing algorithms. One weakness of design is that it can only answer the question What is computable? in the positive. If I can devise an algorithm, then the problem is solvable. However, failing to ﬁnd an algorithm does not mean that a problem is unsolvable. It may mean that I’m just not smart enough, or I haven’t hit upon the right idea yet. This is where analysis comes in. Analysis is the process of examining algorithms and problems mathematically. Computer scientists have shown that some seemingly simple problems are not solvable by any algorithm. Other problems are intractable. The algorithms that solve these problems take too long or require too much memory to be of practical value. Analysis of algorithms is an important part of computer science; throughout this book we will touch on some of the fundamental principles. Chapter 13 has examples of unsolvable and intractable problems. Some problems are too complex or ill-deﬁned to lend themselves to analysis. In such cases, computer scientists rely on experimentation; they actually implement systems and then study the resulting behavior. Even when theoretical analysis is done, experimentation is often needed in order to verify and reﬁne the Slide 11: 1.4. HARDWARE BASICS 3 Output Devices CPU Input Devices Main Memory Secondary Memory Figure 1.1: Functional View of a Computer. analysis. For most problems, the bottom-line is whether a working, reliable system can be built. Often we require empirical testing of the system to determine that this bottom-line has been met. As you begin writing your own programs, you will get plenty of opportunities to observe your solutions in action. 1.4 Hardware Basics You don’t have to know all the details of how a computer works to be a successful programmer, but understanding the underlying principles will help you master the steps we go through to put our programs into action. It’s a bit like driving a car. Knowing a little about internal combustion engines helps to explain why you have to do things like ﬁll the gas tank, start the engine, step on the accelerator, etc. You could learn to drive by just memorizing what to do, but a little more knowledge makes the whole process much more understandable. Let’s take a moment to “look under the hood” of your computer. Although different computers can vary signiﬁcantly in speciﬁc details, at a higher level all modern digital computers are remarkably similar. Figure 1.1 shows a functional view of a computer. The central processing unit (CPU) is the “brain” of the machine. This is where all the basic operations of the computer are carried out. The CPU can perform simple arithmetic operations like adding two numbers and can also do logical operations like testing to see if two numbers are equal. The memory stores programs and data. The CPU can only directly access information that is stored in main memory (called RAM for Random Access Memory). Main memory is fast, but it is also volatile. That is, when the power is turned off, the information in the memory is lost. Thus, there must also be some secondary memory that provides more permanent storage. In a modern personal computer, this is usually some sort of magnetic medium such as a hard disk (also called a hard drive) or ﬂoppy. Humans interact with the computer through input and output devices. You are probably familiar with common devices such as a keyboard, mouse, and monitor (video screen). Information from input devices is processed by the CPU and may be shufﬂed off to the main or secondary memory. Similarly, when information needs to be displayed, the CPU sends it to one or more output devices. So what happens when you ﬁre up your favorite game or word processing program? First, the instructions that comprise the program are copied from the (more) permanent secondary memory into the main memory of the computer. Once the instructions are loaded, the CPU starts executing the program. Technically the CPU follows a process called the fetch execute cycle. The ﬁrst instruction is retrieved from memory, decoded to ﬁgure out what it represents, and the appropriate action carried out. Then the next instruction is fetched, decoded and executed. The cycle continues, instruction after instruction. This is really all the computer does from the time that you turn it on until you turn it off again: fetch, decode, execute. It doesn’t seem very exciting, does it? But the computer can execute this stream of simple instructions with blazing speed, zipping through millions of instructions each second. Put enough simple instructions together in just the right way, and the computer does amazing things. Slide 12: 4 CHAPTER 1. COMPUTERS AND PROGRAMS 1.5 Programming Languages Remember that a program is just a sequence of instructions telling a computer what to do. Obviously, we need to provide those instructions in a language that a computer can understand. It would be nice if we could just tell a computer what to do using our native language, like they do in science ﬁction movies. (“Computer, how long will it take to reach planet Alphalpha at maximum warp?”) Unfortunately, despite the continuing efforts of many top-ﬂight computer scientists (including your author), designing a computer to understand human language is still an unsolved problem. Even if computers could understand us, human languages are not very well suited for describing complex algorithms. Natural language is fraught with ambiguity and imprecision. For example, if I say: “I saw the man in the park with the telescope,” did I have the telescope, or did the man? And who was in the park? We understand each other most of the time only because all humans share a vast store of common knowledge and experience. Even then, miscommunication is commonplace. Computer scientists have gotten around this problem by designing notations for expressing computations in an exact, and unambiguous way. These special notations are called programming languages. Every structure in a programming language has a precise form (its syntax) and a precise meaning (its semantics). A programming language is something like a code for writing down the instructions that a computer will follow. In fact, programmers often refer to their programs as computer code, and the process of writing an algorithm in a programming language is called coding. Python is one example of a programming language. It is the language that we will use throughout this book. You may have heard of some other languages, such as C++, Java, Perl, Scheme, or BASIC. Although these languages differ in many details, they all share the property of having well-deﬁned, unambiguous syntax and semantics. All of the languages mentioned above are examples of high-level computer languages. Although they are precise, they are designed to be used and understood by humans. Strictly speaking, computer hardware can only understand very low-level language known as machine language. Suppose we want the computer to add two numbers. The instructions that the CPU actually carries out might be something like this. load the number from memory location 2001 into the CPU load the number from memory location 2002 into the CPU Add the two numbers in the CPU store the result into location 2003 This seems like a lot of work to add two numbers, doesn’t it? Actually, it’s even more complicated than this because the instructions and numbers are represented in binary notation (as sequences of 0s and 1s). In a high-level language like Python, the addition of two numbers can be expressed more naturally: c = a + b. That’s a lot easier for us to understand, but we need some way to translate the high-level language into the machine language that the computer can execute. There are two ways to do this: a high-level language can either be compiled or interpreted. A compiler is a complex computer program that takes another program written in a high-level language and translates it into an equivalent program in the machine language of some computer. Figure 1.2 shows a block diagram of the compiling process. The high-level program is called source code, and the resulting machine code is a program that the computer can directly execute. The dashed line in the diagram represents the execution of the machine code. An interpreter is a program that simulates a computer that understands a high-level language. Rather than translating the source program into a machine language equivalent, the interpreter analyzes and executes the source code instruction by instruction as necessary. Figure 1.3 illustrates the process. The difference between interpreting and compiling is that compiling is a one-shot translation; once a program is compiled, it may be run over and over again without further need for the compiler or the source code. In the interpreted case, the interpreter and the source are needed every time the program runs. Compiled programs tend to be faster, since the translation is done once and for all, but interpreted languages lend themselves to a more ﬂexible programming environment as programs can be developed and run interactively. The translation process highlights another advantage that high-level languages have over machine language: portability. The machine language of a computer is created by the designers of the particular CPU. Slide 13: 1.6. THE MAGIC OF PYTHON Source Code (Program) 5 Compiler Machine Code Running Inputs Program Outputs Figure 1.2: Compiling a High-Level Language Source Code (Program) Computer Running an Outputs Interpreter Inputs Figure 1.3: Interpreting a High-Level Language. Each kind of computer has its own machine language. A program for a Pentium CPU won’t run on a Macintosh that sports a PowerPC. On the other hand, a program written in a high-level language can be run on many different kinds of computers as long as there is a suitable compiler or interpreter (which is just another program). For example, if I design a new computer, I can also program a Python interpreter for it, and then any program written in Python can be run on my new computer, as is. 1.6 The Magic of Python Now that you have all the technical details, it’s time to start having fun with Python. The ultimate goal is to make the computer do our bidding. To this end, we will write programs that control the computational processes inside the machine. You have already seen that there is no magic in this process, but in some ways programming feels like magic. The computational processes inside the computer are like magical spirits that we can harness for our work. Unfortunately, those spirits only understand a very arcane language that we do not know. What we need is a friendly Genie that can direct the spirits to fulﬁll our wishes. Our Genie is a Python interpreter. We can give instructions to the Python interpreter, and it directs the underlying spirits to carry out our demands. We communicate with the Genie through a special language of spells and incantations (i.e., Python). The best way to start learning about Python is to let our Genie out of the bottle and try some spells. You can start the Python interpreter in an interactive mode and type in some commands to see what happens. When you ﬁrst start the interpreter program, you may see something like the following: Python 2.1 (#1, Jun 21 2001, 11:39:00) [GCC pgcc-2.91.66 19990314 (egcs-1.1.2 release)] on linux2 Type "copyright", "credits" or "license" for more information. >>> The is a Python prompt indicating that the Genie is waiting for us to give it a command. In programming languages, a complete command is called a statement. Here is a sample interaction with the Python interpreter. ¡¡ Slide 14: 6 >>> print "Hello, World" Hello, World >>> print 2 + 3 5 >>> print "2 + 3 =", 2 + 3 2+3=5 CHAPTER 1. COMPUTERS AND PROGRAMS Here I have tried out three examples using the Python print statement. The ﬁrst statement asks Python to display the literal phrase Hello, World. Python responds on the next line by printing the phrase. The second print statement asks Python to print the sum of 2 and 3. The third print combines these two ideas. Python prints the part in quotes “2 + 3 =” followed by the result of adding 2 + 3, which is 5. This kind of interaction is a great way to try out new things in Python. Snippets of interactive sessions are sprinkled throughout this book. When you see the Python prompt in an example, that should tip you off that an interactive session is being illustrated. It’s a good idea to ﬁre up Python and try the examples for yourself. Usually we want to move beyond snippets and execute an entire sequence of statements. Python lets us put a sequence of statements together to create a brand-new command called a function. Here is an example of creating a new function called hello. >>> def hello(): print "Hello" print "Computers are Fun" >>> The ﬁrst line tells Python that we are deﬁning a new function called hello. The following lines are indented to show that they are part of the hello function. The blank line (obtained by hitting the <Enter> key twice) lets Python know that the deﬁnition is ﬁnished, and the interpreter responds with another prompt. Notice that the deﬁnition did not cause anything to happen. We have told Python what should happen when the hello function is used as a command; we haven’t actually asked Python to perform it yet. A function is invoked by typing its name. Here’s what happens when we use our hello command. >>> hello() Hello Computers are Fun >>> Do you see what this does? The two print statements from the hello function are executed in sequence. You may be wondering about the parentheses in the deﬁnition and use of hello. Commands can have changeable parts called parameters that are placed within the parentheses. Let’s look at an example of a customized greeting using a parameter. First the deﬁnition: >>> def greet(person): print "Hello", person print "How are you?" Now we can use our customized greeting. >>> greet("John") Hello John How are you? >>> greet("Emily") Hello Emily How are you? >>> ¡¡ Slide 15: 1.6. THE MAGIC OF PYTHON 7 Can you see what is happening here? When we use greet we can send different names to customize the result. We will discuss parameters in detail later on. For the time being, our functions will not use parameters, so the parentheses will be empty, but you still need to include them when deﬁning and using functions. One problem with entering functions interactively at the Python prompt like this is that the deﬁnitions go away when we quit Python. If we want to use them again the next time, we have to type them all over again. Programs are usually created by typing deﬁnitions into a separate ﬁle called a module or script. This ﬁle is saved on a disk so that it can be used over and over again. A module ﬁle is just a text ﬁle, and you can create one using any program for editing text, like a notepad or word processor program (provided you save your program as a “plain text” ﬁle). A special type of program known as a programming environment simpliﬁes the process. A programming environment is speciﬁcally designed to help programmers write programs and includes features such as automatic indenting, color highlighting, and interactive development. The standard Python distribution includes a programming environment called Idle that you may use for working on the programs in this book. Let’s illustrate the use of a module ﬁle by writing and running a complete program. Our program will illustrate a mathematical concept known as chaos. Here is the program as we would type it into Idle or some other editor and save in a module ﬁle: # File: chaos.py # A simple program illustrating chaotic behavior. def main(): print "This program illustrates a chaotic function" x = input("Enter a number between 0 and 1: ") for i in range(10): x = 3.9 * x * (1 - x) print x main() This ﬁle should be saved with with the name chaos.py. The .py extension indicates that this is a Python module. You can see that this particular example contains lines to deﬁne a new function called main. (Programs are often placed in a function called main.) The last line of the ﬁle is the command to invoke this function. Don’t worry if you don’t understand what main actually does; we will discuss it in the next section. The point here is that once we have a program in a module ﬁle, we can run it any time we want. This program can be run in a number of different ways that depend on the actual operating system and programming environment that you are using. If you are using a windowing system, you can run a Python program by (double-)clicking on the module ﬁle’s icon. In a command-line situation, you might type a command like python chaos.py. If you are using Idle (or another programming environment) you can run a program by opening it in the editor and then selecting a command like import, run, or execute. One method that should always work is to start the Python interpreter and then import the ﬁle. Here is how that looks. >>> import chaos This program illustrates a chaotic function Enter a number between 0 and 1: .25 0.73125 0.76644140625 0.698135010439 0.82189581879 0.570894019197 0.955398748364 0.166186721954 0.540417912062 0.9686289303 0.118509010176 >>> Slide 16: 8 CHAPTER 1. COMPUTERS AND PROGRAMS Typing the ﬁrst line import chaos tells the Python interpreter to load the chaos module from the ﬁle chaos.py into main memory. Notice that I did not include the .py extension on the import line; Python assumes the module will have a .py extension. As Python imports the module ﬁle, each line executes. It’s just as if we had typed them one-by-one at the interactive Python prompt. The def in the module causes Python to create the main function. When Python encounters the last line of the module, the main function is invoked, thus running our program. The running program asks the user to enter a number between 0 and 1 (in this case, I typed “.25”) and then prints out a series of 10 numbers. When you ﬁrst import a module ﬁle in this way, Python creates a companion ﬁle with a .pyc extension. In this example, Python creates another ﬁle on the disk called chaos.pyc. This is an intermediate ﬁle used by the Python interpreter. Technically, Python uses a hybrid compiling/interpreting process. The Python source in the module ﬁle is compiled into more primitive instructions called byte code. This byte code (the .pyc) ﬁle is then interpreted. Having a .pyc ﬁle available makes importing a module faster the second time around. However, you may delete the byte code ﬁles if you wish to save disk space; Python will automatically re-create them as needed. A module only needs to be imported into a session once. After the module has been loaded, we can run the program again by asking Python to execute the main command. We do this by using a special dot notation. Typing chaos.main() tells Python to invoke the main function in the chaos module. Continuing with our example, here is how it looks when we rerun the program with 26 as the input. >>> chaos.main() Enter a number between 0 and 1: .26 0.75036 0.73054749456 0.767706625733 0.6954993339 0.825942040734 0.560670965721 0.960644232282 0.147446875935 0.490254549376 0.974629602149 >>> 1.7 Inside a Python Program The output from the chaos program may not look very exciting, but it illustrates a very interesting phenomenon known to physicists and mathematicians. Let’s take a look at this program line by line and see what it does. Don’t worry about understanding every detail right away; we will be returning to all of these ideas in the next chapter. The ﬁrst two lines of the program start with the # character: # File: chaos.py # A simple program illustrating chaotic behavior. These lines are called comments. They are intended for human readers of the program and are ignored by Python. The Python interpreter always skips any text from the pound sign (#) through the end of a line. The next line of the program begins the deﬁnition of a function called main. def main(): Strictly speaking, it would not be necessary to create a main function. Since the lines of a module are executed as they are loaded, we could have written our program without this deﬁnition. That is, the module could have looked like this: Slide 17: 1.7. INSIDE A PYTHON PROGRAM # File: chaos.py # A simple program illustrating chaotic behavior. print "This program illustrates a chaotic function" x = input("Enter a number between 0 and 1: ") for i in range(10): x = 3.9 * x * (1 - x) print x 9 This version is a bit shorter, but it is customary to place the instructions that comprise a program inside of a function called main. One immediate beneﬁt of this approach was illustrated above; it allows us to (re)run the program by simply invoking chaos.main(). We don’t have to reload the module from the ﬁle in order to run it again, which would be necessary in the main-less case. The ﬁrst line inside of main is really the beginning of our program. print "This program illustrates a chaotic function" This line causes Python to print a message introducing the program when it runs. Take a look at the next line of the program. x = input("Enter a number between 0 and 1: ") Here x is an example of a variable. A variable is used to give a name to a value so that we can refer to it at other points in the program. The entire line is an input statement. When Python gets to this statement, it displays the quoted message Enter a number between 0 and 1: and then pauses, waiting for the user to type something on the keyboard and press the <Enter> key. The value that the user types is then stored as the variable x. In the ﬁrst example shown above, the user entered .25, which becomes the value of x. The next statement is an example of a loop. for i in range(10): A loop is a device that tells Python to do the same thing over and over again. This particular loop says to do something 10 times. The lines indented underneath the loop heading are the statements that are done 10 times. These form the body of the loop. x = 3.9 * x * (1 - x) print x The effect of the loop is exactly the same as if we had written the body of the loop 10 times: x = 3.9 print x x = 3.9 print x x = 3.9 print x x = 3.9 print x x = 3.9 print x x = 3.9 print x x = 3.9 print x x = 3.9 print x x = 3.9 * x * (1 - x) * x * (1 - x) * x * (1 - x) * x * (1 - x) * x * (1 - x) * x * (1 - x) * x * (1 - x) * x * (1 - x) * x * (1 - x) Slide 18: 10 print x x = 3.9 * x * (1 - x) print x CHAPTER 1. COMPUTERS AND PROGRAMS Obviously using the loop instead saves the programmer a lot of trouble. But what exactly do these statements do? The ﬁrst one performs a calculation. x = 3.9 * x * (1 - x) This is called an assignment statement. The part on the right side of the = is a mathematical expression. Python uses the * character to indicate multiplication. Recall that the value of x is 0 25 (from the input statement). The computed value is 3 9 0 25¡ 1 0 25¡ or 0 73125. Once the value on the righthand side is computed, it is stored back (or assigned) into the variable that appears on the lefthand side of the =, in this case x. The new value of x (0 73125) replaces the old value (0 25). The second line in the loop body is a type of statement we have encountered before, a print statement. print x When Python executes this statement the current value of x is displayed on the screen. So, the ﬁrst number of output is 0.73125. Remember the loop executes 10 times. After printing the value of x, the two statements of the loop are executed again. x = 3.9 * x * (1 - x) print x Of course, now x has the value 0 73125, so the formula computes a new value of x as 3 9 0 73125¡ 1 0 73125¡ , which is 0 76644140625. Can you see how the current value of x is used to compute a new value each time around the loop? That’s where the numbers in the example run came from. You might try working through the steps of the program yourself for a different input value (say 0 5). Then run the program using Python and see how well you did impersonating a computer. ¢ ¢ 1.8 Chaos and Computers I said above that the chaos program illustrates an interesting phenomenon. What could be interesting about a screen full of numbers? If you try out the program for yourself, you’ll ﬁnd that, no matter what number you start with, the results are always similar: the program spits back 10 seemingly random numbers between 0 and 1. As the program runs, the value of x seems to jump around, well, chaotically. The function computed by this program has the general form: k x¡ 1 x¡ , where k in this case is 3.9. This is called a logistic function. It models certain kinds of unstable electronic circuits and is also sometimes used to predict population under limiting conditions. Repeated application of the logistic function can produce chaos. Although our program has a well deﬁned underlying behavior, the output seems unpredictable. An interesting property of chaotic functions is that very small differences in the initial value can lead to large differences in the result as the formula is repeatedly applied. You can see this in the chaos program by entering numbers that differ by only a small amount. Here is the output from a modiﬁed program that shows the results for initial values of 0 25 and 0 26 side by side. input 0.25 0.26 --------------------------0.731250 0.750360 0.766441 0.730547 0.698135 0.767707 0.821896 0.695499 0.570894 0.825942 0.955399 0.560671 ¢ Slide 19: 1.9. EXERCISES 0.166187 0.540418 0.968629 0.118509 0.960644 0.147447 0.490255 0.974630 11 With very similar starting values, the outputs stay similar for a few iterations, but then differ markedly. By about the ﬁfth iteration, there no longer seems to be any relationship between the two models. These two features of our chaos program, apparent unpredictability and extreme sensitivity to initial values, are the hallmarks of chaotic behavior. Chaos has important implications for computer science. It turns out that many phenomena in the real world that we might like to model and predict with our computers exhibit just this kind of chaotic behavior. You may have heard of the so-called butterﬂy effect. Computer models that are used to simulate and predict weather patterns are so sensitive that the effect of a single butterﬂy ﬂapping its wings in New Jersey might make the difference of whether or not rain is predicted in Peoria. It’s very possible that even with perfect computer modeling, we might never be able to measure existing weather conditions accurately enough to predict weather more than a few days in advance. The measurements simply can’t be precise enough to make the predictions accurate over a longer time frame. As you can see, this small program has a valuable lesson to teach users of computers. As amazing as computers are, the results that they give us are only as useful as the mathematical models on which the programs are based. Computers can give incorrect results because of errors in programs, but even correct programs may produce erroneous results if the models are wrong or the initial inputs are not accurate enough. 1.9 Exercises 1. Compare and contrast the following pairs of concepts from the chapter. (a) Hardware vs. Software (b) Algorithm vs. Program (c) Programming Language vs. Natural Language (d) High-Level Language vs. Machine Language (e) Interpreter vs. Compiler (f) Syntax vs. Semantics 2. List and explain in your own words the role of each of the ﬁve basic functional units of a computer depicted in Figure 1.1. 3. Write a detailed algorithm for making a peanut butter and jelly sandwich (or some other simple everyday activity). 4. As you will learn in a later chapter, many of the numbers stored in a computer are not exact values, but rather close approximations. For example, the value 0.1, might be stored as 0.10000000000000000555. Usually, such small differences are not a problem; however, given what you have learned about chaotic behavior in Chapter 1, you should realize the need for caution in certain situations. Can you think of examples where this might be a problem? Explain. 5. Trace through the Chaos program from Section 1.6 by hand using 0 15 as the input value. Show the sequence of output that results. 6. Enter and run the Chaos program from Section 1.6 using whatever Python implementation you have available. Try it out with various values of input to see that it functions as described in the chapter. 7. Modify the Chaos program from Section 1.6 using 2.0 in place of 3.9 as the multiplier in the logistic function. Your modiﬁed line of code should look like this: Slide 20: 12 x = 2.0 * x * (1 - x) CHAPTER 1. COMPUTERS AND PROGRAMS Run the program for various input values and compare the results to those obtained from the original program. Write a short paragraph describing any differences that you notice in the behavior of the two versions. 8. Modify the Chaos program from Section 1.6 so that it prints out 20 values instead of 10. 9. (Advanced) Modify the Chaos program so that it accepts two inputs and then prints a table with two columns similar to the one shown in Section 1.8. (Note: You will probably not be able to get the columns to line up as nicely as those in the example. Chapter 4 discusses how to print numbers with a ﬁxed number of decimal places.) Slide 21: Chapter 2 Writing Simple Programs As you saw in the previous chapter, it is easy to run programs that have already been written. The hard part is actually coming up with the program in the ﬁrst place. Computers are very literal, and they must be told what to do right down to the last detail. Writing large programs is a daunting task. It would be almost impossible without a systematic approach. 2.1 The Software Development Process The process of creating a program is often broken down into stages according to the information that is produced in each phase. In a nutshell, here’s what you should do. Formulate Requirements Figure out exactly what the problem to be solved is. Try to understand as much as possible about it. Until you really know what the problem is, you cannot begin to solve it. Determine Speciﬁcations Describe exactly what your program will do. At this point, you should not worry about how your program will work, but rather with deciding exactly what it will accomplish. For simple programs this involves carefully describing what the inputs and outputs of the program will be and how they relate to each other. Create a Design Formulate the overall structure of the program. This is where the how of the program gets worked out. The main task is to design the algorithm(s) that will meet the speciﬁcations. Implement the Design Translate the design into a computer language and put it into the computer. In this book, we will be implementing our algorithms as Python programs. Test/Debug the Program Try out your program and see if it works as expected. If there are any errors (often called bugs), then you should go back and ﬁx them. The process of locating and ﬁxing errors is called debugging a program. Maintain the Program Continue developing the program in response to the needs of your users. Most programs are never really ﬁnished; they keep evolving over years of use. 2.2 Example Program: Temperature Converter Let’s go through the steps of the software development process with a simple real-world example. Suzie Programmer has a problem. Suzie is an American computer science student spending a year studying in Europe. She has no problems with language, as she is ﬂuent in many languages (including Python). Her problem is that she has a hard time ﬁguring out the temperature in the morning so that she knows how to dress for the day. Suzie listens to the weather report each morning, but the temperatures are given in degrees Celsius, and she is used to Fahrenheit. 13 Slide 22: 14 CHAPTER 2. WRITING SIMPLE PROGRAMS Fortunately, Suzie has an idea to solve the problem. Being a computer science major, she never goes anywhere without her laptop computer. She thinks it might be possible that a computer program could help her out. Suzie begins with the requirements of her problem. In this case, the problem is pretty clear: the radio announcer gives temperatures in degrees Celsius, but Suzie only comprehends temperatures that are in degrees Fahrenheit. That’s the crux of the problem. Next, Suzie considers the speciﬁcations of a program that might help her out. What should the input be? She decides that her program will allow her to type in the temperature in degrees Celsius. And the output? The program will display the temperature converted into degrees Fahrenheit. Now she needs to specify the exact relationship of the output to the input. Suzie does some quick ﬁguring to derive the formula F 9 5¡ C 32 (Can you see how?). That seems an adequate speciﬁcation. Notice that this describes one of many possible programs that could solve this problem. If Suzie had background in the ﬁeld of Artiﬁcial Intelligence (AI), she might consider writing a program that would actually listen to the radio announcer to get the current temperature using speech recognition algorithms. For output, she might have the computer control a robot that goes to her closet and picks an appropriate outﬁt based on the converted temperature. This would be a much more ambitious project, to say the least! Certainly the robot program would also solve the problem identiﬁed in the requirements. The purpose of speciﬁcation is to decide exactly what this particular program will do to solve a problem. Suzie knows better than to just dive in and start writing a program without ﬁrst having a clear idea of what she is trying to build. Suzie is now ready to design an algorithm for her problem. She immediately realizes that this is a simple algorithm that follows a standard pattern: Input, Process, Output (IPO). Her program will prompt the user for some input information (the Celsius temperature), process it to convert to a Fahrenheit temperature, and then output the result by displaying it on the computer screen. Suzie could write her algorithm down in a computer language. However, the precision of writing it out formally tends to stiﬂe the creative process of developing the algorithm. Instead, she writes her algorithm using pseudocode. Pseudocode is just precise English that describes what a program does. It is meant to communicate algorithms without all the extra mental overhead of getting the details right in any particular programming language. Here is Suzie’s completed algorithm: Input the temperature in degrees Celsius (call it celsius) Calculate fahrenheit as 9/5 celsius + 32 Output fahrenheit The next step is to translate this design into a Python program. This is straightforward, as each line of the algorithm turns into a corresponding line of Python code. # convert.py # A program to convert Celsius temps to Fahrenheit # by: Suzie Programmer def main(): celsius = input("What is the Celsius temperature? ") fahrenheit = 9.0 / 5.0 * celsius + 32 print "The temperature is", fahrenheit, "degrees Fahrenheit." main() See if you can ﬁgure out what each line of this program does. Don’t worry if some parts are a bit confusing. They will be discussed in detail in the next section. After completing her program, Suzie tests it to see how well it works. She uses some inputs for which she knows the correct answers. Here is the output from two of her tests. What is the Celsius temperature? 0 The temperature is 32.0 degrees fahrenheit. ¢ ¡ Slide 23: 2.3. ELEMENTS OF PROGRAMS What is the Celsius temperature? 100 The temperature is 212.0 degrees fahrenheit. 15 You can see that Suzie used the values of 0 and 100 to test her program. It looks pretty good, and she is satisﬁed with her solution. Apparently, no debugging is necessary. 2.3 Elements of Programs Now that you know something about the programming process, you are almost ready to start writing programs on your own. Before doing that, though, you need a more complete grounding in the fundamentals of Python. The next few sections will discuss technical details that are essential to writing correct programs. This material can seem a bit tedious, but you will have to master these basics before plunging into more interesting waters. 2.3.1 Names You have already seen that names are an important part of programming. We give names to modules (e.g., convert) and to the functions within modules (e.g., main). Variables are used to give names to values (e.g., celsius and fahrenheit). Technically, all these names are called identiﬁers. Python has some rules about how identiﬁers are formed. Every identiﬁer must begin with a letter or underscore (the “ ” character) which may be followed by any sequence of letters, digits, or underscores. This implies that a single identiﬁer cannot contain any spaces. According to these rules, all of the following are legal names in Python: x celsius spam spam2 SpamAndEggs Spam_and_Eggs Identiﬁers are case-sensitive, so spam, Spam, sPam, and SPAM are all different names to Python. For the most part, programmers are free to choose any name that conforms to these rules. Good programmers always try to choose names that describe the thing being named. One other important thing to be aware of is that some identiﬁers are part of Python itself. These names are called reserved words and cannot be used as ordinary identiﬁers. The complete list of Python reserved words is shown in Table 2.1. and assert break class continue def del elif else except exec ﬁnally for from global if import in is lambda not or pass print raise return try while yield Table 2.1: Python Reserved Words. 2.3.2 Expressions Programs manipulate data. The fragments of code that produce or calculate new data values are called expressions. So far our program examples have dealt mostly with numbers, so I’ll use numeric data to illustrate expressions. Slide 24: 16 CHAPTER 2. WRITING SIMPLE PROGRAMS The simplest kind of expression is a literal. A literal is used to indicate a speciﬁc value. In chaos.py you can ﬁnd the numbers 3.9 and 1. The convert.py program contains 9.0, 5.0, and 32. These are all examples of numeric literals, and their meaning is obvious: 32 represents, well, 32. A simple identiﬁer can also be an expression. We use identiﬁers as variables to give names to values. When an identiﬁer appears in an expression, this value is retrieved to provide a result for the expression. Here is an interaction with the Python interpreter that illustrates the use of variables as expressions. >>> x = 5 >>> x 5 >>> print x 5 >>> print spam Traceback (innermost last): File "<pyshell#34>", line 1, in ? print spam NameError: spam >>> First the variable x is assigned the value 5 (using the numeric literal 5). The next line has Python evaluate the expression x. Python spits back 5, which is the value that was just assigned to x. Of course, we get the same result when we put x in a print statement. The last example shows what happens when we use a variable that has not been assigned a value. Python cannot ﬁnd a value, so it reports a Name Error. This says that there is no value with that name. A variable must always be assigned a value before it can be used in an expression. More complex and interesting expressions can be constructed by combining simpler expressions with operators. For numbers, Python provides the normal set of mathematical operations: addition, subtraction, multiplication, division, and exponentiation. The corresponding Python operators are: +, -, , /, and . Here are some examples of complex expressions from chaos.py and convert.py 3.9 x * (1 - x) 9.0 / 5.0 * celsius + 32 Spaces are irrelevant within an expression. The last expression could have been written 9.0/5.0celsius+32 and the result would be exactly the same. Usually it’s a good idea to place some spaces in expressions to make them easier to read. Python’s mathematical operators obey the same rules of precedence and associativity that you learned in your math classes, including using parentheses to modify the order of evaluation. You should have little trouble constructing complex expressions in your own programs. Do keep in mind that only the round parentheses are allowed in expressions, but you can nest them if necessary to create expressions like this. ((x1 - x2) / 2n) + (spam / k3) If you are reading carefully, you may be curious why, in her temperature conversion program, Suzie Programmer chose to write 9.0/5.0 rather than 9/5. Both of these are legal expressions, but they give different results. This mystery will be discussed in Chapter 3. If you can’t stand the wait, try them out for yourself and see if you can ﬁgure out what’s going on. 2.4 Output Statements Now that you have the basic building blocks, identiﬁer and expression, you are ready for a more complete description of various Python statements. You already know that you can display information on the screen using Python’s print statement. But what exactly can be printed? Python, like all programming languages, has a precise set of rules for the syntax (form) and semantics (meaning) of each statement. Computer scientists have developed sophisticated notations called meta-languages for describing programming languages. In this book we will rely on a simple template notation to illustrate the syntax of statements. Here are the possible forms of the print statement: Slide 25*: 2.5. ASSIGNMENT STATEMENTS print print <expr> print <expr>, <expr>, ..., <expr> print <expr>, <expr>, ..., <expr>, 17 In a nutshell, these templates show that a print statement consists of the keyword print followed by zero or more expressions, which are separated by commas. The angle bracket notation ( ) is used to indicate “slots” that are ﬁlled in by other fragments of Python code. The name inside the brackets indicate what is missing; expr stands for an expression. The ellipses (“...”) indicate an indeﬁnite series (of expressions, in this case). You don’t actually type the dots. The fourth version shows that a print statement may be optionally ended with a comma. That is all there is to know about the syntax of print. As far as semantics, a print statement displays information in textual form. Any supplied expressions are evaluated left to right, and the resulting values are displayed on a single line of output in a left-to-right fashion. A single blank space character is placed between the displayed values. Normally, successive print statements will display on separate lines of the screen. A bare print (ﬁrst version above) can be used to get a blank line of output. If a print statement ends with a comma (fourth version), a ﬁnal space is appended to the line, but the output does not advance to the next line. Using this method, multiple print statements can be used to generate a single line of output. Putting it all together, this sequence of print statements print print print print print print 3+4 3, 4, 3 + 4 3, 4, 3+4 "The answer is", 3 + 4 produces this output 7 347 347 The answer is 7 That last print statement may be a bit confusing. According to the syntax templates above, print requires a sequence of expressions. That means "The answer is" must be an expression. In fact, it is an expression, but it doesn’t produce a number. Instead, it produces another kind of data called a string. A sequence of characters enclosed in quotes is a string literal. Strings will be discussed in detail in a later chapter. For now, consider this a convenient way of labeling output. 2.5 Assignment Statements 2.5.1 Simple Assignment One of the most important kinds of statements in Python is the assignment statement. We’ve already seen a number of these in our previous examples. The basic assignment statement has this form: <variable> = <expr> Here variable is an identiﬁer and expr is an expression. The semantics of the assignment is that the expression on the right side is evaluated to produce a value, which is then associated with the variable named on the left side. Here are some of the assignments we’ve already seen. x = 3.9 x * (1 - x) fahrenheit = 9.0 / 5.0 * celsius + 32 x=5 ¡ Slide 26: 18 CHAPTER 2. WRITING SIMPLE PROGRAMS A variable can be assigned many times. It always retains the value of the most recent assignment. Here is an interactive Python session that demonstrates the point: >>> >>> 0 >>> >>> 7 >>> >>> 8 myVar = 0 myVar myVar = 7 myVar myVar = myVar + 1 myVar The last assignment statement shows how the current value of a variable can be used to update its value. In this case I simply added one to the previous value. The chaos.py program from Chapter 1 did something similar, though a bit more complex. Remember, the values of variables can change; that’s why they’re called variables. 2.5.2 Assigning Input The purpose of an input statement is to get some information from the user of a program and store it into a variable. Some programming languages have a special statement to do this. In Python, input is accomplished using an assignment statement combined with a special expression called input. This template shows the standard form. <variable> = input(<prompt>) Here prompt is an expression that serves to prompt the user for input; this is almost always a string literal (i.e., some text inside of quotation marks). When Python encounters an input expression, it evaluates the prompt and displays the result of the prompt on the screen. Python then pauses and waits for the user to type an expression and press the Enter key. The expression typed by the user is then evaluated to produce the result of the input. This sounds complicated, but most uses of input are straightforward. In our example programs, input statements are used to get numbers from the user. x = input("Please enter a number between 0 and 1: ") celsius = input("What is the Celsius temperature? ") If you are reading programs carefully, you probably noticed the blank space inside the quotes at the end of these prompts. I usually put a space at the end of a prompt so that the input that the user types does not start right next to the prompt. Putting a space in makes the interaction easier to read and understand. Although these two examples speciﬁcally prompt the user to enter a number, a number is just a numeric literal—a simple Python expression. In fact, any valid expression would be just as acceptable. Consider the following interaction with the Python interpreter. >>> ans = input("Enter an expression: ") Enter an expression: 3 + 4 * 5 >>> print ans 23 >>> Here, when prompted to enter an expression, the user typed “3 + 4 * 5.” Python evaluated this expression and stored the value in the variable ans. When printed, we see that ans got the value 23 as expected. In a way, the input is like a delayed expression. The example interaction produced exactly the same result as if we had simply done ans = 3 + 4 * 5. The difference is that the expression was supplied at the time the statement was executed instead of being determined when the statement was written by the programmer. Thus, the user can supply formulas for a program to evaluate. Slide 27: 2.5. ASSIGNMENT STATEMENTS 19 2.5.3 Simultaneous Assignment There is an alternative form of the assignment statement that allows us to calculate several values all at the same time. It looks like this: <var>, <var>, ..., <var> = <expr>, <expr>, ..., <expr> This is called simultaneous assignment. Semantically, this tells Python to evaluate all the expressions on the right-hand side and then assign these values to the corresponding variables named on the left-hand side. Here’s an example. sum, diff = x+y, x-y Here sum would get the sum of x and y and diff would get the difference. This form of assignment seems strange at ﬁrst, but it can prove remarkably useful. Here’s an example. Suppose you have two variables x and y and you want to swap the values. That is, you want the value currently stored in x to be in y and the value that is currently in y to be stored in x. At ﬁrst, you might think this could be done with two simple assignments. x=y y=x This doesn’t work. We can trace the execution of these statements step-by-step to see why. Suppose x and y start with the values 2 and 4. Let’s examine the logic of the program to see how the variables change. The following sequence uses comments to describe what happens to the variables as these two statements are executed. # # x # y # variables initial values =y now =x final x 2 4 4 y 4 4 4 See how the ﬁrst statement clobbers the original value of x by assigning to it the value of y? When we then assign x to y in the second step, we just end up with two copies of the original y value. One way to make the swap work is to introduce an additional variable that temporarily remembers the original value of x. temp = x x=y y = temp Let’s walk-through this sequence to see how it works. # variables # initial values temp = x # x=y # y = temp # x 2 2 4 4 y 4 4 4 2 temp no value yet 2 2 2 As you can see from the ﬁnal values of x and y, the swap was successful in this case. This sort of three-way shufﬂe is common in other programming languages. In Python, the simultaneous assignment statement offers an elegant alternative. Here is a simpler Python equivalent: x, y = y, x Slide 28: 20 CHAPTER 2. WRITING SIMPLE PROGRAMS Because the assignment is simultaneous, it avoids wiping out one of the original values. Simultaneous assignment can also be used to get multiple values from the user in a single input. Consider this program for averaging exam scores: # avg2.py # A simple program to average two exam scores # Illustrates use of multiple input def main(): print "This program computes the average of two exam scores." score1, score2 = input("Enter two scores separated by a comma: ") average = (score1 + score2) / 2.0 print "The average of the scores is:", average main() The program prompts for two scores separated by a comma. Suppose the user types 86, 92. The effect of the input statement is then the same as if we had done this assignment: score1, score2 = 86, 92 We have gotten a value for each of the variables in one fell swoop. This example used just two values, but it could be generalized to any number of inputs. Of course, we could have just gotten the input from the user using separate input statements. score1 = input("Enter the first score: ") score2 = input("Enter the second score: ") In some ways this may be better, as the separate prompts are more informative for the user. In this example the decision as to which approach to take is largely a matter of taste. Sometimes getting multiple values in a single input provides a more intuitive user interface, so it is nice technique to have in your toolkit. 2.6 Deﬁnite Loops You already know that programmers use loops to execute a sequence of statements several times in succession. The simplest kind of loop is called a deﬁnite loop. This is a loop that will execute a deﬁnite number of times. That is, at the point in the program when the loop begins, Python knows how many times to go around (or iterate) the body of the loop. For example, the Chaos program from Chapter 1 used a loop that always executed exactly ten times. for i in range(10): x = 3.9 * x * (1 - x) print x This particular loop pattern is called a counted loop, and it is built using a Python for statement. Before considering this example in detail, let’s take a look at what for loops are all about. A Python for loop has this general form. for <var> in <sequence>: <body> The body of the loop can be any sequence of Python statements. The start and end of the body is indicated by its indentation under the loop heading (the for <var> in <sequence>: part). The meaning of the for statement is a bit awkward to explain in words, but is very easy to understand, once you get the hang of it. The variable after the keyword for is called the loop index. It takes on Slide 29: 2.6. DEFINITE LOOPS 21 each successive value in the sequence, and the statements in the body are executed once for each value. Usually, the sequence portion is a list of values. You can build a simple list by placing a sequence of expressions in square brackets. Some interactive examples help to illustrate the point: >>> for i in [0,1,2,3]: print i 0 1 2 3 >>> for odd in [1, 3, 5, 7, 9]: print odd * odd 1 9 25 49 81 You can see what is happening in these two examples. The body of the loop is executed using each successive value in the list. The length of the list determines the number of times the loop will execute. In the ﬁrst example, the list contains the four values 0 through 3, and these successive values of i are simply printed. In the second example, odd takes on the values of the ﬁrst ﬁve odd natural numbers, and the body of the loop prints the squares of these numbers. Now, let’s go back to the example which began this section (from chaos.py) Look again at the loop heading: for i in range(10): Comparing this to the template for the for loop shows that the last portion, range(10) must be some kind of sequence. Let’s see what the Python interpreter tells us. >>> range(10) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] Do you see what is happening here? The range function is a built-in Python command that simply produces a list of numbers. The loop using range(10) is exactly equivalent to one using a list of 10 numbers. for i in [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]: In general, range(<expr>) will produce a list of numbers that starts with 0 and goes up to, but not including, the value of <expr>. If you think about it, you will see that the value of the expression determines the number of items in the resulting list. In chaos.py we did not even care what values the loop index variable used (since the value of i was not referred to anywhere in the loop body). We just needed a list of length 10 to make the body execute 10 times. As I mentioned above, this pattern is called a counted loop, and it is a very common way to use deﬁnite loops. When you want to do something in your program a certain number of times, use a for loop with a suitable range. for <variable> in range(<expr>): The value of the expression determines how many times the loop executes. The name of the index variable doesn’t really matter much; programmers often use i or j as the loop index variable for counted loops. Just be sure to use an identiﬁer that you are not using for any other purpose. Otherwise you might accidentally wipe out a value that you will need later. Slide 30: 22 CHAPTER 2. WRITING SIMPLE PROGRAMS The interesting and useful thing about loops is the way that they alter the “ﬂow of control” in a program. Usually we think of computers as executing a series of instructions in strict sequence. Introducing a loop causes Python to go back and do some statements over and over again. Statements like the for loop are called control structures because they control the execution of other parts of the program. Some programmers ﬁnd it helpful to think of control structures in terms of pictures called ﬂowcharts. A ﬂowchart is a diagram that uses boxes to represent different parts of a program and arrows between the boxes to show the sequence of events when the program is running. Figure 2.1 depicts the semantics of the for loop as a ﬂowchart. more items in <sequence> no yes <var> = next item <body> Figure 2.1: Flowchart of a for loop. If you are having trouble understanding the for loop, you might ﬁnd it useful to study the ﬂowchart. The diamond shape box in the ﬂowchart represents a decision in the program. When Python gets the the loop heading, it checks to see if there are any (more) items left if the sequence. If the answer is yes, the value of the loop index variable is set to the next item in the sequence, and then the loop body is executed. Once the body is complete, the program goes back to the loop heading and checks for another value in the sequence. The loop quits when there are no more items, and the program moves on to the statements that come after the loop. 2.7 Example Program: Future Value Let’s close the chapter with one more example of the programming process in action. We want to develop a program to determine the future value of an investment. Let’s start with an analysis of the problem (requirements). You know that money that is deposited in a bank account earns interest, and this interest accumulates as the years pass. How much will an account be worth ten years from now? Obviously it depends on how much money we start with (the principal) and how much interest the account earns. Given the principal and the interest rate, a program should be able to calculate the value of the investment ten years into the future. We continue by developing the exact speciﬁcations for the program. Recall, this is a description of what the program will do. What exactly should the inputs be? We need the user to enter the initial amount to invest, the principal. We will also need some indication of how much interest the account earns. This depends both on the interest rate and how often the interest is compounded. One simple way of handling this is to have the user enter an annualized percentage rate. Whatever the actual interest rate and compounding frequency, the annualized rate tells us how much the investment accrues in one year. If the annualized interest is 3%, then a Slide 31: 2.7. EXAMPLE PROGRAM: FUTURE VALUE 23 $100 investment will grow to $103 in one year’s time. How should the user represent an annualized rate of 3%? There are a number of reasonable choices. Let’s assume the user supplies a decimal, so the rate would be entered as 0.03. This leads us to the following speciﬁcation. Program Future Value Inputs principal The amount of money being invested in dollars. apr The annualized percentage rate expressed as a decimal fraction. Output The value of the investment 10 years into the future. Relationship Value after one year is given by principal 1 apr¡ . This formula needs to be applied 10 times. Next we design an algorithm for the program. We’ll use pseudocode, so that we can formulate our ideas without worrying about all the rules of Python. Given our speciﬁcation, the algorithm seems straightforward. Print an introduction Input the amount of the principal (principal) Input the annualized percentage rate (apr) Repeat 10 times: principal = principal * (1 + apr) Output the value of principal Now that we’ve thought the problem all the way through to pseudocode, it’s time to put our new Python knowledge to work and develop a program. Each line of the algorithm translates into a statement of Python. Print an introduction (print statement, Section 2.4) print "This program calculates the future value of a 10-year investment" Input the amount of the principal (input statement, Section 2.5.2) principal = input("Enter the initial principal: Input the annualized percentage rate (input statement, Section 2.5.2) apr = input("Enter the annualized interest rate: Repeat 10 times: (counted loop, Section 2.6) for i in range(10): Calculate principal = principal * (1 + apr) (simple assignment statement, Section 2.5.1) principal = principal * (1 + apr) Output the value of the principal (print statement, Section 2.4) print "The amount in 10 years is:", principal ¢ ") ") All of the statement types in this program have been discussed in detail in this chapter. If you have any questions, you should go back and review the relevant descriptions. Notice especially the counted loop pattern is used to apply the interest formula 10 times. That about wraps it up. Here is the completed program. # futval.py # A program to compute the value of an investment # carried 10 years into the future Slide 32: 24 # by: John M. Zelle CHAPTER 2. WRITING SIMPLE PROGRAMS def main(): print "This program calculates the future value of a 10-year investment." principal = input("Enter the initial principal: ") apr = input("Enter the annualized interest rate: ") for i in range(10): principal = principal * (1 + apr) print "The amount in 10 years is:", principal main() Notice that I have added a few blank lines to separate the Input, Processing, and Output portions of the program. Strategically placed “white space” can help make your programs more readable. That’s about it for this example; I leave the testing and debugging as an exercise for you. 2.8 Exercises 1. List and describe in your own words the six steps in the software development process. 2. Write out the chaos.py program (Section 1.6) and identify the parts of the program as follows: Circle each identiﬁer. Underline each expression. Put a comment at the end of each line indicating the type of statement on that line (output, assignment, input, loop, etc.) 3. A user-friendly program should print an introduction that tells the user what the program does. Modify the convert.py program (Section 2.2) to print an introduction. 4. Modify the avg2.py program (Section 2.5.3) to ﬁnd the average of three exam scores. 5. Modify the futval.py program (Section 2.7) so that the number of years for the investment is also a user input. Make sure to change the ﬁnal message to reﬂect the correct number of years. 6. Modify the convert.py program (Section 2.2) with a loop so that it executes 5 times before quitting (i.e., it converts 5 temperatures in a row). 7. Modify the convert.py program (Section 2.2) so that it computes and prints a table of Celsius temperatures and the Fahrenheit equivalents every 10 degrees from 0C to 100C. 8. Write a program that converts from Fahrenheit to Celsius. 9. Modify the futval.py program (Section 2.7) so that it computes the actual purchasing power of the investment, taking inﬂation into account. The yearly rate of inﬂation will be a second input. The adjustment is given by this formula: principal = principal/(1 + inflation) Slide 33: Chapter 3 Computing with Numbers When computers were ﬁrst developed, they were seen primarily as number crunchers, and that is still an important application. As you have seen, problems that involve mathematical formulas are easy to translate into Python programs. This chapter takes a closer look at computations involving numeric calculations. 3.1 Numeric Data Types The information that is stored and manipulated by computer programs is generically referred to as data. Different kinds of data will be stored and manipulated in different ways. Consider this program that calculates the value of loose change. # change.py # A program to calculate the value of some change in dollars def main(): print "Change Counter" print print "Please enter the count of each coin type." quarters = input("Quarters: ") dimes = input("Dimes: ") nickels = input("Nickels: ") pennies = input("Pennies: ") total = quarters * .25 + dimes * .10 + nickels * .05 + pennies * .01 print print "The total value of your change is", total main() Here is an example of the output. Change Counter Please enter the count of each coin type. Quarters: 5 Dimes: 3 Nickels: 4 Pennies: 6 The total value of your change is 1.81 This program actually manipulates two different kinds of numbers. The values entered by the user (5, 3, 4, 6) are are whole numbers; they don’t have any fractional part. The values of the coins (.25, .10, .05, .01) 25 Slide 34: 26 CHAPTER 3. COMPUTING WITH NUMBERS are decimal fractions. Inside the computer, whole numbers and numbers that have fractional components are represented differently. Technically, we say that these are two different data types. The data type of an object determines what values it can have and what operations can be performed on it. Whole numbers are represented using the integer data type (int for short). Values of type int can be positive or negative whole numbers. Numbers that can have fractional parts are represented as ﬂoating point (or ﬂoat) values. So how do we tell whether a number is an int or a ﬂoat? A numeric literal that does not contain a decimal point produces an int value, while a literal that has a decimal point is represented by a ﬂoat (even if the fractional part is 0). Python provides a special function called type that tells us the data type of any value. Here is an interaction with the Python interpreter showing the difference between int and ﬂoat literals. >>> type(3) <type ’int’> >>> type(3.14) <type ’float’> >>> type(3.0) <type ’float’> >>> myInt = -32 >>> type(myInt) <type ’int’> >>> myFloat = 32.0 >>> type(myFloat) <type ’float’> You may be wondering why there are two different data types for numbers. One reason has to do with program style. Values that represent counts can’t be fractional; we can’t have 3 1 quarters, for example. Using 2 an int value tells the reader of a program that the value can’t be a fraction. Another reason has to do with the efﬁciency of various operations. The underlying algorithms that perform computer arithmetic are simpler, and therefore faster, for ints than the more general algorithms required for ﬂoat values. You should be warned that the ﬂoat type only stores approximations. There is a limit to the precision, or accuracy, of the stored values. Since ﬂoat values are not exact, while ints always are, your general rule of thumb should be: if you don’t absolutely need fractional values, use an int. operator ¢ ¢ % abs() operation addition subtraction multiplication division exponentiation remainder absolute value Table 3.1: Python built-in numeric operations. A value’s data type determines what operations can be used on it. As we have seen, Python supports the usual mathematical operations on numbers. Table 3.1 summarizes these operations. Actually, this table is somewhat misleading since the two numeric data types have their own operations. When addition is performed on ﬂoats, the computer performs a ﬂoating point addition. Whereas, with ints, the computer performs an integer addition. Consider the following interaction with Python: >>> 3.0 + 4.0 7.0 >>> 3 + 4 7 ¡ ¡ Slide 35: 3.2. USING THE MATH LIBRARY >>> 3.0 * 4.0 12.0 >>> 3 * 4 12 >>> 10.0 / 3.0 3.33333333333 >>> 10 / 3 3 >>> 10 % 3 1 >>> abs(5) 5 >>> abs(-3.5) 3.5 27 Notice how operations on ﬂoats produce ﬂoats, and operations on ints produce ints. Most of the time, we don’t have to worry about what type of operation is being performed; for example, integer addition produces pretty much the same result as ﬂoating point addition. However, in the case of division, the results are quite different. Integer division always produces an integer, discarding any fractional result. Think of integer division as “gozinta.” The expression, 10 / 3 produces 3 because three gozinta (goes into) ten three times (with a remainder of one). The third to last example shows the remainder operation (%) in action. The remainder of dividing 10 by 3 is 1. The last two examples illustrate taking the absolute value of an expression. You may recall from Chapter 2 that Suzie Programmer used the expression 9.0 / 5.0 in her temperature conversion program rather than 9 / 5. Now you know why. The former gives the correct multiplier of 1 8, while the latter yields just 1, since 5 gozinta 9 just once. 3.2 Using the Math Library Besides the operations listed in Table 3.1, Python provides many other useful mathematical functions in a special math library. A library is just a module that contains some useful deﬁnitions. Our next program illustrates the use of this library to compute the roots of quadratic equations. A quadratic equation has the form ax2 bx c 0. Such an equation has two solutions for the value of x given by the quadratic formula: b b2 4ac x 2a Let’s write a program that can ﬁnd the solutions to a quadratic equation. The input to the program will be the values of the coefﬁcients a, b, and c. The outputs are the two values given by the quadratic formula. Here’s a program that does the job. # quadratic.py # A program that computes the real roots of a quadratic equation. # Illustrates use of the math library. # Note: this program crashes if the equation has no real roots. import math # Makes the math library available. ¢ def main(): print "This program finds the real solutions to a quadratic" print a, b, c = input("Please enter the coefficients (a, b, c): ") discRoot = math.sqrt(b * b - 4 * a * c) ¡ ¢ ¢ ¢ ¢ Slide 36: 28 root1 = (-b + discRoot) / (2 * a) root2 = (-b - discRoot) / (2 * a) CHAPTER 3. COMPUTING WITH NUMBERS print print "The solutions are:", root1, root2 main() This program makes use of the square root function sqrt from the math library module. The line at the top of the program: import math tells Python that we are using the math module. Importing a module makes whatever is deﬁned in it available to the program. To compute x, we use math.sqrt(x). You may recall this dot notation from Chapter 1. This tells Python to use the sqrt function that “lives” in the math module. In the quadratic program we calculate b2 4 ac with the line discRoot = math.sqrt(b * b - 4 * a * c) Here is how the program looks in action: This program finds the real solutions to a quadratic Please enter the coefficients (a, b, c): 3, 4, -2 The solutions are: 0.387425886723 -1.72075922006 This program is ﬁne as long as the quadratics we try to solve have real solutions. However, some inputs will cause the program to crash. Here’s another example run: This program finds the real solutions to a quadratic Please enter the coefficients (a, b, c): 1, 2, 3 Traceback (innermost last): File "<stdin>", line 1, in ? File "quadratic.py", line 13, in ? discRoot = math.sqrt(b * b - 4 * a * c) OverflowError: math range error The problem here is that b2 4 a c 0, and the sqrt function is unable to compute the square root of a negative number. Python prints a math range error. Right now, we don’t have the tools to ﬁx this problem, so we will just have to assume that the user gives us solvable equations. Actually, quadratic.py did not need to use the math library. We could have taken the square root using exponentiation . (Can you see how?) Using math.sqrt is somewhat more efﬁcient and allowed me to illustrate the use of the math library. In general, if your program requires a common mathematical function, the math library is the ﬁrst place to look. Table 3.2 shows some of the other functions that are available in the math library. 3.3 Accumulating Results: Factorial Suppose you have a root beer sampler pack containing six different kinds of root beer. Drinking the various ﬂavors in different orders might affect how good they taste. If you wanted to try out every possible ordering, how many different orders would there be? It turns out the answer is a surprisingly large number, 720. Do you know where this number comes from? The value 720 is the factorial of 6. In mathematics, factorial is often denoted with an exclamation (“!”). The factorial of a whole number n 1¡ . This happens to be the number of distinct arrangements for n items. is deﬁned as n! n n 1¡ n 2¡ Given six items, we compute 6! 6¡ 5¡ 4¡ 3¡ 2¡ 1¡ 720 possible arrangements. ¢ ¢ ¡ ¢ ¢ ¡ Slide 37*: 3.3. ACCUMULATING RESULTS: FACTORIAL Python pi e sin(x) cos(x) tan(x) asin(x) acos(x) atan(x) log(x) log10(x) exp(x) ceil(x) floor(x) Mathematics π e sin x cos x tan x arcsinx arccosx arctanx ln x log10 x ex x¡ x£ ¢ 29 English An approximation of pi. An approximation of e. The sine of x. The cosine of x. The tangent of x. The inverse of sine x. The inverse of cosine x. The inverse of tangent x. The natural (base e) logarithm of x The common (base 10) logarithm of x. The exponential of x. The smallest whole number x The largest whole number x Table 3.2: Some math library functions. Let’s write a program that will compute the factorial of a number entered by the user. The basic outline of our program follows an Input-Process-Output pattern. Input number to take factorial of, n Compute factorial of n, fact Output fact Obviously, the tricky part here is in the second step. How do we actually compute the factorial? Let’s try one by hand to get an idea for the process. In computing the factorial of 6, we ﬁrst multiply 6 5 30. Then we take that result and do another multiplication 30 4 120. This result is multiplied by three 120 3 360. Finally, this result is multiplied by 2 360 2 720. According to the deﬁnition, we then multiply this result by 1, but that won’t change the ﬁnal value of 720. Now let’s try to think about the algorithm more generally. What is actually going on here? We are doing repeated multiplications, and as we go along, we keep track of the running product. This is a very common algorithmic pattern called an accumulator. We build up, or accumulate, a ﬁnal value piece by piece. To accomplish this in a program, we will use an accumulator variable and a loop structure. The general pattern looks like this. Initialize the accumulator variable Loop until final result is reached update the value of accumulator variable Realizing this is the pattern that solves the factorial problem, we just need to ﬁll in the details. We will be accumulating the factorial. Let’s keep it in a variable called fact. Each time through the loop, we need to multiply fact by one of the factors n¤ n 1¡ 1. It looks like we should use a for loop that iterates over this sequence of factors. For example, to compute the factorial of 6, we need a loop that works like this. fact = 1 for factor in [6,5,4,3,2,1]: fact = fact factor Take a minute to trace through the execution of this loop and convince yourself that it works. When the loop body ﬁrst executes, fact has the value 1 and factor is 6. So, the new value of fact is 1 6 6. The next time through the loop, factor will be 5, and fact is updated to 6 5 30. The pattern continues for each successive factor until the ﬁnal result of 720 has been accumulated. The initial assignment of 1 to fact before the loop is essential to get the loop started. Each time through the loop body (including the ﬁrst), the current value of fact is used to compute the next value. The ¤ ¦ ¤ ¥ ¢ Slide 38: 30 CHAPTER 3. COMPUTING WITH NUMBERS initialization ensures that fact has a value on the very ﬁrst iteration. Whenever you use the accumulator pattern, make sure you include the proper initialization. Forgetting it is a common mistake of beginning programmers. Of course, there are many other ways we could have written this loop. As you know from math class, multiplication is commutative and associative, so it really doesn’t matter what order we do the multiplications in. We could just as easily go the other direction. You might also notice that including 1 in the list of factors is unnecessary, since multiplication by 1 does not change the result. Here is another version that computes the same result. fact = 1 for factor in [2,3,4,5,6]: fact = fact * factor Unfortunately, neither of these loops solves the original problem. We have hand-coded the list of factors to compute the factorial of six. What we really want is a program that can compute the factorial of any given input n. We need some way to generate an appropriate list from the value of n. Luckily, this is quite easy to do using the Python range function. Recall that range(n) produces a list of numbers starting with 0 and continuing up to, but not including, n. There are other variations of range that can be used to produce different sequences. With two parameters, range(start,n) produces a sequence that starts with the value start and continues up to, but not including, n. A third version range(start, n, step) is like the two parameter version, except that it uses step as the increment between numbers. Here are some examples. >>> range(10) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] >>> range(5,10) [5, 6, 7, 8, 9] >>> range(5, 10, 3) [5, 8] Given our input value n we have a couple of different range commands that produce an appropriate list of factors for computing the factorial of n. To generate them from smallest to largest (a la our second loop), we could use range(2,n+1). Notice how I used n+1 as the second parameter, since the range will go up to, but not including this value. We need the +1 to make sure that n itself is included as the last factor. Another possibility is to generate the factors in the other direction (a la our ﬁrst loop) using the threeparameter version of range and a negative step to cause the counting to go backwards: range(n,1,-1). This one produces a list starting with n and counting down (step -1) to, but not including 1. Here then is one possible version of the factorial program. # factorial.py # Program to compute the factorial of a number # Illustrates for loop with an accumulator def main(): n = input("Please enter a whole number: ") fact = 1 for factor in range(n,1,-1): fact = fact * factor print "The factorial of", n, "is", fact main() Of course there are numerous other ways this program could have been written. I have already mentioned changing the order of factors. Another possibility is to initialize fact to n and then use factors starting at n 1 (as long as n 0). You might try out some of these variations and see which you like best. ¢ Slide 39: 3.4. THE LIMITS OF INT 31 3.4 The Limits of Int So far, I have talked about numeric data types as representations of familiar numbers such as integers and decimal fractions. It is important to keep in mind, however, that these numeric types are just representations, and they do not always behave exactly like the numbers that they represent. We can see an example of this as we test out our new factorial program. >>> import factorial Please enter a whole number: 6 The factorial of 6 is 720 >>> factorial.main() Please enter a whole number: 10 The factorial of 10 is 3628800 >>> factorial.main() Please enter a whole number: 13 Traceback (innermost last): File "<stdin>", line 1, in ? File "factorial.py", line 9, in main fact = fact * factor OverflowError: integer multiplication Everything seems ﬁne until we try to compute the factorial of 13. When computing 13! the program prints out an OverflowError message. What is going on here? The problem is that this program is representing whole numbers using Python’s int data type. Unfortunately, ints are not exactly like mathematical integers. There are inﬁnitely many integers, but only a ﬁnite range of ints. Inside the computer, ints are stored in a ﬁxed-sized binary representation. To make sense of this, we need to look at what’s going on at the hardware level. Computer memory is composed of electrical “switches,” each of which can be in one of two possible states, basically on or off. Each switch represents a binary digit or bit of information. One bit can encode two possibilities, usually represented with the numerals 0 (for off) and 1 (for on). A sequence of bits can be used to represent more possibilities. With two bits, we can represent four things. bit 2 0 0 1 1 bit 1 0 1 0 1 Three bits allow us to represent eight different values by adding a zero or one to each of the four two-bit patterns. bit 3 bit 2 bit 1 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 You can see the pattern here. Each extra bit doubles the number of distinct patterns. In general, n bits can represent 2n different values. The number of bits that a particular computer uses to represent an int depends on the design of the CPU. Typical PCs today use 32 bits. That means there are 232 possible values. These values are centered at 0 to Slide 40: 32 CHAPTER 3. COMPUTING WITH NUMBERS 32 represent a range of positive and negative integers. Now 22 231 . So, the range of integers that can be 31 31 represented in a 32 bit int value is 2 2 1. The reason for the 1 on the high end is to account for the representation of 0 in the top half of the range. Let’s try out some expressions in Python to test this analysis. Remember that is the Python exponentiation operator. >>> 2 30 1073741824 >>> 2 31 Traceback (innermost last): File "<stdin>", line 1, in ? OverflowError: integer pow() Python can calculate 230 , but “blows up” trying to compute 231 . You can see that the overﬂow happens somewhere between the 30th and 31st power of two. That is consistent with our analysis that the largest int is 231 1. Suppose we try to display the largest int. >>> 2 31 - 1 Traceback (innermost last): File "<stdin>", line 1, in ? OverflowError: integer pow() Our ﬁrst try didn’t work. Can you see why? Python evaluates this expression by ﬁrst trying to calculate 2 31. That calculation produces the error before Python has a chance to subtract one. We need to be a little cleverer and sneak up on the value from underneath. We can use the fact that 231 230 230 . Strategically subtracting one from each side gives us 231 1 230 1 230. By subtracting one in the middle of the computation, we can ensure that the intermediate value never gets bigger than the ﬁnal result. Here’s what Python says: >>> 2 30 - 1 + 2 30 2147483647 By the way, this expression illustrates another way that Python ints differ from the integers that they represent. In normal arithmetic, there is no difference between 2 31 1 and 230 1 230. They both represent the same value. In computer arithmetic, however, one is computable and the other is not! Representations of numbers do not always obey all the properties that we take for granted with numbers. Now that we have a numeric value, we can directly test our conjecture that this is the largest int. >>> 2147483647 2147483647 >>> 2147483648 OverflowError: integer literal too large There you have it. The largest int that can be represented in 32 bits is 2¤ 147¤ 483¤ 647. Now you know exactly why our program for factorial can’t compute 13!. This value is larger than the limit of 2¤ 147¤ 483¤ 647. Naturally, the next step is to ﬁgure out a way around this limitation. ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ 3.5 Handling Large Numbers: Long Ints As long as our factorial program relies on the int data type, we will not be able to ﬁnd the factorial of larger numbers. We need to use another numeric type. You might ﬁrst think of using a ﬂoat instead. This does not really solve our problem. Here is an example run of a modiﬁed factorial program that initializes fact to the ﬂoat 1 0. Slide 41: 3.5. HANDLING LARGE NUMBERS: LONG INTS Please enter a whole number. 15 The factorial of 15 is 1.307674368e+12 33 We do not get an overﬂow error, but we also do not get an exact answer. A very large (or very small) ﬂoating point value is printed out using exponential, or scientiﬁc, notation. The e+12 at the end means that the result is equal to 1 307674368 10 12. You can think of the +12 at the end as a marker that shows where the decimal point should be placed. In this case, it must move 12 places to the right to get the actual value. However, there are only 9 digits to the right of the decimal, so we have “lost” the last three digits. Remember, ﬂoats are approximations. Using a ﬂoat allows us to represent a much larger range of values, but the amount of precision is still ﬁxed. In fact, a computer stores ﬂoating point numbers as a pair of ﬁxedlength (binary) integers. One integer represents the string of digits in the value, and the second represents the exponent value that keeps track of where the whole part ends and the fractional part begins. Fortunately, Python provides a better solution for large, exact values in the form of a third numeric type long int. A long int is not a ﬁxed size, but expands to accommodate whatever value it holds. The only limit is the amount of memory the computer has available to it. To get a long int, you put an “L” sufﬁx on a numeric literal. So, the literal 5 is an int representation of the number ﬁve, but 5L is a long int representation of the number ﬁve. Of course, for a number this small, there is no reason to use a long int. However, using a long int causes Python to use long int operations, and our value can grow to any size. Here are some examples that illustrate: >>> 2L 2L >>> 2L 31 2147483648L >>> type(100L) <type ’long int’> >>> 10000000000000000000000000000000000000L + 25 10000000000000000000000000000000000025L Notice how calculations involving a long int produce a long int result. Using long ints allows us to compute with really large numbers. We can modify our factorial program to use long int by simply initializing fact to 1L. Since we start with a long int, each successive multiplication will produce another long int. # factorial2.py def main(): n = input("Please enter a whole number: ") fact = 1L # Use a long int here for factor in range(n,0,-1): fact = fact * factor print "The factorial of", n, "is", fact Now we can take the factorial of arbitrarily large inputs. >>> import factorial2 Please enter a whole number: 13 The factorial of 13 is 6227020800 >>> factorial2.main() Please enter a whole number: 100 The factorial of 100 is 933262154439441526816992388562667004907159682 643816214685929638952175999932299156089414639761565182862536979208272 23758251185210916864000000000000000000000000 Slide 42: 34 CHAPTER 3. COMPUTING WITH NUMBERS If you have an older version of Python (prior to version 2.0), these answers will be printed with an “L” appended. This is Python’s way of saying that the number is a long int. In newer versions, this artifact is automatically removed by the print statement. In the next chapter, you will learn how you could take care of getting the “L” out yourself. Now we have a factorial program that can compute interestingly large results. Long ints are pretty cool; you might be tempted to use them all the time. The down-side of using long ints is that these representations are less efﬁcient than the plain int data type. The operations needed to do arithmetic on ints is built into the CPU of the computer. When doing operations on long ints, Python has to employ algorithms that simulate long arithmetic using the computer’s built-in ﬁxed-length operations. As a result, long int arithmetic is much slower than int arithmetic. Unless you need very large values, ints are preferred. 3.6 Type Conversions Sometimes values of one data type need to be converted into another. You have seen that combining an int with an int produces an int, and combining a ﬂoat with a ﬂoat creates another ﬂoat. But what happens if we write an expression that mixes an int with a ﬂoat? For example, what should the value of x be after this assignment statement? x = 5.0 / 2 If this is ﬂoating point division, then the result should be the ﬂoat value 2 5. If integer division is performed, the result is 2. Before reading ahead for the answer, take a minute to consider how you think Python should handle this situation. In order to make sense of the expression 5.0 / 2, Python must either change 5.0 to 5 and perform integer division or convert 2 to 2.0 and perform ﬂoating point division. In general, converting a ﬂoat to an int is a dangerous step, because some information (the fractional part) will be lost. On the other hand, an int can be safely turned into a ﬂoat just by adding a fractional part of 0. So, in mixed-typed expressions, Python will automatically convert ints to ﬂoats and perform ﬂoating point operations to produce a ﬂoat result. Sometimes we may want to perform a type conversion ourselves. This is called an explicit type conversion. For example, suppose we are writing a program that ﬁnds the average of some numbers. Our program would ﬁrst sum up the numbers and then divide by n, the count of how many numbers there are. The line of code to compute the average might look like this. average = sum / n Unfortunately, this line may not always produce the result we intend. Consider a speciﬁc example. The numbers to be averaged are the ints 4, 5, 6, 7. The sum variable will hold 22, also an int, and dividing by 4 gives the answer 5, not 5.5. Remember, an int divided by an int always produces an int. To solve this problem, we need to tell Python to convert one of the operands to a ﬂoating point value. average = float(sum) / n The float() function converts an int into a ﬂoat. We only need to convert the numerator, because this produces a mixed-type expression, and Python will automatically convert the denominator. Notice that putting the float() around the entire expression would not work. average = float(sum/n) In this form, sum and n could both be ints causing Python to perform an integer division and then convert the resulting quotient to a ﬂoat. Of course, this ﬂoat would always end in 0, since it is being converted from an int. That is not what we want. Python also provides int() and long() functions that can be used to convert numbers into ints and longs, respectively. Here are a few examples. Slide 43: 3.7. EXERCISES >>> 4 >>> 3 >>> 3L >>> 3.0 >>> 3 >>> 3 int(4.5) int(3.9) long(3.9) float(int(3.3)) int(float(3.3)) int(float(3)) 35 As you can see, converting to an int or long int simply discards the fractional part of a ﬂoat; the value is truncated, not rounded. If you want a rounded result, you can add 0.5 to the value before using int(), assuming the value is positive. A more general way of rounding off numbers is to use the built-in round function which rounds a ﬂoat off to the nearest whole value. >>> round(3.14) 3.0 >>> round(-3.14) -3.0 >>> round(3.5) 4.0 >>> round(-3.5) -4.0 >>> int(round(-3.14)) -3 Notice that round returns a ﬂoat. The last example shows how the result can then be turned into an int value, if necessary, by using int(). 3.7 Exercises 1. Show the result of evaluating each expression. Be sure that the value is in the proper form to indicate its type (int, long int, or ﬂoat). If the expression is illegal, explain why. (a) 4.0 / 10.0 + 3.5 * 2 (b) 10 % 4 + 6 / 2 (c) abs(4 - 20 / 3) ** 3 (d) sqrt(4.5 - 5.0) + 7 * 3 (e) 3 * 10 / 3 + 10 % 3 (f) 3L 3 2. Translate each of the following mathematical expressions into an equivalent Python expression. You may assume that the math library has been imported (via import math). n n¡ 1¢ 2 4πr2 ¢ ¢ (a) 3 (b) (c) (d) (e) £ 4¡ r cos a¡ y2¡ y1 x2¡ x1 r sina¡ 5 2 2 Slide 44*: 36 CHAPTER 3. COMPUTING WITH NUMBERS 3. Show the list of numbers that would be generated by each of the following range expressions. (a) range(5) (b) range(3, 10) (c) range(4, 13, 3) (d) range(15, 5, -2) (e) range(5, 3) 4. Show the output that would be generated by each of the following program fragments. (a) for i in range(1, 11): print ii (b) for i in [1,3,5,7,9]: print i, ":", i*3 print i (c) x = 2 y = 10 for j in range(0, y, x): print j, print x + y print "done" (d) ans = 0 for i in range(1, 11): ans = ans + ii print i print ans 5. Write a program to calculate the volume and surface area of a sphere from its radius, given as input. Here are some formulas that might be useful: V 4 3πr 3 A 4πr2 6. Write a program that calculates the cost per square inch of a circular pizza, given its diameter and price. A πr2 7. Write a program that determines the molecular weight of a hydrocarbon based on the number of hydrogen, carbon, and oxygen atoms. You should use the following weights: Atom H C O Weight (grams / mole) 1.0079 12.011 15.9994 ¡ 8. Write a program that determines the distance to a lighting strike based on the time elapsed between the ﬂash and the sound of thunder. The speed of sound is approximately 1100 ft/sec and 1 mile is 5280 ft. 9. The Konditorei coffee shop sells coffee at $10.50 a pound plus the cost of shipping. Each order ships for $0.86 per pound + $1.50 ﬁxed cost for overhead. Write a program that calculates the cost of an order. 10. Two points in a plane are speciﬁed using the coordinates (x1,y1) and (x2,y2). Write a program that calculates the slope of a line through two (non-vertical) points entered by the user. m y2¡ y1 x2¡ x1 11. Write a program that accepts two points (see previous problem) and determines the distance between x2 x1¡ 2 y2 y1¡ 2 them. d ¢ ¢ ¢ £ Slide 45: 3.7. EXERCISES 37 12. The Gregorian Epact is the number of days between Jan. 1st and the previous 1st quarter moon phase. This value is used to ﬁgure out the date of Easter. It is calculated by these formulas (using int arith8 C 4¡ C 8C 13¡ 25¡ 11 year%19¡ %30 Write a program metic): C year 100 epact that prompts the user for a 4-digit year and then outputs the value of the epact. 13. Write a program to calculate the area of a triangle given the length of its three sides a, b, and c. b s a 2 cA s s a¡ s b¡ s c¡ 14. Write a program to determine the length of a ladder required to reach a given height when leaned height against a house. The height and angle of the ladder are given as inputs. len sin angle 15. Write a program to ﬁnd the sum of the ﬁrst n natural numbers, where the value of n is provided by the user. 16. Write a program to ﬁnd the sum of the squares for the ﬁrst n natural numbers. 17. Write a program to sum a series of numbers entered by the user. The program should ﬁrst prompt the user for how many numbers are to be summed. It should then input each of the numbers and print a total sum. 18. Write a program that ﬁnds the average of a series of numbers entered by the user. As in the previous problem, the program will ﬁrst ask the user how many numbers there are. Note: the average should always be a ﬂoat, even if the user inputs are all ints. 19. Write a program that approximates the value of π by summing the terms of this series: 4 1 4 3 4 5 4 7 4 9 4 11 The program should prompt the user for n, the number of terms to sum and then output the sum of the ﬁrst n terms of this series. 20. A Fibonacci sequence is a sequence of numbers where each successive number is the sum of the previous two. The classic Fibonacci sequence begins: 1, 1, 2, 3, 5, 8, 13, . Write a program that computes the nth Fibonacci number where n is a value input by the user. For example, if n = 6, then the result is 8. Note: Fibonacci numbers grow very rapidly; your program should be able to handle very large numbers. 21. You have seen that the math library contains a function that computes the square root of numbers. In this exercise, you are to write your own algorithm for computing square roots. One way to solve this problem is to use a guess-and-check approach. You ﬁrst guess what the square root might be and then see how close your guess is. You can use this information to make another guess and continue guessing until you have found the square root (or a close approximation to it). One particularly good way of making guesses is to use Newton’s method. Suppose x is the number we want the root of, and guess is the current guessed answer. The guess can be improved by using x guess guess 2 ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¢ ¢ £ ¡ ¢ ¡ ¢ ¢ ¢ ¡ ¢ ¡ as the next guess. Write a program that implements Newton’s method. The program should prompt the user for the value to ﬁnd the square root of (x) and the number of times to improve the guess. Starting with a guess value of x/2, your program should loop the speciﬁed number of times applying Newton’s method and report the ﬁnal value of guess. You might also print out the value of math.sqrt(x) for comparison. Slide 46: 38 CHAPTER 3. COMPUTING WITH NUMBERS Slide 47: Chapter 4 Computing with Strings So far, we have been discussing programs designed to manipulate numbers. These days we know that computers are also useful for working with other kinds of data. In fact, the most common use for most personal computers is word processing. The data in this case is text. Text is represented in programs by the string data type, which is the subject of this chapter. 4.1 The String Data Type A string is a sequence of characters. In Chapter 2 you learned that a string literal is a sequence of characters in quotations. Python also allows strings to be delimited by single quotes (apostrophes). There’s no difference— just be sure to use a matching set. Strings can be stored in variables, just like numbers. Here are some examples illustrating these two forms of string literals. >>> str1 = "Hello" >>> str2 = ’spam’ >>> print str1, str2 Hello spam >>> type(str1) <type ’string’> >>> type(str2) <type ’string’> You already know how to print strings. Some programs also need to get string input from the user (e.g., a name). Getting string-valued input requires a bit of care. Remember that the input statement treats whatever the user types as an expression to be evaluated. Consider the following interaction. >>> firstName = input("Please Please enter your name: John Traceback (innermost last): File "<pyshell#8>", line 1, firstName = input("Please File "<string>", line 0, in NameError: John enter your name: ") in ? enter your name: ") ? Something has gone wrong here. Can you see what the problem is? Remember, an input statement is just a delayed expression. When I entered the name, “John”, this had the exact same effect as executing this assignment statement: firstName = John This statement says, “look up the value of the variable John and store that value in firstName.” Since John was never given a value, Python cannot ﬁnd any variable with that name and responds with a NameError. One way to ﬁx this problem is to type quotes around a string input so that it evaluates as a string literal. 39 Slide 48: 40 CHAPTER 4. COMPUTING WITH STRINGS >>> firstName = input("Please enter your name: ") Please enter your name: "John" >>> print "Hello", firstName Hello John This works, but it is not a very satisfactory solution. We shouldn’t have to burden the users of our programs with details like typing quotes around their names. Python provides a better mechanism. The raw input function is exactly like input except it does not evaluate the expression that the user types. The input is simply handed to the program as a string of text. Revisiting our example, here is how it looks with raw input: >>> firstName = raw_input("Please enter your name: ") Please enter your name: John >>> print "Hello", firstName Hello John Notice that this example works as expected without having to type quotes around the input. If you want to get textual input from the user, raw input is the way to do it. So far, we have seen how to get strings as input, assign them to variables and print them out. That’s enough to write a parrot program, but not to do any serious text-based computing. For that, we need some string operations. The rest of this section takes you on a tour of the more important Python string operations. In the following section, we’ll put these ideas to work in some example programs. While the idea of numeric operations may be old hat to you from your math studies, you may not have thought about string operations before. What kinds of things can we do with strings? For starters, remember what a string is: a sequence of characters. One thing we might want to do is access the individual characters that make up the string. In Python, this can be done through the operation of indexing. We can think of the positions in a string as being numbered, starting from the left with 0. Figure 4.1 H 0 e 1 l 2 l 3 o 4 5 B 6 o 7 b 8 Figure 4.1: Indexing of the string ”Hello Bob” illustrates with the string “Hello Bob.” Indexing is used in string expressions to access a speciﬁc character position in the string. The general form for indexing is <string>[<expr>]. The value of the expression determines which character is selected from the string. Here are some interactive indexing examples: >>> >>> ’H’ >>> Hl >>> >>> B greet = "Hello Bob" greet[0] print greet[0], greet[2], greet[4] o x=8 print greet[x-2] Notice that, in a string of n characters, the last character is at position n 1, because the indexes start at 0. Indexing returns a string containing a single character from a larger string. It is also possible to access a contiguous sequence of characters or substring from a string. In Python, this is accomplished through an operation called slicing. You can think of slicing as a way of indexing a range of positions in the string. Slicing takes the form string [ start : end ]. Both start and end should be int-valued expressions. A slice produces the substring starting at the position given by start and running up to, but not including, position end. Continuing with our interactive example, here are some slices. ¢ Slide 49: 4.2. SIMPLE STRING PROCESSING >>> greet[0:3] ’Hel’ >>> greet[5:9] ’ Bob’ >>> greet[:5] ’Hello’ >>> greet[5:] ’ Bob’ >>> greet[:] ’Hello Bob’ 41 The last three examples show that if either expression is missing, the start and end of the string are the assumed defaults. The ﬁnal expression actually hands back the entire string. Indexing and slicing are useful operations for chopping strings into smaller pieces. The string data type also supports operations for putting strings together. Two handy operators are concatenation (+) and repetition (). Concatenation builds a string by “gluing” two strings together. Repetition builds a string by multiple concatenations of a string with itself. Another useful function is len, which tells how many characters are in a string. Here are some examples: >>> "spam" + "eggs" ’spameggs’ >>> "Spam" + "And" + "Eggs" ’SpamAndEggs’ >>> 3 "spam" ’spamspamspam’ >>> "spam" * 5 ’spamspamspamspamspam’ >>> (3 * "spam") + ("eggs" * 5) ’spamspamspameggseggseggseggseggs’ >>> len("spam") 4 >>> len("SpamAndEggs") 11 >>> These basic string operations are summarized in Table 4.1. Operator + * string [ ] len( string ) string [ : ] Meaning Concatenation Repetition Indexing length slicing Table 4.1: Python string operations 4.2 Simple String Processing Now that you have an idea what string operations can do, we’re ready to write some programs. Our ﬁrst example is a program to compute the usernames for a computer system. Many computer systems use a username and password combination to authenticate system users. The system administrator must assign a unique username to each user. Often, usernames are derived from the user’s actual name. One scheme for generating usernames is to use the user’s ﬁrst initial followed by up Slide 50: 42 CHAPTER 4. COMPUTING WITH STRINGS to seven letters of the user’s last name. Using this method, the username for Elmer Thudpucker would be “ethudpuc,” and John Smith would just be “jsmith.” We want to write a program that reads a person’s name and computes the corresponding username. Our program will follow the basic input-process-output pattern. The result is simple enough that we can skip the algorithm development and jump right to the code. The outline of the algorithm is included as comments in the ﬁnal program. # username.py # Simple string processing program to generate usernames. def main(): print "This program generates computer usernames." print # get user’s first and last names first = raw_input("Please enter your first name (all lowercase): ") last = raw_input("Please enter your last name (all lowercase): ") # concatenate first initial with 7 chars of the last name. uname = first[0] + last[:7] # output the username print "Your username is:", uname main() This program ﬁrst uses raw input to get strings from the user. Then indexing, slicing, and concatenation are combined to produce the username. Here’s an example run. This program generates computer usernames. Please enter your first name (all lowercase): elmer Please enter your last name (all lowercase): thudpucker Your username is: ethudpuc As you can see, computing with strings is very similar to computing with numbers. Here is another problem that we can solve with string operations. Suppose we want to print the abbreviation of the month that corresponds to a given month number. The input to the program is an int that represents a month number (1–12), and the output is the abbreviation for the corresponding month. For example, if the input is 3, then the output should be Mar, for March. At ﬁrst, it might seem that this program is beyond your current ability. Experienced programmers recognize that this is a decision problem. That is, we have to decide which of 12 different outputs is appropriate, based on the number given by the user. We will not cover decision structures until later; however, we can write the program now by some clever use of string slicing. The basic idea is to store all the month names in a big string. months = "JanFebMarAprMayJunJulAugSepOctNovDec" We can lookup a particular month by slicing out the appropriate substring. The trick is computing where to slice. Since each month is represented by three letters, if we knew where a given month started in the string, we could easily extract the abbreviation. monthAbbrev = months[pos:pos+3] This would get us the substring of length three that starts in the position indicated by pos. How do we compute this position? Let’s try a few examples and see what we ﬁnd. Remember that string indexing starts at 0. Slide 51: 4.3. STRINGS AND SECRET CODES month Jan Feb Mar Apr number 1 2 3 4 position 0 3 6 9 43 Of course, the positions all turn out to be multiples of 3. To get the correct multiple, we just subtract 1 from the month number and then multiply by 3. So for 1 we get 1 1¡ 3 0 3 0 and for 12 we have 12 1¡ 3 11 3 33 Now we’re ready to code the program. Again, the ﬁnal result is short and sweet; the comments document the algorithm we’ve developed. # month.py # A program to print the abbreviation of a month, given its number def main(): # months is used as a lookup table months = "JanFebMarAprMayJunJulAugSepOctNovDec" n = input("Enter a month number (1-12): ") # compute starting position of month n in months pos = (n-1) * 3 # Grab the appropriate slice from months monthAbbrev = months[pos:pos+3] # print the result print "The month abbreviation is", monthAbbrev + "." main() Notice the last line of this program uses string concatenation to put a period at the end of the month abbreviation. Here is a sample of program output. Enter a month number (1-12): 4 The month abbreviation is Apr. 4.3 Strings and Secret Codes 4.3.1 String Representation Hopefully, you are starting to get the hang of computing with textual (string) data. However, you might still be wondering how computers actually manipulate strings. In the previous chapter, you saw how computers store numbers in binary notation (sequences of zeros and ones); the computer CPU contains circuitry to do arithmetic with these representations. Textual information is represented in exactly the same way. Underneath, when the computer is manipulating text, it is really no different from number crunching. To understand this, you might think in terms of messages and secret codes. Consider the age-old grade school dilemma. You are sitting in class and want to pass a note to a friend across the room. Unfortunately, the note must pass through the hands, and in front of the curious eyes, of many classmates before it reaches its ﬁnal destination. And, of course, there is always the risk that the note could fall into enemy hands (the teacher’s). So you and your friend need to design a scheme for encoding the contents of your message. One approach is to simply turn the message into a sequence of numbers. You could choose a number to correspond to each letter of the alphabet and use the numbers in place of letters. Without too much ¢ ¢ Slide 52: 44 CHAPTER 4. COMPUTING WITH STRINGS imagination, you might use the numbers 1-26 to represent the letters a–z. Instead of the word “sourpuss,” you would write “18, 14, 20, 17, 15, 20, 18, 18.” To those who don’t know the code, this looks like a meaningless string of numbers. For you and your friend, however, it represents a word. This is how a computer represents strings. Each character is translated into a number, and the entire string is stored as a sequence of (binary) numbers in computer memory. It doesn’t really matter what number is used to represent any given character as long as the computer is consistent about the encoding/decoding process. In the early days of computing, different designers and manufacturers used different encodings. You can imagine what a headache this was for people transferring data between different systems. Consider the situation that would result if, say, PCs and MacIntosh computers each used their own encoding. If you type a term paper on a PC and save it as a text ﬁle, the characters in your paper are represented as a certain sequence of numbers. Then if the ﬁle was read into your instructor’s MacIntosh computer, the numbers would be displayed on the screen as different characters from the ones you typed. The result would look like gibberish! To avoid this sort of problem, computer systems today use industry standard encodings. One important standard is called ASCII (American Standard Code for Information Interchange). ASCII uses the numbers 0 through 127 to represent the characters typically found on an (American) computer keyboard, as well as certain special values known as control codes that are used to coordinate the sending and receiving of information. For example, the capital letters A–Z are represented by the values 65–90, and the lowercase versions have codes 97–122. One problem with the ASCII encoding, as its name implies, is that it is American-centric. It does not have symbols that are needed in many other languages. Extended ASCII encodings have been developed by the International Standards Organization to remedy this situation. Most modern systems are moving to support of UniCode a much larger standard that includes support for the characters of all written languages. Newer versions of Python include support for UniCode as well as ASCII. Python provides a couple built-in functions that allow us to switch back and forth between characters and the numeric values used to represent them in strings. The ord function returns the numeric (“ordinal”) code of a single-character string, while chr goes the other direction. Here are some interactive examples: >>> 97 >>> 65 >>> ’a’ >>> ’Z’ ord("a") ord("A") chr(97) chr(90) 4.3.2 Programming an Encoder Let’s return to the note-passing example. Using the Python ord and chr functions, we can write some simple programs that automate the process of turning messages into strings of numbers and back again. The algorithm for encoding the message is simple. get the message to encode for each character in the message: print the letter number of the character Getting the message from the user is easy, a raw input will take care of that for us. message = raw_input("Please enter the message to encode: ") Implementing the loop requires a bit more effort. We need to do something for each character of the message. Recall that a for loop iterates over a sequence of objects. Since a string is a kind of sequence, we can just use a for loop to run-through all the characters of the message. for ch in message: Slide 53: 4.3. STRINGS AND SECRET CODES 45 Finally, we need to convert each character to a number. The simplest approach is to use the ASCII number (provided by ord) for each character in the message. Here is the ﬁnal program for encoding the message: # text2numbers.py # A program to convert a textual message into a sequence of # numbers, utlilizing the underlying ASCII encoding. def main(): print "This program converts a textual message into a sequence" print "of numbers representing the ASCII encoding of the message." print # Get the message to encode message = raw_input("Please enter the message to encode: ") print print "Here are the ASCII codes:" # Loop through the message and print out the ASCII values for ch in message: print ord(ch), # use comma to print all on one line. print main() We can use the program to encode important messages. This program converts a textual message into a sequence of numbers representing the ASCII encoding of the message. Please enter the message to encode: What a Sourpuss! Here are the ASCII codes: 87 104 97 116 32 97 32 83 111 117 114 112 117 115 115 33 One thing to notice about this result is that even the space character has a corresponding ASCII code. It is represented by the value 32. 4.3.3 Programming a Decoder Now that we have a program to turn a message into a sequence of numbers, it would be nice if our friend on the other end had a similar program to turn the numbers back into a readable message. Let’s solve that problem next. Our decoder program will prompt the user for a sequence of numbers representing ASCII codes and then print out the text message corresponding to those codes. This program presents us with a couple of challenges; we’ll address these as we go along. The overall outline of the decoder program looks very similar to the encoder program. One change in structure is that the decoding version will collect the characters of the message in a string and print out the entire message at the end of the program. To do this, we need to use an accumulator variable, a pattern we saw in the factorial program from the previous chapter. Here is the decoding algorithm: get the sequence of numbers to decode message = "" for each number in the input: convert the number to the appropriate character Slide 54: 46 CHAPTER 4. COMPUTING WITH STRINGS add the character to the end of message print the message Before the loop, the accumulator variable message is initialized to be an empty string, that is a string that contains no characters (""). Each time through the loop a number from the input is converted into an appropriate character and appended to the end of the message constructed so far. The algorithm seems simple enough, but even the ﬁrst step presents us with a problem. How exactly do we get the sequence of numbers to decode? We don’t even know how many numbers there will be. To solve this problem, we are going to rely on some more string manipulation operations. First, we will read the entire sequence of numbers as a single string using raw input. Then we will split the big string into a sequence of smaller strings, each of which represents one of the numbers. Finally, we can iterate through the list of smaller strings, convert each into a number, and use that number to produce the corresponding ASCII character. Here is the complete algorithm: get the sequence of numbers as a string, inString split inString into a sequence of smaller strings message = "" for each of the smaller strings: change the string of digits into the number it represents append the ASCII character for that number to message print message This looks complicated, but Python provides some functions that do just what we need. We saw in Chapter 3 that Python provides a standard math library containing useful functions for computing with numbers. Similarly, the string library contains many functions that are useful in stringmanipulation programs. For our decoder, we will make use of the split function. This function is used to split a string into a sequence of substrings. By default, it will split the string wherever a space occurs. Here’s an example: >>> import string >>> string.split("Hello string library!") [’Hello’, ’string’, ’library!’] You can see how split has turned the original string "Hello string library!" into a list of three strings: "Hello", "string" and "library!". By the way, the split function can be used to split a string at places other than spaces by supplying the character to split on as a second parameter. For example, if we have a string of numbers separated by commas, we could split on the commas. >>> string.split("32,24,25,57", ",") [’32’, ’24’, ’25’, ’57’] Since our decoder program should accept the same format that was produced by the encoder program, namely a sequence of numbers with spaces between, the default version of split works nicely. >>> string.split("87 104 97 116 32 97 32 83 111 117 114 112 117 115 115 33") [’87’, ’104’, ’97’, ’116’, ’32’, ’97’, ’32’, ’83’, ’111’, ’117’, ’114’, ’112’, ’117’, ’115’, ’115’, ’33’] Notice that the resulting list is not a sequence of numbers, it is a sequence of strings. It just so happens these strings contain only digits and could be interpreted as numbers. All that we need now is a way of converting a string containing digits into a Python number. One way to accomplish this is with the Python eval function. This function takes any string and evaluates it as if it were a Python expression. Here are some interactive examples of eval: >>> numStr = "500" >>> eval(numStr) 500 Slide 55: 4.3. STRINGS AND SECRET CODES >>> eval("345.67") 345.67 >>> eval("3+4") 7 >>> x = 3.5 >>> y = 4.7 >>> eval("x * y") 16.45 >>> x = eval(raw_input("Enter a number ")) Enter a number 3.14 >>> print x 3.14 47 The last pair of statements shows that the eval of a raw input produces exactly what we would expect from an input expression. Remember, input evaluates what the user types, and eval evaluates whatever string it is given. Using split and eval we can write our decoder program. # numbers2text.py # A program to convert a sequence of ASCII numbers into # a string of text. import string # include string library for the split function. def main(): print "This program converts a sequence of ASCII numbers into" print "the string of text that it represents." print # Get the message to encode inString = raw_input("Please enter the ASCII-encoded message: ") # Loop through each substring and build ASCII message message = "" for numStr in string.split(inString): asciiNum = eval(numStr) # convert digit string to a number message = message + chr(asciiNum) # append character to message print "The decoded message is:", message main() Study this program a bit, and you should be able to understand exactly how it accomplishes its task. The heart of the program is the loop. for numStr in string.split(inString): asciiNum = eval(numStr) message = message + chr(asciiNum) The split function produces a sequence of strings, and numStr takes on each successive (sub)string in the sequence. I called the loop variable numStr to emphasize that its value is a string of digits that represents some number. Each time through the loop, the next substring is converted to a number by evaling it. This number is converted to the corresponding ASCII character via chr and appended to the end of the accumulator, message. When the loop is ﬁnished, every number in inString has been processed and message contains the decoded text. Here is an example of the program in action: Slide 56: 48 CHAPTER 4. COMPUTING WITH STRINGS >>> import numbers2text This program converts a sequence of ASCII numbers into the string of text that it represents. Please enter the ASCII-encoded message: 83 116 114 105 110 103 115 32 97 114 101 32 70 117 110 33 The decoded message is: Strings are Fun! 4.3.4 Other String Operations Now we have a couple programs that can encode and decode messages as sequences of ASCII values. These programs turned out to be quite simple due to the power both of Python’s string data type and its built-in operations as well as extensions that can be found in the string library. Python is a very good language for writing programs that manipulate textual data. Table 4.2 lists some of the other useful functions of the string library. Note that many of these functions, like split, accept additional parameters to customize their operation. Python also has a number of other standard libraries for text-processing that are not covered here. You can consult the online documentation or a Python reference to ﬁnd out more. Function capitalize(s) capwords(s) center(s, width) count(s, sub) find(s, sub) join(list) ljust(s, width) lower(s) lstrip(s) replace(s,oldsub,newsub) rfind(s, sub) rjust(s,width) rstrip(s) split(s) upper(s) Meaning Copy of s with only the ﬁrst character capitalized Copy of s with ﬁrst character of each word capitalized Center s in a ﬁeld of given width Count the number of occurrences of sub in s Find the ﬁrst position where sub occurs in s Concatenate list of strings into one large string Like center, but s is left-justiﬁed Copy of s in all lowercase characters Copy of s with leading whitespace removed Replace all occurrences of oldsub in s with newsub Like find, but returns the rightmost position Like center, but s is right-justiﬁed Copy of s with trailing whitespace removed Split s into a list of substrings (see text). Copy of s with all characters converted to upper case Table 4.2: Some components of the Python string library 4.3.5 From Encoding to Encryption We have looked at how computers represent strings as a sort of encoding problem. Each character in a string is represented by a number that is stored in the computer in a binary representation. You should realize that there is nothing really secret about this code at all. In fact, we are simply using an industry-standard mapping of characters into numbers. Anyone with a little knowledge of computer science would be able to crack our code with very little effort. The process of encoding information for the purpose of keeping it secret or transmitting it privately is called encryption. The study of encryption methods is an increasingly important subﬁeld of mathematics and computer science known as cryptography. For example, if you shop over the Internet, it is important that your personal information such as social security number or credit card number is transmitted using encodings that keep it safe from potential eavesdroppers on the network. Our simple encoding/decoding programs use a very weak form of encryption known as a substitution cipher. Each character of the original message, called the plaintext, is replaced by a corresponding symbol (in our case a number) from a cipher alphabet. The resulting code is called the ciphertext . Slide 57: 4.4. OUTPUT AS STRING MANIPULATION 49 Even if our cipher were not based on the well-known ASCII encoding, it would still be easy to discover the original message. Since each letter is always encoded by the same symbol, a code-breaker could use statistical information about the frequency of various letters and some simple trial and error testing to discover the original message. Such simple encryption methods may be sufﬁcient for grade-school note passing, but they are certainly not up to the task of securing communication over global networks. Modern approaches to encryption start by translating a message into numbers, much like our encoding program. Then sophisticated mathematical algorithms are employed to transform these numbers into other numbers. Usually, the transformation is based on combining the message with some other special value called the key. In order to decrypt the message, the party on the receiving end needs to have the appropriate key so that the encoding can be reversed to recover the original message. Encryption approaches come in two ﬂavors: private key and public key. In a private key system the same key is used for encrypting and decrypting messages. All parties that wish to communicate need to know the key, but it must be kept secret from the outside world. This is usual system that people think of when considering secret codes. In public key systems, there are separate but related keys for encrypting and decrypting. Knowing the encryption key does not allow you to decrypt messages or discover the decryption key. In a public key system, the encryption key can be made publicly available, while the decryption key is kept private. Anyone can safely send a message using the public key for encryption. Only the party holding the decryption key will be able to decipher it. For example, a secure web site can send your web browser its public key, and the browser can use it to encode your credit card information before sending it on the Internet. Then only the company that is requesting the information will be able to decrypt and read it using the proper private key. 4.4 Output as String Manipulation Even programs that we may not view as primarily doing text-manipulation often need to make use of string operations. For example, a program to do a ﬁnancial analysis must produce a nicely formatted report of the results. Much of this report will be in the form of text that is used to label and explain numbers, charts, tables and ﬁgures. In this section, we’ll look at techniques for generating nicely formatted text-based output. 4.4.1 Converting Numbers to Strings In the ASCII decoding program, we used the eval function to convert from a string data type into a numeric data type. Recall that eval evaluates a string as a Python expression. It is very general and can be used to turn strings into nearly any other Python data type. It is also possible to go the other direction and turn many Python data types into strings using the str function. Here are a couple of simple examples. >>> str(500) ’500’ >>> value = 3.14 >>> str(value) ’3.14’ >>> print "The value is", str(value) + "." The value is 3.14. Notice particularly the last example. By turning value into a string, we can use string concatenation to put a period at the end of a sentence. If we didn’t ﬁrst turn value into a string, Python would interpret the + as a numerical operation and produce an error, because “.” is not a number. Adding eval and str to the type-conversion operations discussed in Chapter 3, we now have a complete set of operations for converting values among various Python data types. Table 4.3 summarizes these ﬁve Python type conversion functions. One common reason for converting a number into a string is so that string operations can be used to control the way the value is printed. For example, in the factorial program from last chapter we saw that Python long ints have the letter “L” appended to them. In versions of Python prior to 2.0, the “L” showed up Slide 58: 50 Function float(<expr>) int(<expr>) long(<expr> str(<expr>) eval(<string>) CHAPTER 4. COMPUTING WITH STRINGS Meaning Convert expr to a ﬂoating point value. Convert expr to an integer value. Convert expr to a long integer value. Return a string representation of expr. Evaluate string as an expression. Table 4.3: Type Conversion Functions whenever a long int was printed. However, it is easy to remove this artifact using some straightforward string manipulation. factStr = str(fact) print factStr[0:len(factStr)-1] Can you see how this code turns the long int into a string and then uses slicing to remove the “L?” The print statement prints every position in the string up to, but not including, the ﬁnal “L,” which is in position length - 1. As an aside, Python also allows sequences to be indexed from the back by using negative numbers as indexes. Thus -1 is the last position in a string, -2 is the second to last, etc. Using this convention, we can slice off the last character without ﬁrst computing the length of the string. print str(fact)[:-1] This version uses str to turn fact into a string and then immediately slices it “in place” from the beginning (0 is the default start) up to, but not including, the last position. 4.4.2 String Formatting As you have seen, basic string operations can be used to build nicely formatted output. This technique is useful for simple formatting, but building up a complex output through slicing and concatenation of smaller strings can be tedious. Python provides a powerful string formatting operation that makes the job much easier. Let’s start with a simple example. Here is a run of the change counting program from last chapter. Change Counter Please enter the count of each coin type. How many quarters do you have? 6 How many dimes do you have? 0 How many nickels do you have? 0 How many pennies do you have? 0 The total value of your change is 1.5 Notice that the ﬁnal value is given as a fraction with only one decimal place. This looks funny, since we expect the output to be something like $1.50. We can ﬁx this problem by changing the very last line of the program as follows. print "The total value of your change is $%0.2f" % (total) Now the program prints this message: The total value of your change is $1.50 Let’s try to make some sense of this. The percent sign % is Python’s string formatting operator. In general, the string formatting operator is used like this: <template-string> % (<values>) Slide 59: 4.4. OUTPUT AS STRING MANIPULATION 51 Percent signs inside the template-string mark “slots” into which the values are inserted. There must be exactly one slot for each value. Each of the slots is described by a format speciﬁer that tells Python how the value for that slot should appear. Returning to the example, the template contains a single speciﬁer at the end: %0.2f. The value of total will be inserted into the template in place of the speciﬁer. The speciﬁer also tells Python that total is a ﬂoating point number that should be rounded to two decimal places. To understand the formatting, we need to look at the structure of the speciﬁer. A formatting speciﬁer has this general form: %<width>.<precision><type-char> The speciﬁer starts with a % and ends with a character that indicates the data type of the value being inserted. We will use three different format types: decimal, float, and string. Decimal mode is used to display ints as base-10 numbers. (Python allows ints to be printed using a number of different bases; we will only use the normal decimal representation.) Float and string formats are obviously used for the corresponding data types. The width and precision portions of a speciﬁer are optional. If present, width tells how many spaces to use in displaying the value. If a value requires more room than is given in width, Python will just expand the width so that the value ﬁts. You can use a 0 width to indicate “use as much space as needed.” Precision is used with ﬂoating point values to indicate the desired number of digits after the decimal. The example speciﬁer %0.2f tells Python to insert a ﬂoating point value using as much space as necessary and rounding it to two decimal places. The easiest way to get the hang of formatting is just to play around with some examples in the interactive environment. >>> "Hello %s %s, you may have won $%d!" % ("Mr.", "Smith", 10000) ’Hello Mr. Smith, you may have already won $10000!’ >>> ’This int, %5d, was placed in a field of width 5’ % (7) ’This int, 7, was placed in a field of width 5’ >>> ’This int, %10d, was placed in a field of width 10’ % (7) ’This int, 7, was placed in a field of width 10’ >>> ’This float, %10.5f, has width 10 and precision 5.’ % (3.1415926) ’This float, 3.14159, has width 10 and precision 5.’ >>> ’This float, %0.5f, has width 0 and precision 5.’ % (3.1415926) ’This float, 3.14159, has width 0 and precision 5.’ >>> "Compare %f and %0.20f" % (3.14, 3.14) ’Compare 3.140000 and 3.14000000000000012434’ A couple points are worth noting here. If the width in the speciﬁer is larger than needed, the value is right-justiﬁed in the ﬁeld by default. You can left-justify the value in the ﬁeld by preceding the width with a minus sign (e.g., %-8.3f). The last example shows that if you print enough digits of a ﬂoating point number, you will almost always ﬁnd a “surprise.” The computer can’t represent 3.14 exactly as a ﬂoating point number. The closest value it can represent is ever so slightly larger than 3.14. If not given an explicit precision, Python will print the number out to a few decimal places. The slight extra extra amount shows up if you print lots of digits. Generally, Python only displays a closely rounded version of a ﬂoat. Using explicit formatting allows you to see the full result down to the last bit. 4.4.3 Better Change Counter Let’s close our formatting discussion with one more example program. Given what you have learned about ﬂoating point numbers, you might be a little uneasy about using them to represent money. Slide 60: 52 CHAPTER 4. COMPUTING WITH STRINGS Suppose you are writing a computer system for a bank. Your customers would not be too happy to learn that a check went through for an amount “very close to $107.56.” They want to know that the bank is keeping precise track of their money. Even though the amount of error in a given value is very small, the small errors can be compounded when doing lots of calculations, and the resulting error could add up to some real cash. That’s not a satisfactory way of doing business. A better approach would be to make sure that our program used exact values to represent money. We can do that by keeping track of the money in cents and using an int (or long int) to store it. We can then convert this into dollars and cents in the output step. If total represents the value in cents, then we can get the number of dollars by total / 100 and the cents from total % 100. Both of these are integer calculations and, hence, will give us exact results. Here is the updated program: # change2.py # A program to calculate the value of some change in dollars # This version represents the total cash in cents. def main(): print "Change Counter" print print "Please enter the count of each coin type." quarters = input("Quarters: ") dimes = input("Dimes: ") nickels = input("Nickels: ") pennies = input("Pennies: ") total = quarters * 25 + dimes * 10 + nickels * 5 + pennies print print "The total value of your change is $%d.%02d" \ % (total/100, total%100) main() I have split the ﬁnal print statement across two lines. Normally a statement ends at the end of the line. Sometimes it is nicer to break a long statement into smaller pieces. A backslash at the end of a line is one way to indicate that a statement is continued on the following line. Notice that the backslash must appear outside of the quotes; otherwise, it would be considered part of the string literal. The string formatting in the print statement contains two slots, one for dollars as an int and one for cents as an int. This example also illustrates one additional twist on format speciﬁers. The value of cents is printed with the speciﬁer %02d. The zero in front of the width tells Python to pad the ﬁeld (if necessary) with zeroes instead of spaces. This ensures that a value like 10 dollars and 5 cents prints as $10.05 rather than $10. 5. 4.5 File Processing I began the chapter with a reference to word-processing as an application of the string data type. One critical feature of any word processing program is the ability to store and retrieve documents as ﬁles on disk. In this section, we’ll take a look at ﬁle input and output, which, as it turns out, is really just another form of string processing. 4.5.1 Multi-Line Strings Conceptually, a ﬁle is a sequence of data that is stored in secondary memory (usually on a disk drive). Files can contain any data type, but the easiest ﬁles to work with are those that contain text. Files of text have the advantage that they can be read and understood by humans, and they are easily created and edited using general-purpose text editors and word processors. In Python, text ﬁles can be very ﬂexible, since it is easy to convert back and forth between strings and other types. Slide 61: 4.5. FILE PROCESSING 53 You can think of a text ﬁle as a (possibly long) string that happens to be stored on disk. Of course, a typical ﬁle generally contains more than a single line of text. A special character or sequence of characters is used to mark the end of each line. There are numerous conventions for end-of-line markers. Python uses a single character called newline as a marker. You can think of newline as the character produced when you press the Enter key on your keyboard. Although a newline is a single character, it is represented in Python (and many other computer languages) using the special notation ’ n’. Other special characters are represented using a similar notation (i.e., ’ t’ for Tab ). Let’s take a look at a concrete example. Suppose you type the following lines into a text editor exactly as shown here. Hello World Goodbye 32 When stored to a ﬁle, you get this sequence of characters. Hello\nWorld\n\nGoodbye 32\n Notice that the blank line becomes a bare newline in the resulting ﬁle/string. By the way, by embedding newline characters into output strings, you can produce multiple lines of output with a single print statement. Here is the example from above printed interactively. >>> print "Hello\nWorld\n\nGoodbye 32\n" Hello World Goodbye 32 >>> If you simply ask Python to evaluate a string containing newline characters, you will just get the embedded newline representation back again. >>>"Hello\nWorld\n\nGoodbye 32\n" ’Hello\nWorld\n\nGoodbye 32\n’ It’s only when a string is printed that the special characters affect how the string is displayed. 4.5.2 File Processing The exact details of ﬁle-processing differ substantially among programming languages, but virtually all languages share certain underlying ﬁle manipulation concepts. First, we need some way to associate a ﬁle on disk with a variable in a program. This process is called opening a ﬁle. Once a ﬁle has been opened, it is manipulated through the variable we assign to it. Second, we need a set of operations that can manipulate the ﬁle variable. At the very least, this includes operations that allow us to read the information from a ﬁle and write new information to a ﬁle. Typically, the reading and writing operations for text ﬁles are similar to the operations for text-based, interactive input and output. Finally, when we are ﬁnished with a ﬁle, it is closed. Closing a ﬁle makes sure that any bookkeeping that was necessary to maintain the correspondence between the ﬁle on disk and the ﬁle variable is ﬁnished up. For example, if you write information to a ﬁle variable, the changes might not show up on the disk version until the ﬁle has been closed. This idea of opening and closing ﬁles is closely related to how you might work with ﬁles in an application program like a word processor. However, the concepts are not exactly the same. When you open a ﬁle in a program like Microsoft Word, the ﬁle is actually read from the disk and stored into RAM. In programming Slide 62: 54 CHAPTER 4. COMPUTING WITH STRINGS terminology, the ﬁle is opened for reading and the the contents of the ﬁle are then read into memory via ﬁle reading operations. At this point, the ﬁle is closed (again in the programming sense). As you “edit the ﬁle,” you are really making changes to data in memory, not the ﬁle itself. The changes will not show up in the ﬁle on the disk until you tell the application to “save” it. Saving a ﬁle also involves a multi-step process. First, the original ﬁle on the disk is reopened, this time in a mode that allows it to store information—the ﬁle on disk is opened for writing. Doing so actually erases the old contents of the ﬁle. File writing operations are then used to copy the current contents of the in-memory version into the new ﬁle on the disk. From your perspective, it appears that you have edited an existing ﬁle. From the program’s perspective, you have actually opened a ﬁle, read its contents into memory, closed the ﬁle, created a new ﬁle (having the same name), written the (modiﬁed) contents of memory into the new ﬁle, and closed the new ﬁle. Working with text ﬁles is easy in Python. The ﬁrst step is to associate a ﬁle with a variable using the open function. <filevar> = open(<name>, <mode>) Here name is a string that provides the name of the ﬁle on the disk. The mode parameter is either the string "r" or "w" depending on whether we intend to read from the ﬁle or write to the ﬁle. For example, to open a ﬁle on our disk called “numbers.dat” for reading, we could use a statement like the following. infile = open("numbers.dat", "r") Now we can use the variable infile to read the contents of numbers.dat from the disk. Python provides three related operations for reading information from a ﬁle: <filevar>.read() <filevar>.readline() <filevar>.readlines() The read operation returns the entire contents of the ﬁle as a single string. If the ﬁle contains more than one line of text, the resulting string has embedded newline characters between the lines. Here’s an example program that prints the contents of a ﬁle to the screen. # printfile.py # Prints a file to the screen. def main(): fname = raw_input("Enter filename: ") infile = open(fname,’r’) data = infile.read() print data main() The program ﬁrst prompts the user for a ﬁlename and then opens the ﬁle for reading through the variable infile. You could use any name for the variable, I used infile to emphasize that the ﬁle was being used for input. The entire contents of the ﬁle is then read as one large string and stored in the variable data. Printing data causes the contents to be displayed. The readline operation can be used to read one line from a ﬁle; that is, it reads all the characters up through the next newline character. Each time it is called, readline returns the next line from the ﬁle. This is analogous to raw input which reads characters interactively until the user hits the Enter key; each call to raw input get another line from the user. One thing to keep in mind, however, is that the string returned by readline will always end with a newline character, whereas raw input discards the newline character. As a quick example, this fragment of code prints out the ﬁrst ﬁve lines of a ﬁle. Slide 63: 4.5. FILE PROCESSING infile = open(someFile, ’r’) for i in range(5): line = infile.readline() print line[:-1] 55 Notice the use of slicing to strip off the newline character at the end of the line. Since print automatically jumps to the next line (i.e., it outputs a newline), printing with the explicit newline at the end would put an extra blank line of output between the lines of the ﬁle. As an alternative to readline, when you want to loop through all the (remaining) lines of a ﬁle, you can use readlines. This operation returns a sequence of strings representing the lines of the ﬁle. Used with a for loop, it is a particularly handy way to process each line of a ﬁle. infile = open(someFile, ’r’) for line in infile.readlines(): # process the line here infile.close() Opening a ﬁle for writing prepares that ﬁle to receive data. If no ﬁle with the given name exists, a new ﬁle will be created. A word of warning: if a ﬁle with the given name does exist, Python will delete it and create a new, empty ﬁle. When writing to a ﬁle, make sure you do not clobber any ﬁles you will need later! Here is an example of opening a ﬁle for output. outfile = open("mydata.out", "w") We can put data into a ﬁle, using the write operation. <file-var>.write(<string>) This is similar to print, except that write is a little less ﬂexible. The write operation takes a single parameter, which must be a string, and writes that string to the ﬁle. If you want to start a new line in the ﬁle, you must explicitly provide the newline character. Here’s a silly example that writes two lines to a ﬁle. outfile = open("example.out", ’w’) count = 1 outfile.write("This is the first line\n") count = count + 1 outfile.write("This is line number %d" % (count)) outfile.close() Notice the use of string formatting to write out the value of the variable count. If you want to output something that is not a string, you must ﬁrst convert it; the string formatting operator is often a handy way to do this. This code will produce a ﬁle on disk called “example.out” containing the following two lines: This is the first line This is line number 2 If “example.out” existed before executing this fragment, it’s old contents were destroyed. 4.5.3 Example Program: Batch Usernames To see how all these pieces ﬁt together, let’s redo the username generation program. Our previous version created usernames interactively by having the user type in his or her name. If we were setting up accounts for a large number of users, the process would probably not be done interactively, but in batch mode. In batch processing, program input and output is done through ﬁles. Our new program is designed to process a ﬁle of names. Each line of the input ﬁle will contain the ﬁrst and last names of a new user separated by one or more spaces. The program produces an output ﬁle containing a line for each generated username. Slide 64: 56 CHAPTER 4. COMPUTING WITH STRINGS # userfile.py # Program to create a file of usernames in batch mode. import string def main(): print "This program creates a file of usernames from a" print "file of names." # get the file names infileName = raw_input("What file are the names in? ") outfileName = raw_input("What file should the usernames go in? ") # open the files infile = open(infileName, ’r’) outfile = open(outfileName, ’w’) # process each line of the input file for line in infile.readlines(): # get the first and last names from line first, last = string.split(line) # create the username uname = string.lower(first[0]+last[:7]) # write it to the output file outfile.write(uname+’\n’) # close both files infile.close() outfile.close() print "Usernames have been written to", outfileName main() There are a few things worth noticing in this program. I have two ﬁles open at the same time, one for input (infile) and one for output (outfile). It’s not unusual for a program to operate on several ﬁles simultaneously. Also, when creating the username, I used the lower function from the string library. This ensures that the username is all lower case, even if the input names are mixed case. Finally, you should also notice the line that writes the username to the ﬁle. outfile.write(uname+’\n’) Concatenating the newline character is necessary to indicate the end of line. Without this, all of the usernames would run together in one giant line. 4.5.4 Coming Attraction: Objects Have you noticed anything strange about the syntax of the ﬁle processing examples? To apply an operation to a ﬁle, we use dot notation. For example, to read from infile we type infile.read(). This is different from the normal function application that we have used before. Afterall, to take the absolute value of a variable x, we type abs(x), not x.abs(). In Python, a ﬁle is an example of an object. Objects combine both data and operations together. An object’s operations, called methods, are invoked using the dot notation. That’s why the syntax looks a bit different. Slide 65: 4.6. EXERCISES 57 For completeness, I should mention that strings are also objects in Python. If you have a relatively new version of Python (2.0 or later), you can use string methods in place of the string library functions that we discussed earlier. For example, myString.split() is equivalent to string.split(myString) If this object stuff sounds confusing right now, don’t worry; Chapter 5 is all about the power of objects (and pretty pictures, too). 4.6 Exercises 1. Given the initial statements: import string s1 = "spam" s2 = "ni!" Show the result of evaluating each of the following string expressions. (a) "The Knights who say, " + s2 (b) 3 * s1 + 2 * s2 (c) s1[1] (d) s1[1:3] (e) s1[2] + s2[:2] (f) s1 + s2[-1] (g) string.upper(s1) (h) string.ljust(string.upper(s2),4) * 3 2. Given the same initial statements as in the previous problem, show a Python expression that could construct each of the following results by performing string operations on s1 and s2. (a) "NI" (b) "ni!spamni!" (c) "Spam Ni! (d) "span" (e) ["sp","m"] (f) "spm" 3. Show the output that would be generated by each of the following program fragments. (a) for ch in "aardvark": print ch (b) for w in string.split("Now is the winter of our discontent..."): print w (c) for w in string.split("Mississippi", "i"): print w, Spam Ni! Spam Ni!" Slide 66: 58 CHAPTER 4. COMPUTING WITH STRINGS (d) msg = "" for s in string.split("secret","e"): msg = msg + s print msg (e) msg = "" for ch in "secret": msg = msg + chr(ord(ch)+1) print msg 4. Show the string that would result from each of the following string formatting operations. If the operation is not legal, explain why. (a) "Looks like %s and %s for breakfast" % ("spam", "eggs") (b) "There is %d %s %d %s" % (1,"spam", 4, "you") (c) "Hello %s" % ("Suzie", "Programmer") (d) "%0.2f %0.2f" % (2.3, 2.3468) (e) "%7.5f %7.5f" % (2.3, 2.3468) (f) "Time left %02d:%05.2f" % (1, 37.374) (g) "%3d" % ("14") 5. Explain why public key encryption is more useful for securing communications on the Internet than private (shared) key encryption. 6. A certain CS professor gives 5-point quizzes that are graded on the scale 5-A, 4-B, 3-C, 2-D, 1-F, 0-F. Write a program that accepts a quiz score as an input and prints out the corresponding grade. 7. A certain CS professor gives 100-point exams that are graded on the scale 90–100:A, 80–89:B, 70– 79:C, 60–69:D, 60:F. Write a program that accepts an exam score as input and prints out the corresponding grade. 8. An acronym is a word formed by taking the ﬁrst letters of the words in a phrase and making a word from them. For example, RAM is an acronym for “random access memory.” Write a program that allows the user to type in a phrase and outputs the acronym for that phrase. Note: the acronym should be all uppercase, even if the words in the phrase are not capitalized. 9. Numerologists claim to be able to determine a person’s character traits based on the “numeric value” of a name. The value of a name is determined by summing up the values of the letters of the name where ’a’ is 1, ’b’ is 2, ’c’ is 3 etc., up to ’z’ being 26. For example, the name “Zelle” would have the value 26 5 12 12 5 60 (which happens to be a very auspicious number, by the way). Write a program that calculates the numeric value of a single name provided as input. 10. Expand your solution to the previous problem to allow the calculation of a complete name such as “John Marvin Zelle” or “John Jacob Jingleheimer Smith.” The total value is just the sum of the numeric value for each name. 11. A Caesar cipher is a simple substitution cipher based on the idea of shifting each letter of the plaintext message a ﬁxed number (called the key) of positions in the alphabet. For example, if the key value is 2, the word “Sourpuss” would be encoded as “Uqwtrwuu.” The original message can be recovered by “reencoding” it using the negative of the key. Write a program that can encode and decode Caesar ciphers. The input to the program will be a string of plaintext and the value of the key. The output will be an encoded message where each character in the original message is replaced by shifting it key characters in the ASCII character set. For example, if ch is a character in the string and key is the amount to shift, then the character that replaces ch can be calculated as: chr(ord(ch) + key). ¢ ¢ ¢ ¢ Slide 67: 4.6. EXERCISES 59 12. One problem with the previous exercise is that it does not deal with the case when we “drop off the end” of the alphabet (or ASCII encodings). A true Caesar cipher does the shifting in a circular fashion where the next character after “z” is “a”. Modify your solution to the previous problem to make it circular. You may assume that the input consists only of letters and spaces. 13. Write a program that counts the number of words in a sentence entered by the user. 14. Write a program that calculates the average word length in a sentence entered by the user. 15. Write an improved version of the Chaos program from Chapter 1 that allows a user to input two initial values and the number of iterations and then prints a nicely formatted table showing how the values change over time. For example, if the starting values were .25 and .26 with 10 iterations, the table might look like this: index 0.25 0.26 ---------------------------1 0.731250 0.750360 2 0.766441 0.730547 3 0.698135 0.767707 4 0.821896 0.695499 5 0.570894 0.825942 6 0.955399 0.560671 7 0.166187 0.960644 8 0.540418 0.147447 9 0.968629 0.490255 10 0.118509 0.974630 16. Write an improved version of the future value program from Chapter 2. Your program will prompt the user for the amount of the investment, the annualized interest rate, and the number of years of the investment. The program will then output a nicely formatted table that tracks the value of the investment year by year. Your output might look something like this: Years Value ---------------0 $2000.00 1 $2200.00 2 $2420.00 3 $2662.00 4 $2928.20 5 $3221.02 6 $3542.12 7 $3897.43 17. Redo any of the previous programming problems to make them batch oriented (using text ﬁles for input and output). 18. Word count. A common utility on Unix/Linux systems is a small program called “wc.” This program analyzes a ﬁle to determine the number of lines, words, and characters contained therein. Write your own version of wc. The program should accept a ﬁle name as input and then print three numbers showing the count of lines, words, and characters in the ﬁle. Slide 68: 60 CHAPTER 4. COMPUTING WITH STRINGS Slide 69: Chapter 5 Objects and Graphics So far we have been writing programs that use the built-in Python data types for numbers and strings. We saw that each data type could represent a certain set of values, and each had a set of associated operations. Basically, we viewed the data as passive entities that were manipulated and combined via active operations. This is a traditional way to view computation. To build complex systems, however, it helps to take a richer view of the relationship between data and operations. Most modern computer programs are built using an object-oriented (OO) approach. Object orientation is not easily deﬁned. It encompasses a number of principles for designing and implementing software, principles that we will return to numerous times throughout course of this book. This chapter provides a basic introduction to object concepts by way of some computer graphics. 5.1 The Object of Objects The basic idea of object-oriented development is to view a complex system as the interaction of simpler objects. The word objects is being used here in a speciﬁc technical sense. Part of the challenge of OO programming is ﬁguring out the vocabulary. You can think of an OO object as a sort of active data type that combines both data and operations. To put it simply, objects know stuff (they contain data), and they can do stuff (they have operations). Objects interact by sending each other messages. A message is simply a request for an object to perform one of its operations. Consider a simple example. Suppose we want to develop a data processing system for a college or university. We will need to keep track of considerable information. For starters, we must keep records on the students who attend the school. Each student could be represented in the program as an object. A student object would contain certain data such as name, ID number, courses taken, campus address, home address, GPA, etc. Each student object would also be able to respond to certain requests. For example, to send out a mailing, we would need to print an address for each student. This task might be handled by a printCampusAddress operation. When a particular student object is sent the printCampusAddress message, it prints out its own address. To print out all the addresses, a program would loop through the collection of student objects and send each one in turn the printCampusAddress message. Objects may refer to other objects. In our example, each course in the college might also be represented by an object. Course objects would know things such as who the instructor is, what students are in the course, what the prerequisites are, and when and where the course meets. One example operation might be addStudent, which causes a student to be enrolled in the course. The student being enrolled would be represented by the appropriate student object. Instructors would be another kind of object, as well as rooms, and even times. You can see how successive reﬁnement of these ideas could lead to a rather sophisticated model of the information structure of the college. As a beginning programmer, you’re probably not yet ready to tackle a college information system. For now, we’ll study objects in the context of some simple graphics programming. 61 Slide 70: 62 CHAPTER 5. OBJECTS AND GRAPHICS 5.2 Graphics Programming Modern computer applications are not limited to the sort of textual input and output that we have been using so far. Most of the applications that you are familiar with probably have a so-called Graphical User Interface (GUI) that provides visual elements like windows, icons (representative pictures), buttons and menus. Interactive graphics programming can be very complicated; entire textbooks are devoted to the intricacies of graphics and graphical interfaces. Industrial-strength GUI applications are usually developed using a dedicated graphics programming framework. Python comes with its own standard GUI module called Tkinter. As GUI frameworks go, Tkinter is one of the simplest to use, and Python is great language for developing real-world GUIs. Even so, taking the time to learn Tkinter would detract from the more fundamental task of learning the principles of programming and design that are the focus of this book. To make learning easier, I have written a graphics library (graphics.py) for use with this book. This library is freely available as a Python module ﬁle1 and you are welcome to use it as you see ﬁt. Using the library is as easy as placing a copy of the graphics.py ﬁle in the same folder as your graphics program(s). Alternatively, you can put graphics.py in the system directory where other Python libraries are stored so that it can be used from any folder on the system. The graphics library makes it easy to write simple graphics programs. As you do, you will be learning principles of object-oriented programming and computer graphics that can be applied in more sophisticated graphical programming environments. The details of the graphics module will be explored in later sections. Here we’ll concentrate on a basic hands-on introduction to whet your appetite. As usual, the best way to start learning new concepts is to roll up your sleeves and try out some examples. The ﬁrst step is to import the graphics module. Assuming you have placed graphics.py in an appropriate place, you can import the graphics commands into an interactive Python session. >>> import graphics Next we need to create a place on the screen where the graphics will appear. That place is a graphics window or GraphWin, which is provided by the graphics module. >>> win = graphics.GraphWin() This command creates a new window on the screen. The window will have the title “Graphics Window.” The GraphWin may overlap your Python interpreter window, so you might have to resize the Python window to make both fully visible. Figure 5.1 shows an example screen view. The GraphWin is an object, and we have assigned it to the the variable called win. We can manipulate the window object through this variable, similar to the way that ﬁle objects are manipulated through ﬁle variables. For example, when we are ﬁnished with a window, we can destroy it. This is done by issuing the close command. >>> win.close() Typing this command causes the window to vanish from the screen. We will be working with quite a few commands from the graphics library, and it gets tedious having to type the graphics. notation every time we use one. Python has an alternative form of import that can help out. from graphics import * The from statement allows you to load speciﬁc deﬁnitions from a library module. You can either list the names of deﬁnitions to be imported or use an asterisk, as shown, to import everything deﬁned in the module. The imported commands become directly available without having to preface them with the module name. After doing this import, we can create a GraphWin more simply. win = GraphWin() 1 See Appendix A for information on how to obtain the graphics library and other supporting materials for this book. Slide 71: 5.2. GRAPHICS PROGRAMMING 63 Figure 5.1: Screen shot with a Python window and a GraphWin All of the rest of the graphics examples will assume that the entire graphics module has been imported using from. Let’s try our hand at some drawing. A graphics window is actually a collection of tiny points called pixels (short for picture elements). By controlling the color of each pixel, we control what is displayed in the window. By default, a GraphWin is 200 pixels tall and 200 pixels wide. That means there are 40,000 pixels in the GraphWin. Drawing a picture by assigning a color to each individual pixel would be a daunting challenge. Instead, we will rely on a library of graphical objects. Each type of object does its own bookkeeping and knows how to draw itself into a GraphWin. The simplest object in the graphics module is a Point. In geometry, a point is a dimensionless location in space. A point is located by reference to a coordinate system. Our graphics object Point is similar; it can represent a location in a GraphWin. We deﬁne a point by supplying x and y coordinates x¤ y¡ . The x value represents the horizontal location of the point, and the y value represents the vertical. Traditionally, graphics programmers locate the point 0¤ 0¡ in the upper-left corner of the window. Thus x values increase from left to right, and y values increase from top to bottom. In the default 200 x 200 GraphWin, the lower-right corner has the coordinates 199¤ 199¡ . Drawing a Point sets the color of the corresponding pixel in the GraphWin. The default color for drawing is black. Here is a sample interaction with Python illustrating the use of Points. >>> >>> 50 >>> 60 >>> >>> >>> >>> p = Point(50,60) p.getX() p.getY() win = GraphWin() p.draw(win) p2 = Point(140,100) p2.draw(win) The ﬁrst line creates a Point located at 50¤ 60¡ . After the Point has been created, its coordinate values can be accessed by the operations getX and getY. A Point is drawn into a window using the draw operation. In this example, two different point objects (p and p2) are created and drawn into the GraphWin called win. Figure 5.2 shows the resulting graphical output. In addition to points, the graphics library contains commands for drawing lines, circles, rectangles, ovals, Slide 72: 64 CHAPTER 5. OBJECTS AND GRAPHICS Figure 5.2: Graphics window with two points drawn. polygons and text. Each of these objects is created and drawn in a similar fashion. Here is a sample interaction to draw various shapes into a GraphWin. >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> #### Open a graphics window win = GraphWin(’Shapes’) #### Draw a red circle centered at point (100,100) with radius 30 center = Point(100,100) circ = Circle(center, 30) circ.setFill(’red’) circ.draw(win) #### Put a textual label in the center of the circle label = Text(center, "Red Circle") label.draw(win) #### Draw a square using a Rectangle object rect = Rectangle(Point(30,30), Point(70,70)) rect.draw(win) #### Draw a line segment using a Line object line = Line(Point(20,30), Point(180, 165)) line.draw(win) #### Draw an oval using the Oval object oval = Oval(Point(20,150), Point(180,199)) oval.draw(win) Try to ﬁgure out what each of these statements does. If you type them in as shown, the ﬁnal result will look like Figure 5.3. 5.3 Using Graphical Objects Some of the examples in the above interactions may look a bit strange to you. To really understand the graphics module, we need to take an object-oriented point of view. Recall, objects combine data with operations. Computation is performed by asking an object to carry out one of its operations. In order to make use of objects, you need to know how to create them and how to request operations. In the interactive examples above, we manipulated several different kinds of objects: GraphWin, Point, Circle, Oval, Line, Text, and Rectangle. These are examples of classes. Every object is an instance of some class, and the class describes the properties the instance will have. Borrowing a biological metaphor, when we say that Fido is a dog, we are actually saying that Fido is a speciﬁc individual in the larger class of all dogs. In OO terminology, Fido is an instance of the dog class. Slide 73: 5.3. USING GRAPHICAL OBJECTS 65 Figure 5.3: Various shapes from the graphics module. Because Fido is an instance of this class, we expect certain things. Fido has four legs, a tail, a cold, wet nose and he barks. If Rex is a dog, we expect that he will have similar properties, even though Fido and Rex may differ in speciﬁc details such as size or color. The same ideas hold for our computational objects. We can create two separate instances of Point, say p and p2. Each of these points has an x and y value, and they both support the same set of operations like getX and draw. These properties hold because the objects are Points. However, different instances can vary in speciﬁc details such as the values of their coordinates. To create a new instance of a class, we use a special operation called a constructor. A call to a constructor is an expression that creates a brand new object. The general form is as follows. <class-name>(<param1>, <param2>, ...) Here <class-name> is the name of the class that we want to create a new instance of, e.g., Circle or Point. The expressions in the parentheses are any parameters that are required to initialize the object. The number and type of the parameters depends on the class. A Point requires two numeric values, while a GraphWin can be constructed without any parameters. Typically, a constructor is used on the right side of an assignment statement, and the resulting object is immediately assigned to a variable on the left side that is then used to manipulate the object. To take a concrete example, let’s look at what happens when we create a graphical point. Here is a constructor statement from the interactive example above. p = Point(50,60) The constructor for the Point class requires two parameters giving the x and y coordinates for the new point. These values are stored as instance variables inside of the object. In this case, Python creates an instance of Point having an x value of 50 and a y value of 60. The resulting point is then assigned to the variable p. A conceptual diagram of the result is shown in Figure 5.4. Note that, in this diagram as well as similar ones later on, only the most salient details are shown. Points also contain other information such as their color and which window (if any) they are drawn in. Most of this information is set to default values when the Point is created. To perform an operation on an object, we send the object a message. The set of messages that an object responds to are called the methods of the object. You can think of methods as functions that live inside of the object. A method is invoked using dot-notation. <object>.<method-name>(<param1>, <param2>, ...) The number and type of the parameters is determined by the method being used. Some methods require no parameters at all. You can ﬁnd numerous examples of method invocation in the interactive examples above. As examples of parameterless methods, consider these two expressions. Slide 74: 66 p: x: y: CHAPTER 5. OBJECTS AND GRAPHICS Point 50 60 Figure 5.4: Conceptual picture of the result of p = Point(50,60). The variable p refers to a freshly created Point having the given coordinates. p.getX() p.getY() The getX and getY methods return the x and y values of a point, respectively. Methods such as these are sometimes called accessors, because they allow us to access information from the instance variables of the object. Other methods change the values of an object’s instance variables, hence changing the state of the object. All of the graphical objects have a move method. Here is a speciﬁcation: move(dx, dy): Moves the object dx units in the x direction and dy units in the y direction. To move the point p to the right 10 units, we could use this statement. p.move(10,0) This changes the x instance variable of p by adding 10 units. If the point is currently drawn in a GraphWin, move will also take care of erasing the old image and drawing it in its new position. Methods that change the state of an object are sometimes called mutators. The move method must be supplied with two simple numeric parameters indicating the distance to move the object along each dimension. Some methods require parameters that are themselves complex objects. For example, drawing a Circle into a GraphWin involves two objects. Let’s examine a sequence of commands that does this. circ = Circle(Point(100,100), 30) win = GraphWin() circ.draw(win) The ﬁrst line creates a Circle with a center located at the Point 100¤ 100¡ and a radius of 30. Notice that we used the Point constructor to create a location for the ﬁrst parameter to the Circle constructor. The second line creates a GraphWin. Do you see what is happening in the third line? This is a request for the Circle object circ to draw itself into the GraphWin object win. The visible effect of this statement is a circle in the GraphWin centered at 100¤ 100¡ and having a radius of 30. Behind the scenes, a lot more is happening. Remember, the draw method lives inside the circ object. Using information about the center and radius of the circle from the instance variables, the draw method issues an appropriate sequence of lowlevel drawing commands (a sequence of method invocations) to the GraphWin. A conceptual picture of the interactions among the Point, Circle and GraphWin objects is shown in Figure 5.5. Fortunately, we don’t usually have to worry about these kinds of details; they’re all taken care of by the graphical objects. We just create objects, call the appropriate methods, and let them do the work. That’s the power of object-oriented programming. There is one subtle “gotcha” that you need to keep in mind when using objects. It is possible for two different variables to refer to exactly the same object; changes made to the object through one variable will also be visible to the other. Suppose we are trying to write a sequence of code that draws a smiley face. We want to create two eyes that are 20 units apart. Here is a sequence of code intended to draw the eyes. Slide 75: 5.3. USING GRAPHICAL OBJECTS circ: Circle center: radius: draw( . . . ) 30 x: y: Point 67 100 100 Low-level drawing commands win: . . . GraphWin Figure 5.5: Object interactions to draw a circle. ## Incorrect way to create two circles. leftEye = Circle(Point(80, 50), 5) leftEye.setFill(’yellow’) leftEye.setOutline(’red’) rightEye = leftEye rightEye.move(20,0) The basic idea is to create the left eye and then copy that into a right eye which is then moved over 20 units. This doesn’t work. The problem here is that only one Circle object is created. The assignment rightEye = leftEye simply makes rightEye refer to the very same circle as leftEye. Figure 5.6 shows the situation. When the Circle is moved in the last line of code, both rightEye and leftEye refer to it in its new location on the right side. This situation where two variables refer to the same object is called aliasing, and it can sometimes produce rather unexpected results. leftEye: Circle center: radius: 10 x: y: rightEye: Point 80 50 Figure 5.6: Variables leftEye and rightEye are aliases. One solution to this problem would be to create a separate circle for each eye. ## A correct way to create two circles. leftEye = Circle(Point(80, 50), 5) leftEye.setFill(’yellow’) Slide 76: 68 leftEye.setOutline(’red’) rightEye = Circle(Point(100, 50), 5) rightEye.setFill(’yellow’) rightEye.setOutline(’red’) CHAPTER 5. OBJECTS AND GRAPHICS This will certainly work, but it’s cumbersome. We had to write duplicated code for the two eyes. That’s easy to do using a “cut and paste” approach, but it’s not very elegant. If we decide to change the appearance of the eyes, we will have to be sure to make the changes in two places. The graphics library provides a better solution; all graphical objects support a clone method that makes a copy of the object. Using clone, we can rescue the original approach. ## Correct way to create two circles, using clone. leftEye = Circle(Point(80, 50), 5) leftEye.setFill(’yellow’) leftEye.setOutline(’red’) rightEye = leftEye.clone() # rightEye is an exact copy of the left rightEye.move(20,0) Strategic use of cloning can make some graphics tasks much easier. 5.4 Graphing Future Value Now that you have some idea of how to use objects from the graphics module, we’re ready to try some real graphics programming. One of the most important uses of graphics is providing a visual representation of data. They say a picture is worth a thousand words; it is almost certainly better than a thousand numbers. Few programs that manipulate numeric data couldn’t be improved with a bit of graphical output. Remember the program in Chapter 2 that computed the future value of a ten year investment? Let’s try our hand at creating a graphical summary. Programming with graphics requires careful planning. You’ll probably want pencil and paper handy to draw some diagrams and scratch out calculations as we go along. As usual, we begin by considering the speciﬁcation of exactly what the program will do. The original program futval.py had two inputs, the amount of money to be invested and the annualized rate of interest. Using these inputs, the program calculated the change in principal year by year for ten years using the formula principal principal 1 apr¡ . It then printed out the ﬁnal value of the principal. In the graphical version, the output will be a ten-year bar graph where the height of successive bars represents the value of the principal in successive years. Let’s use a concrete example for illustration. Suppose we invest $2000 at 10% interest. This table shows the growth of the investment over a ten-year period: Years 0 1 2 3 4 5 6 7 8 9 10 ¢ Value $2,000.00 $2,200.00 $2,420.00 $2,662.00 $2,928.20 $3,221.02 $3,542.12 $3,897.43 $4,287.18 $4,715.90 $5,187.49 Our program will display this information in a bar graph. Figure 5.7 shows the data in graphical form. The graph actually contains eleven bars. The ﬁrst bar shows the original value of the principal. For reference, let’s number these bars according the the number of years of interest accrued, 0–10. Here is a rough design for the program. Slide 77: 5.4. GRAPHING FUTURE VALUE 69 Figure 5.7: Bar graph showing growth of $2,000 at 10% interest Print an introduction Get value of principal and apr from user Create a GraphWin Draw scale labels on left side of window Draw bar at position 0 with height corresponding to principal For successive years 1 through 10 Calculate principal = principal * (1 + apr) Draw a bar for this year having a height corresponding to principal Wait for user to press Enter The pause created by the last step is necessary to keep the graphics window displayed so that we can interpret the results. Without such a pause, the program would end and the GraphWin would vanish with it. While this design gives us the broad brush strokes for our algorithm, there are some very important details that have been glossed over. We must decide exactly how big the graphics window will be and how we will position the objects that appear in this window. For example, what does it mean to draw, say, a bar for year ﬁve with height corresponding to $3,221.02? Let’s start with the size of the GraphWin. Recall that the size of a window is given in terms of the number of pixels in each dimension. Computer screens are also measured in terms of pixels. The number of pixels or resolution of the screen is determined by the monitor and graphics card in the computer you use. The lowest resolution screen you are likely to encounter these days is a so-called standard VGA screen that is 640 x 480 pixels. Most screens are considerably larger. Let’s make the GraphWin one quarter the size of a 640 x 480 screen, or 320 x 240. That should allow all users to see the graphical output as well as the textual output from our program. Given this analysis, we can ﬂesh out a bit of our design. The third line of the design should now read: Create a 320 x 240 GraphWin titled ‘‘Investment Growth Chart’’ You may be wondering how this will translate into Python code. You have already seen that the GraphWin constructor allows an optional parameter to specify the title of the window. You may also supply width and height parameters to control the size of the window. Thus, the command to create the output window will be: win = GraphWin("Investment Growth Chart", 320, 240) Next we turn to the problem of printing labels along the left edge of our window. To simplify the problem, we will assume the graph is always scaled to a maximum of $10,000 with the ﬁve labels “0.0K” to “10.0K” as shown in the example window. The question is how should the labels be drawn? We will need some Text Slide 78: 70 CHAPTER 5. OBJECTS AND GRAPHICS objects. When creating Text, we specify the anchor point (the point the text is centered on) and the string to use as the label. The label strings are easy. Our longest label is ﬁve characters, and the labels should all line up on the right side of a column, so the shorter strings will be padded on the left with spaces. The placement of the labels is chosen with a bit of calculation and some trial and error. Playing with some interactive examples, it seems that a string of length ﬁve looks nicely positioned in the horizontal direction placing the center 20 pixels in from the left edge. This leaves just a bit of whitespace at the margin. In the vertical direction, we have just over 200 pixels to work with. A simple scaling would be to have 100 pixels represent $5,000. That means our ﬁve labels should be spaced 50 pixels apart. Using 200 pixels for the range 0–10,000 leaves 240 200 40 pixels to split between the top and bottom margins. We might want to leave a little more margin at the top to accommodate values that grow beyond $10,000. A little experimentation suggests that putting the “ 0.0K” label 10 pixels from the bottom (position 230) seems to look nice. Elaborating our algorithm to include these details, the single step Draw scale labels on left side of window becomes a sequence of steps Draw Draw Draw Draw Draw label label label label label " 0.0K" " 2.5K" " 5.0K" " 7.5K" "10.0K" at at at at at (20, (20, (20, (20, (20, 230) 180) 130) 80) 30) ¢ The next step in the original design calls for drawing the bar that corresponds to the initial amount of the principal. It is easy to see where the lower left corner of this bar should be. The value of $0.0 is located vertically at pixel 230, and the labels are centered 20 pixels in from the left edge. Adding another 20 pixels gets us to the right edge of the labels. Thus the lower left corner of the 0th bar should be at location 40¤ 230¡ . Now we just need to ﬁgure out where the opposite (upper right) corner of the bar should be so that we can draw an appropriate rectangle. In the vertical direction, the height of the bar is determined by the value of principal. In drawing the scale, we determined that 100 pixels is equal to $5,000. This means that we have 100 5000 0 02 pixels to the dollar. This tells us, for example, that a principal of $2,000 should 40 pixels. In general, the y position of the upper-right corner will be produce a bar of height 2000 02¡ given by 230 principal¡ 0 02¡ . (Remember that 230 is the 0 point, and the y coordinates decrease going up). How wide should the bar be? The window is 320 pixels wide, but 40 pixels are eaten up by the labels on the left. That leaves us with 280 pixels for 11 bars 280 11 25 4545. Let’s just make each bar 25 pixels; that will give us a bit of margin on the right side. So, the right edge of our ﬁrst bar will be at position 40 25 65. We can now ﬁll the details for drawing the ﬁrst bar into our algorithm. Draw a rectangle from (40, 230) to (65, 230 - principal * 0.02) ¢ ¡ ¢ ¡ At this point, we have made all the major decisions and calculations required to ﬁnish out the problem. All that remains is to percolate these details into the rest of the algorithm. Figure 5.8 shows the general layout of the window with some of the dimensions we have chosen. Let’s ﬁgure out where the lower-left corner of each bar is going to be located. We chose a bar width of 25, so the bar for each successive year will start 25 pixels farther right than the previous year. We can use a variable year to represent the year number and calculate the x coordinate of the lower left corner as year¡ 25¡ 40. (The 40 leaves space on the left edge for the labels.) Of course, the y coordinate of this point is still 230 (the bottom of the graph). To ﬁnd the upper-right corner of a bar, we add 25 (the width of the bar) to the x value of the lower-left corner. The y value of the upper right corner is determined from the (updated) value of principal exactly as we determined it for the ﬁrst bar. Here is the reﬁned algorithm. for year running from a value of 1 up through 10: Calculate principal = principal * (1 + apr) ¢ ¢ Slide 79: 5.4. GRAPHING FUTURE VALUE (0,0) 71 10.0K 7.5K 240 5.0K 2.5K 50 0.0K 10 40 320 (40,230) 25 (315,230) (319,239) Figure 5.8: Position of elements in future value bar graph. Calculate xll = 25 * year + 40 Calculate height = principal * 0.02 Draw a rectangle from (xll, 230) to (xll+25, 230 - height) The variable xll stands for x lower left—the x value of the lower left corner of the bar. Putting all of this together produces the detailed algorithm shown below. Print an introduction Get value of principal and apr from user Create a 320x240 GraphWin titled ‘‘Investment Growth Chart’’ Draw label " 0.0K" at (20,230) Draw label " 2.5K" at (20, 180) Draw label " 5.0K" at (20, 130) Draw label " 7.5K" at (20, 80) Draw label "10.0K" at (20, 30) Draw a rectangle from (40, 230) to (65, 230 - principal * 0.02) for year running from a value of 1 up through 10: Calculate principal = principal * (1 + apr) Calculate xll = 25 * year + 40 Draw a rectangle from (xll, 230) to (xll+25, 230 - principal * 0.02) Wait for user to press Enter Whew! That was a lot of work, but we are ﬁnally ready to translate this algorithm into actual Python code. The translation is straightforward using objects from the graphics module. Here’s the program: # futval_graph.py from graphics import * def main(): # Introduction print "This program plots the growth of a 10-year investment." # Get principal and interest rate principal = input("Enter the initial principal: ") apr = input("Enter the annualized interest rate: ") Slide 80: 72 CHAPTER 5. OBJECTS AND GRAPHICS # Create a graphics window with labels on left edge win = GraphWin("Investment Growth Chart", 320, 240) win.setBackground("white") Text(Point(20, 230), ’ 0.0K’).draw(win) Text(Point(20, 180), ’ 2.5K’).draw(win) Text(Point(20, 130), ’ 5.0K’).draw(win) Text(Point(20, 80), ’ 7.5k’).draw(win) Text(Point(20, 30), ’10.0K’).draw(win) # Draw bar for initial principal height = principal * 0.02 bar = Rectangle(Point(40, 230), Point(65, 230-height)) bar.setFill("green") bar.setWidth(2) bar.draw(win) # Draw bars for successive years for year in range(1,11): # calculate value for the next year principal = principal * (1 + apr) # draw bar for this value xll = year * 25 + 40 height = principal * 0.02 bar = Rectangle(Point(xll, 230), Point(xll+25, 230-height)) bar.setFill("green") bar.setWidth(2) bar.draw(win) raw_input("Press <Enter> to quit.") win.close() main() If you study this program carefully, you will see that I added a number of features to spruce it up a bit. All graphical objects support methods for changing color. I have set the background color of the window to white (by default it’s gray). win.setBackground("white") I have also changed the color of the bar object. The following line asks the bar to color its interior green (because it’s money, you know). bar.setFill("green") You can also change the color of a shape’s outline using the setOutline method. In this case, I have chosen to leave the outline the default black so that the bars stand out from each other. To enhance this effect, this code makes the outline wider (two pixels instead of the default one). bar.setWidth(2) You might also have noted the economy of notation in drawing the labels. Since we don’t ever change the labels, saving them into a variable is unnecessary. We can just create a Text object, tell it to draw itself, and be done with it. Here is an example. Text(Point(20,230), ’ 0.0K’).draw(win) Finally, take a close look at the use of the year variable in the loop. Slide 81: 5.5. CHOOSING COORDINATES for year in range(1,11): 73 The expression range(1,11) produces a sequence of ints 1–10. The loop index variable year marches through this sequence on successive iterations of the loop. So, the ﬁrst time through year is 1, then 2, then 3, etc., up to 10. The value of year is then used to compute the proper position of the lower left corner of each bar. xll = year * 25 + 40 I hope you are starting to get the hang of graphics programming. It’s a bit strenuous, but very addictive. 5.5 Choosing Coordinates The lion’s share of the work in designing the futval graph program was in determining the precise coordinates where things would be placed on the screen. Most graphics programming problems require some sort of a coordinate transformation to change values from a real-world problem into the window coordinates that get mapped onto the computer screen. In our example, the problem domain called for x values representing the year (0–10) and y values representing monetary amounts ($0–$10,000). We had to transform these values to be represented in a 320 x 240 window. It’s nice to work through an example or two to see how this transformation happens, but it makes for tedious programming. Coordinate transformation is an integral and well-studied component of computer graphics. It doesn’t take too much mathematical savvy to see that the transformation process always follows the same general pattern. Anything that follows a pattern can be done automatically. In order to save you the trouble of having to explicitly convert back and forth between coordinate systems, the graphics module provides a simple mechanism to do it for you. When you create a GraphWin you can specify a coordinate system for the window using the setCoords method. The method requires four parameters specifying the coordinates of the lower-left and upper-right corners, respectively. You can then use this coordinate system to place graphical objects in the window. To take a simple example, suppose we just want to divide the window into nine equal squares, Tic-TacToe fashion. This could be done without too much trouble using the default 200 x 200 window, but it would require a bit of arithmetic. The problem becomes trivial if we ﬁrst change the coordinates of the window to run from 0 to 3 in both dimensions. # create a default 200x200 window win = GraphWin("Tic-Tac-Toe") # set coordinates to go from (0,0) in the lower left # to (3,3) in the upper right. win.setCoords(0.0, 0.0, 3.0, 3.0) # Draw vertical lines Line(Point(1,0), Point(1,3)).draw(win) Line(Point(2,0), Point(2,3)).draw(win) # Draw horizontal lines Line(Point(0,1), Point(3,1)).draw(win) Line(Point(0,2), Point(3,2)).draw(win) Another beneﬁt of this approach is that the size of the window can be changed by simply changing the dimensions used when the window is created (e.g. win = GraphWin("Tic-Tac-Toe", 300, 300)). Because the same coordinates span the window (due to setCoords) the objects will scale appropriately to the new window size. Using “raw” window coordinates would require changes in the deﬁnitions of the lines. We can apply this idea to simplify our graphing future value program. Basically, we want our graphics window to go from 0 through 10 (representing years) in the x dimension and from 0 to 10,000 (representing dollars) in the y dimension. We could create just such a window like this. Slide 82: 74 CHAPTER 5. OBJECTS AND GRAPHICS win = GraphWin("Investment Growth Chart", 320, 240) win.setCoords(0.0, 0.0, 10.0, 10000.0) Then creating a bar for any values of year and principal would be simple. Each bar starts at the given year and a baseline of 0 and grows to the next year and a height equal to principal. bar = Rectangle(Point(year, 0), Point(year+1, principal)) There is a small problem with this scheme. Can you see what I have forgotten? The bars will ﬁll the entire window; we haven’t left any room for labels or margins around the edges. This is easily ﬁxed by expanding the coordinates of the window slightly. Since our bars start at 0, we can locate the left side labels at -1. We can add a bit of whitespace around the graph by expanding the coordinates slightly beyond that required for our graph. A little experimentation leads to this window deﬁnition: win = GraphWin("Investment Growth Chart", 320, 240) win.setCoords(-1.75,-200, 11.5, 10400) Here is the program again, using the alternative coordinate system: # futval_graph2.py from graphics import * def main(): # Introduction print "This program plots the growth of" print "a 10-year investment." # Get principal and interest rate principal = input("Enter the initial principal: ") apr = input("Enter the annualized interest rate: ") # Create a graphics window with labels on left edge win = GraphWin("Investment Growth Chart", 320, 240) win.setBackground("white") win.setCoords(-1.75,-200, 11.5, 10400) Text(Point(-1, 0), ’ 0.0K’).draw(win) Text(Point(-1, 2500), ’ 2.5K’).draw(win) Text(Point(-1, 5000), ’ 5.0K’).draw(win) Text(Point(-1, 7500), ’ 7.5k’).draw(win) Text(Point(-1, 10000), ’10.0K’).draw(win) # Draw bar for initial principal bar = Rectangle(Point(0, 0), Point(1, principal)) bar.setFill("green") bar.setWidth(2) bar.draw(win) # Draw a bar for each subsequent year for year in range(1, 11): principal = principal * (1 + apr) bar = Rectangle(Point(year, 0), Point(year+1, principal)) bar.setFill("green") bar.setWidth(2) bar.draw(win) Slide 83: 5.6. INTERACTIVE GRAPHICS raw_input("Press <Enter> to quit.") main() 75 Notice how the complex calculations have been eliminated. This version also makes it easy to change the size of the GraphWin. Changing the window size to 640 x 480 produces a larger, but correctly drawn bar graph. In the original program, all of the calculations would have to be redone to accommodate the new scaling factors in the larger window. Obviously, the second version of our program is much easier to develop and understand. When you are doing graphics programming, give some consideration to choosing a coordinate system that will make your task as simple as possible. 5.6 Interactive Graphics Graphical interfaces can be used for input as well as output. In a GUI environment, users typically interact with their applications by clicking on buttons, choosing items from menus, and typing information into on-screen text boxes. These applications use a technique called event-driven programming. Basically, the program draws a set of interface elements (often called widgets) on the screen, and then waits for the user to do something. When the user moves the mouse, clicks a button or types a key on the keyboard, this generates an event. Basically, an event is an object that encapsulates data about what just happened. The event object is then sent off to an appropriate part of the program to be processed. For example, a click on a button might produce a button event. This event would be passed to the button handling code, which would then perform the appropriate action corresponding to that button. Event-driven programming can be tricky for novice programmers, since it’s hard to ﬁgure out “who’s in charge” at any given moment. The graphics module hides the underlying event-handling mechanisms and provides two simple ways of getting user input in a GraphWin. 5.6.1 Getting Mouse Clicks We can get graphical information from the user via the getMouse method of the GraphWin class. When getMouse is invoked on a GraphWin, the program pauses and waits for the user to click the mouse somewhere in the graphics window. The spot where the user clicks is returned to the program as a Point. Here is a bit of code that reports the coordinates of ten successive mouse clicks. from graphics import * win = GraphWin("Click Me!") for i in range(10): p = win.getMouse() print "You clicked (%d, %d)" % (p.getX(), p.getY()) The value returned by getMouse() is a ready-made Point. We can use it like any other point using accessors such as getX and getY or other methods such as draw and move. Here is an example of an interactive program that allows the user to draw a triangle by clicking on three points in a graphics window. This example is completely graphical, making use of Text objects as prompts. No interaction with a Python text window is required. If you are programming in a Windows environment, you can name this program using a .pyw extension. Then when the program is run, it will not even display the Python shell window. # Program: triangle.pyw from graphics import * def main(): win = GraphWin("Draw a Triangle") Slide 84: 76 CHAPTER 5. OBJECTS AND GRAPHICS win.setCoords(0.0, 0.0, 10.0, 10.0) message = Text(Point(5, 0.5), "Click on three points") message.draw(win) # Get and draw three vertices of triangle p1 = win.getMouse() p1.draw(win) p2 = win.getMouse() p2.draw(win) p3 = win.getMouse() p3.draw(win) # Use Polygon object to draw the triangle triangle = Polygon(p1,p2,p3) triangle.setFill("peachpuff") triangle.setOutline("cyan") triangle.draw(win) # Wait for another click to exit message.setText("Click anywhere to quit.") win.getMouse() main() The three-click triangle illustrates a couple new features of the graphics module. There is no triangle class; however there is a general class Polygon that can be used for any multi-sided, closed shape. The constructor for Polygon accepts any number of points and creates a polygon by using line segments to connect the points in the order given and to connect the last point back to the ﬁrst. A triangle is just a three-sided polygon. Once we have three Points p1, p2, and p3, creating the triangle is a snap. triangle = Polygon(p1, p2, p3) You should also study how the Text object is used to provide prompts. A single Text object is created and drawn near the beginning of the program. message = Text(Point(5, 0.5), "Click on three points") message.draw(win) To change the prompt, we don’t need to create a new Text object, we can just change the text that is displayed. This is done near the end of the program with the setText method. message.setText("Click anywhere to quit.") As you can see, the getMouse method of GraphWin provides a simple way of interacting with the user in a graphics-oriented program. 5.6.2 Handling Textual Input In the triangle example, all of the input was provided through mouse clicks. The graphics module also includes a simple Entry object that can be used to get keyboard input in a GraphWin. An Entry object draws a box on the screen that can contain text. It understands setText and getText methods just like the Text object does. The difference is that the contents of an Entry can be edited by the user. Here’s a version of the temperature conversion program from Chapter 2 with a graphical user interface: # convert_gui1.pyw # Program to convert Celsius to Fahrenheit using a simple # graphical interface. Slide 85: 5.6. INTERACTIVE GRAPHICS 77 from graphics import * def main(): win = GraphWin("Celsius Converter", 300, 200) win.setCoords(0.0, 0.0, 3.0, 4.0) # Draw the interface Text(Point(1,3), " Celsius Temperature:").draw(win) Text(Point(1,1), "Fahrenheit Temperature:").draw(win) input = Entry(Point(2,3), 5) input.setText("0.0") input.draw(win) output = Text(Point(2,1),"") output.draw(win) button = Text(Point(1.5,2.0),"Convert It") button.draw(win) Rectangle(Point(1,1.5), Point(2,2.5)).draw(win) # wait for a mouse click win.getMouse() # convert input celsius = eval(input.getText()) fahrenheit = 9.0/5.0 * celsius + 32 # display output and change button output.setText("%0.1f" % fahrenheit) button.setText("Quit") # wait for click and then quit win.getMouse() win.close() main() When run, this produces a window with an entry box for typing in a Celsius temperature and a “button” for doing the conversion. The button is just for show. The program actually just pauses for a mouse click anywhere in the window. Figure 5.9 shows how the window looks when the program starts. Initially, the input entry box is set to contain the value 0.0. The user can delete this value and type in another temperature. The program pauses until the user clicks the mouse. Notice that the point where the user clicks is not even saved; the getMouse function is just used to pause the program until the user has a chance to enter a value in the input box. The program then processes the input in four steps. First, the text in the input box is converted into a number (via eval). This number is then converted to degrees Fahrenheit. Finally, the resulting number is turned back into a string (via the string formatting operator) for display in the output text area. Figure 5.10 shows how the window looks after the user has typed an input and clicked the mouse. Notice that the converted temperature shows up in the output area, and the label on the button has changed to “Quit” to show that clicking again will exit the program. This example could be made much prettier using some of the options in the graphics library for changing the colors, sizes and line widths of the various widgets. The code for the program is deliberately Spartan to illustrate just the essential elements of GUI design. Although the basic tools getMouse and Entry do not provide a full-ﬂedged GUI environment, we will see in later chapters how these simple mechanisms can support surprisingly rich interactions. Slide 86: 78 CHAPTER 5. OBJECTS AND GRAPHICS Figure 5.9: Initial screen for graphical temperature converter Figure 5.10: Graphical temperature converter after user input. Slide 87: 5.7. GRAPHICS MODULE REFERENCE 79 5.7 Graphics Module Reference The examples in this chapter have touched on most of the elements in the graphics module. This section provides a complete reference to the objects and functions provided in the graphics library. Experienced programmers use these sorts of guides to learn about new libraries. You will probably want to refer back to this section often when you are writing your own graphical programs. 5.7.1 GraphWin Objects A GraphWin object represents a window on the screen where graphical images may be drawn. A program may deﬁne any number of GraphWins. A GraphWin understands the following methods. GraphWin(title, width, height) Constructs a new graphics window for drawing on the screen. The parameters are optional, the default title is “Graphics Window,” and the default size is 200 x 200. plot(x, y, color) Draws the pixel at x¤ y¡ in the window. Color is optional, black is the default. plotPixel(x, y, Color) Draws the pixel at the “raw” position x¤ y¡ ignoring any coordinate transformations set up by setCoords. setBackground(color) Sets the window background to the given color. The initial background is gray. See section 5.7.5 for information on specifying colors. close() Closes the on-screen window. getMouse() Pauses for the user to click a mouse in the window and returns where the mouse was clicked as a Point object. setCoords(xll, yll, xur, yur) Sets the coordinate system of the window. The lower left corner is xll yll¡ and the upper right corner is xur¤ yur¡ . All subsequent drawing will be done with respect to the altered coordinate system (except for plotPixel). flush() Updates the appearance of the window to reﬂect all drawing operations that have been done so far. In normal operation, the window is only updated during “idle” periods. A sequence of drawing commands may end up appearing all at once. If you want to get an animation effect, flush should be called at appropriate places in the program to perform drawing operations incrementally. 5.7.2 Graphics Objects The module provides the following classes of drawable objects: Point, Line, Circle, Oval, Rectangle, Polygon, and Text. All objects are initially created unﬁlled with a black outline. All graphics objects support the following generic set of methods. setFill(color) Sets the interior of the object to the given color. setOutline(color) Sets the outline of the object to the given color. setWidth(pixels) Sets the width of the outline of the object to this many pixels. draw(aGraphWin) Draws the object into the given GraphWin. undraw() Undraws the object from a graphics window. This produces an error if the object is not currently drawn. move(dx,dy) Moves the object dx units in the x direction and dy units in the y direction. If the object is currently drawn, the image is adjusted to the new position. clone() Returns a duplicate of the object. Clones are always created in an undrawn state. Other than that, they are identical to the cloned object. ¤ Slide 88: 80 Point Methods CHAPTER 5. OBJECTS AND GRAPHICS Point(x,y) Constructs a point having the given coordinates. getX() Returns the x coordinate of a point. getY() Returns the y coordinate of a point. Line Methods Line(point1, point2) Constructs a line segment from point1 to point2. setArrow(string) Sets the arrowhead status of a line. Arrows may be drawn at either the ﬁrst point, the last point, or both. Possible values of string are ’first’, ’last’, ’both’, and ’none’. The default setting is ’none’. getCenter() Returns a clone of the midpoint of the line segment. getP1(), getP2() Returns a clone of the corresponding endpoint of the segment. Circle Methods Circle(centerPoint, radius) Constructs a circle with given center point and radius. getCenter() Returns a clone of the center point of the circle. getRadius() Returns the radius of the circle. getP1(), getP2() Returns a clone of the corresponding corner of the circle’s bounding box. These are opposite corner points of a square that circumscribes the circle. Rectangle Methods Rectangle(point1, point2) Constructs a rectangle having opposite corners at point1 and point2. getCenter() Returns a clone of the center point of the rectangle. getP1(), getP2() Returns a clone of corner points originally used to construct the rectangle. Oval Methods Oval(point1, point2) Constructs an oval in the bounding box determined by point1 and point2. getCenter() Returns a clone of the point at the center of the oval. getP1(), getP2 Return a clone of the corresponding point used to construct the oval. Polygon Methods Polygon(point1, point2, point3, ...) Constructs a polygon having the given points as vertices. getPoints() Returns a list containing clones of the points used to construct the polygon. Slide 89: 5.7. GRAPHICS MODULE REFERENCE Text Methods 81 Text(anchorPoint, string) Constructs a text object that displays the given string centered at anchorPoint. The text is displayed horizontally. setText(string) Sets the text of the object to string. getText() Returns the current string. getAnchor() Returns a clone of the anchor point. setFace(family) Changes the font face to the given family. Possible values are: ’helvetica’, ’courier’, ’times roman’, and ’arial’. setSize(point) Changes the font size to the given point size. Sizes from 5 to 36 points are legal. setStyle(style) Changes font to the given style. Possible values are: ’normal’, ’bold’, ’italic’, and ’bold italic’. 5.7.3 Entry Objects Objects of type Entry are displayed as text entry boxes that can be edited by the user of the program. Entry objects support the generic graphics methods move(), draw(graphwin), undraw(), setFill(color), and clone(). The Entry speciﬁc methods are given below. Entry(centerPoint, width) Constructs an Entry having the given center and width. The width is speciﬁed in number of characters of text that can be displayed. getAnchor() Returns a clone of the point where the entry box is centered. getText() Returns the string of text that is currently in the entry box. setText(string) Sets the text in the entry box to the given string. 5.7.4 Displaying Images The graphics module also provides minimal support for displaying certain image formats into a GraphWin. Most platforms will support JPEG, PPM and GIF images. Display is done with an Image object. Images support the generic methods move(dx,dy), draw(graphwin), undraw(), and clone(). Image speciﬁc methods are given below. Image(centerPoint, filename) Constructs an image from contents of the given ﬁle, centered at the given center point. getAnchor() Returns a clone of the point where the image is centered. 5.7.5 Generating Colors Colors are indicated by strings. Most normal colors such as ’red’, ’purple’, ’green’, ’cyan’, etc. should be available. Many colors come in various shades, such as ’red1’, ’red2’,’red3’, ’red4’, which are increasingly darker shades of red. The graphics module also provides a function for mixing your own colors numerically. The function color rgb(red, green, blue) will return a string representing a color that is a mixture of the intensities of red, green and blue speciﬁed. These should be ints in the range 0–255. Thus color rgb(255, 0, 0) is a bright red, while color rgb(130, 0, 130) is a medium magenta. Slide 90: 82 CHAPTER 5. OBJECTS AND GRAPHICS 5.8 Exercises 1. Pick an example of an interesting real-world object and describe it as a programming object by listing its data (attributes, what it “knows”) and its methods (behaviors, what it can “do”). 2. Describe in your own words the object produced by each of the following operations from the graphics module. Be as precise as you can. Be sure to mention such things as the size, position, and appearance of the various objects. You may include a sketch if that helps. (a) Point(130,130) (b) c = Circle(Point(30,40),25) c.setFill(’blue’) c.setOutline(’red’) (c) r = Rectangle(Point(20,20), Point(40,40)) r.setFill(color_rgb(0,255,150)) r.setWidth(3) (d) l = Line(Point(100,100), Point(100,200)) l.setOutline(’red4’) l.setArrow(’first’) (e) Oval(Point(50,50), Point(60,100)) (f) shape = Polygon(Point(5,5), Point(10,10), Point(5,10), Point(10,5)) shape.setFill(’orange’) (g) t = Text(Point(100,100), "Hello World!") t.setFace("courier") t.setSize(16) t.setStyle("italic") 3. Describe what happens when the following interactive graphics program runs. from graphics import * def main(): win = GraphWin() shape = Circle(Point(50,50), 20) shape.setOutline("red") shape.setFill("red") shape.draw(win) for i in range(10): p = win.getMouse() c = shape.getCenter() dx = p.getX() - c.getX() dy = p.getY() - c.getY() shape.move(dx,dy) win.close() 4. Modify the program from the previous problem in the following ways: (a) Make it draw squares instead of circles. (b) Have each successive click draw an additional square on the screen (rather than moving the existing one). (c) Print a message on the window when all the clicks is entered, and wait for a ﬁnal click before closing the window. Slide 91: 5.8. EXERCISES 83 5. An archery target consists of a central circle of yellow surrounded by concentric rings of red, blue, black and white. Each ring has the same “width,” which is the same as the radius of the yellow circle. Write a program that draws such a target. Hint: Objects drawn later will appear on top of objects drawn earlier. 6. Write a program that draws some sort of face. 7. Write a program that draws a winter scene with a Christmas tree and a snowman. 8. Write a program that draws 5 dice on the screen depicting a straight (1, 2, 3, 4, 5 or 2, 3, 4, 5, 6). 9. Modify the graphical future value program so that the input (principal and apr) also are done in a graphical fashion using Entry objects. 10. Circle Intersection. Write a program that computes the intersection of a circle with a horizontal line and displays the information textually and graphically. Input: Radius of the circle and the y-intercept of the line. Output: Draw a circle centered at 0¤ 0¡ with the given radius in a window with coordinates running from -10,-10 to 10,10. Draw a horizontal line across the window with the given y-intercept. Draw the two points of intersection in red. Print out the x values of the points of intersection. Formula: x 11. Line Information. This program allows the user to draw a line segment and then displays some graphical and textual information about the line segment. Input: 2 mouse clicks for the end points of the line segment. Output: Draw the midpoint of the segment in cyan. Draw the line. Print the length and the slope of the line. ¢ Formulas: dx x2 x1 dy y2 y1 slope dy dx length dx2 dy2 ¢ ¡ ¢ £ 12. Rectangle Information. This program displays information about a rectangle drawn by the user. Input: 2 mouse clicks for the opposite corners of a rectangle. Output: Draw the rectangle. Print the perimeter and area of the rectangle. Formulas: length¡ width¡ area perimeter 2 length width¡ ¢ 13. Triangle Information. Same as previous problem, but with 3 clicks for the verticies of a triangle. Formulas: For perimeter, see length from line problem. s s a¡ s b¡ s c¡ where a¤ b¤ and c are the lengths of the sides and s area ¢ ¢ ¢ £ ¢ £ r2 y2 abc 2 Slide 92: 84 14. 5-click house. CHAPTER 5. OBJECTS AND GRAPHICS You are to write a program that allows the user to draw a simple house using ﬁve mouse-clicks. The ﬁrst two clicks will be the opposite corners of the rectangular frame of the house. The third click will indicate the center of the top edge of a rectangular door. The door should have a total width that is 1 5 of the width of the house frame. The sides of the door should extend from the corners of the top down to the bottom of the frame. The fourth click will indicate the center of a square window. The window is half as wide as the door. The last click will indicate the peak of the roof. The edges of the roof will extend from the point at the peak to the corners of the top edge of the house frame. 5 2 3 4 1 Slide 93: Chapter 6 Deﬁning Functions The programs that we have written so far comprise a single function, usually called main. We have also been using pre-written functions and methods including built-in Python functions (e.g., abs), functions from the Python standard libraries (e.g., math.sqrt, string.split), and object methods from the graphics module (e.g., myPoint.getX()). Functions are an important tool for building sophisticated programs. This chapter covers the whys and hows of designing your own functions to make your programs easier to write and understand. 6.1 The Function of Functions In the previous chapter, we looked at a graphic solution to the future value problem. This program makes use of the graphics library to draw a bar chart showing the growth of an investment. Here is the program as we left it: # futval_graph2.py from graphics import * def main(): # Introduction print "This program plots the growth of a 10-year investment." # Get principal and interest rate principal = input("Enter the initial principal: ") apr = input("Enter the annualized interest rate: ") # Create a graphics window with labels on left edge win = GraphWin("Investment Growth Chart", 640, 480) win.setBackground("white") win.setCoords(-1.75,-200, 11.5, 10400) Text(Point(-1, 0), ’ 0.0K’).draw(win) Text(Point(-1, 2500), ’ 2.5K’).draw(win) Text(Point(-1, 5000), ’ 5.0K’).draw(win) Text(Point(-1, 7500), ’ 7.5k’).draw(win) Text(Point(-1, 10000), ’10.0K’).draw(win) # Draw bar for initial principal bar = Rectangle(Point(0, 0), Point(1, principal)) bar.setFill("green") bar.setWidth(2) bar.draw(win) 85 Slide 94: 86 CHAPTER 6. DEFINING FUNCTIONS # Draw a bar for each subsequent year for year in range(1, 11): principal = principal * (1 + apr) bar = Rectangle(Point(year, 0), Point(year+1, principal)) bar.setFill("green") bar.setWidth(2) bar.draw(win) raw_input("Press <Enter> to quit.") This is certainly a workable program, but there is a nagging issue of program style that really should be addressed. Notice that this program draws bars in two different places. The initial bar is drawn just before the loop, and the subsequent bars are drawn inside of the loop. Having similar code like this in two places has some drawbacks. Obviously, one issue is having to write the code twice. A more subtle problem is that the code has to be maintained in two different places. Should we decide to change the color or other facets of the bars, we would have to make sure these changes occurred in both places. Failing to keep related parts of the code in synch is a common problem in program maintenance. Functions can be used to reduce code duplication and make programs more understandable and easier to maintain. Before ﬁxing up the future value program, let’s take look at what functions have to offer. 6.2 Functions, Informally You can think of a function as a subprogram—a small program inside of a program. The basic idea of a function is that we write a sequence of statements and give that sequence a name. The instructions can then be executed at any point in the program by referring to the function name. The part of the program that creates a function is called a function deﬁnition. When a function is subsequently used in a program, we say that the deﬁnition is called or invoked. A single function deﬁnition may be called at many different points of a program. Let’s take a concrete example. Suppose you want to write a program that prints out the lyrics to the “Happy Birthday” song. The standard lyrics look like this. Happy Happy Happy Happy birthday to you! birthday to you! birthday, dear <insert-name>. birthday to you! We’re going to play with this example in the interactive Python environment. You might want to ﬁre up Python and try some of this out for yourself. A simple approach to this problem is to use four print statements. Here’s an interactive session that creates a program for singing Happy Birthday to Fred. >>> def main(): print "Happy print "Happy print "Happy print "Happy birthday to you!" birthday to you!" birthday, dear Fred." birthday to you!" We can then run this program to get our lyrics. >>> main() Happy birthday to you! Happy birthday to you! Happy birthday, dear Fred. Happy birthday to you! Slide 95: 6.2. FUNCTIONS, INFORMALLY 87 Obviously, there is some duplicated code in this program. For such a simple program, that’s not a big deal, but even here it’s a bit annoying to keep retyping the same line. Let’s introduce a function that prints the lyrics of the ﬁrst, second, and fourth lines. >>> def happy(): print "Happy birthday to you!" We have deﬁned a new function called happy. Here is an example of what it does. >>> happy() Happy birthday to you! Invoking the happy command causes Python to print a line of the song. Now we can redo the verse for Fred using happy. Let’s call our new version singFred. >>> def singFred(): happy() happy() print "Happy birthday, dear Fred." happy() This version required much less typing, thanks to the happy command. Let’s try printing the lyrics for Fred just to make sure it works. >>> singFred() Happy birthday to you! Happy birthday to you! Happy birthday, dear Fred. Happy birthday to you! So far, so good. Now suppose that it’s also Lucy’s birthday, and we want to sing a verse for Fred followed by a verse for Lucy. We’ve already got the verse for Fred; we can prepare one for Lucy as well. >>> def singLucy(): happy() happy() print "Happy birthday, dear Lucy." happy() Now we can write a main program that sings to both Fred and Lucy. >>> def main(): singFred() print singLucy() The bare print between the two function calls puts a space between the verses in our output. And here’s the ﬁnal product in action. >>> main() Happy birthday to you! Happy birthday to you! Happy birthday, dear Fred. Happy birthday to you! Happy Happy Happy Happy birthday to you! birthday to you! birthday, dear Lucy. birthday to you! Slide 96: 88 CHAPTER 6. DEFINING FUNCTIONS Well now, that certainly seems to work, and we’ve removed some of the duplication by deﬁning the happy function. However, something still doesn’t feel quite right. We have two functions, singFred and singLucy, that are almost identical. Following this approach, adding a verse for Elmer would have us create a singElmer function that looks just like those for Fred and Lucy. Can’t we do something about the proliferation of verses? Notice that the only difference between singFred and singLucy is the name at the end of the third print statement. The verses are exactly the same except for this one changing part. We can collapse these two functions together by using a parameter. Let’s write a generic function called sing. >>> def sing(person): happy() happy() print "Happy Birthday, dear", person + "." happy() This function makes use of a parameter named person. A parameter is a variable that is initialized when the function is called. We can use the sing function to print a verse for either Fred or Lucy. We just need to supply the name as a parameter when we invoke the function. >>> sing("Fred") Happy birthday to you! Happy birthday to you! Happy Birthday, dear Fred. Happy birthday to you! >>> sing("Lucy") Happy birthday to you! Happy birthday to you! Happy Birthday, dear Lucy. Happy birthday to you! Let’s ﬁnish with a program that sings to all three of our birthday people. >>> def main(): sing("Fred") print sing("Lucy") print sing("Elmer") It doesn’t get much easier than that. Here is the complete program as a module ﬁle. # happy.py def happy(): print "Happy Birthday to you!" def sing(person): happy() happy() print "Happy birthday, dear", person + "." happy() def main(): sing("Fred") print Slide 97: 6.3. FUTURE VALUE WITH A FUNCTION sing("Lucy") print sing("Elmer") 89 6.3 Future Value with a Function Now that you’ve seen how deﬁning functions can help solve the code duplication problem, let’s return to the future value graph. Recall the problem is that bars of the graph are printed at two different places in the program. The code just before the loop looks like this. # Draw bar for initial principal bar = Rectangle(Point(0, 0), Point(1, principal)) bar.setFill("green") bar.setWidth(2) bar.draw(win) And the code inside of the loop is as follows. bar = Rectangle(Point(year, 0), Point(year+1, principal)) bar.setFill("green") bar.setWidth(2) bar.draw(win) Let’s try to combine these two into a single function that draws a bar on the screen. In order to draw the bar, we need some information. Speciﬁcally, we need to know what year the bar will be for, how tall the bar will be, and what window the bar will be drawn in. These three values will be supplied as parameters for the function. Here’s the function deﬁnition. def drawBar(window, year, height): # Draw a bar in window for given year with given height bar = Rectangle(Point(year, 0), Point(year+1, height)) bar.setFill("green") bar.setWidth(2) bar.draw(window) To use this function, we just need to supply values for the three parameters. For example, if win is a GraphWin, we can draw a bar for year 0 and a principal of $2,000 by invoking drawBar like this. drawBar(win, 0, 2000) Incorporating the drawBar function, here is the latest version of our future value program. # futval_graph3.py from graphics import * def drawBar(window, year, height): # Draw a bar in window starting at year with given height bar = Rectangle(Point(year, 0), Point(year+1, height)) bar.setFill("green") bar.setWidth(2) bar.draw(window) def main(): # Introduction print "This program plots the growth of a 10-year investment." Slide 98: 90 CHAPTER 6. DEFINING FUNCTIONS # Get principal and interest rate principal = input("Enter the initial principal: ") apr = input("Enter the annualized interest rate: ") # Create a graphics window with labels on left edge win = GraphWin("Investment Growth Chart", 320, 240) win.setBackground("white") win.setCoords(-1.75,-200, 11.5, 10400) Text(Point(-1, 0), ’ 0.0K’).draw(win) Text(Point(-1, 2500), ’ 2.5K’).draw(win) Text(Point(-1, 5000), ’ 5.0K’).draw(win) Text(Point(-1, 7500), ’ 7.5k’).draw(win) Text(Point(-1, 10000), ’10.0K’).draw(win) # Draw bar for initial principal drawBar(win, 0, principal) # Draw a bar for each subsequent year for year in range(1, 11): principal = principal * (1 + apr) drawBar(win, year, principal) raw_input("Press <Enter> to quit.") You can see how drawBar has eliminated the duplicated code. Should we wish to change the appearance of the bars in the graph, we only need to change the code in one spot, the deﬁnition of drawBar. Don’t worry yet if you don’t understand every detail of this example. You still have some things to learn about functions. 6.4 Functions and Parameters: The Gory Details You may be wondering about the choice of parameters for the drawBar function. Obviously, the year for which a bar is being drawn and the height of the bar are the changeable parts in the drawing of a bar. But, why is window also a parameter to this function? After all, we will be drawing all of the bars in the same window; it doesn’t seem to change. The reason for making window a parameter has to do with the scope of variables in function deﬁnitions. Scope refers to the places in a program where a given variable may be referenced. Remember each function is its own little subprogram. The variables used inside of one function are local to that function, even if they happen to have the same name as variables that appear inside of another function. The only way for a function to see a variable from another function is for that variable to be passed as a parameter. Since the GraphWin (in the variable win) is created inside of main, it is not directly accessible in drawBar. However, the window parameter in drawBar gets assigned the value of win from main when drawBar is called. To see how this happens, we need to take a more detailed look at the function invocation process. A function deﬁnition looks like this. def <name>(<formal-parameters>): <body> The name of the function must be an identiﬁer, and formal-parameters is a (possibly empty) list of variable names (also identiﬁers). The formal parameters, like all variables used in the function, are only accessible in the body of the function. Variables with identical names elswhere in the program are distinct from the formal parameters and variables inside of the function body. A function is called by using its name followed by a list of actual parameters or arguments. <name>(<actual-parameters>) Slide 99: 6.4. FUNCTIONS AND PARAMETERS: THE GORY DETAILS When Python comes to a function call, it initiates a four-step process. 1. The calling program suspends at the point of the call. 91 2. The formal parameters of the function get assigned the values supplied by the actual parameters in the call. 3. The body of the function is executed. 4. Control returns to the point just after where the function was called. Returning to the Happy Birthday example, let’s trace through the singing of two verses. Here is part of the body from main. sing("Fred") print sing("Lucy") ... When Python gets to sing("Fred"), execution of main is temporarily suspended. At this point, Python looks up the deﬁnition of sing and sees that it has a single formal parameter, person. The formal parameter is assigned the value of the actual, so it is as if we had executed this statement: person = "Fred" A snapshot of this situation is shown in Figure 6.1. Notice the variable person inside of sing has just been initialized. def def main(): person = "Fred" sing("Fred") print sing("Lucy") sing(person): happy() happy() print "Happy birthday, dear", person + "." happy() person: "Fred" Figure 6.1: Illustration of control transferring to sing. At this point, Python begins executing the body of sing. The ﬁrst statement is another function call, this one to happy. Python suspends execution of sing and transfers control to the called function. The body of happy consists of a single print. This statement is executed, and then control returns to where it left off in sing. Figure 6.2 shows a snapshot of the execution so far. def def main(): person = "Fred" sing("Fred") print sing("Lucy") def happy(): sing(person): print "Happy Birthday to you!" happy() happy() print "Happy birthday, dear", person + "." happy() person: "Fred" Figure 6.2: Snaphot of completed call to happy. Execution continues in this manner with Python making two more side trips back to happy to complete the execution of sing. When Python get to the end of sing, control then returns to main and continues immediately after the function call. Figure 6.3 shows where we are at that point. Notice that the person Slide 100: 92 def def main(): person = "Fred" sing("Fred") print sing("Lucy") CHAPTER 6. DEFINING FUNCTIONS sing(person): happy() happy() print "Happy birthday, dear", person + "." happy() Figure 6.3: Snaphot of completed call to sing. variable in sing has disappeared. The memory occupied by local function variables is reclaimed when the function ﬁnishes. Local variables do not retain any values from one function execution to the next. The next statement to execute is the bare print statement in main. This produces a blank line in the output. Then Python encounters another call to sing. As before, control transfers to the function deﬁnition. This time the formal parameter is "Lucy". Figure 6.4 shows the situation as sing begins to execute for the second time. def def main(): y" "Luc sing("Fred") on = pers print sing("Lucy") sing(person): happy() happy() print "Happy birthday, dear", person + "." happy() person: "Lucy" Figure 6.4: Snaphot of second call to sing. Now we’ll fast forward to the end. The function body of sing is executed for Lucy (with three side trips through happy) and control returns to main just after the point of the function call. Now we have reached the bottom of our code fragment, as illustrated by Figure 6.5. These three statements in main have caused sing to execute twice and happy to execute six times. Overall, nine total lines of output were generated. def def main(): ucy" sing("Fred") = "L on print pers sing("Lucy") sing(person): happy() happy() print "Happy birthday, dear", person + "." happy() Figure 6.5: Completion of second call to sing. Hopefully you’re getting the hang of how function calls actually work. One point that this example did not address is the use of multiple parameters. When a function deﬁnition has several parameters, the actual parameters are matched up with the formal parameters by position. The ﬁrst actual parameter is assigned to the ﬁrst formal paramter, the second actual is assigned to the second formal, etc. As an example, look again at the use of the drawBar function from the future value program. Here is the call to draw the initial bar. drawBar(win, 0, principal) When Python transfers control to drawBar, these parameters are matched up to the formal parameters in the function heading. def drawBar(window, year, height): Slide 101: 6.5. FUNCTIONS THAT RETURN VALUES The net effect is as if the function body had been prefaced with three assignment statements. window = win year = 0 height = principal 93 You must always be careful when calling a function that you get the actual parameters in the correct order to match the function deﬁnition. 6.5 Functions that Return Values You have seen that parameter passing provides a mechanism for initializing the variables in a function. In a way, parameters act as inputs to a function. We can call a function many times and get different results by changing the input parameters. Sometimes we also want to get information back out of a function. This is accomplished by having functions return a value to the caller. You have already seen numerous examples of this type of function. For example, consider this call to the sqrt function from the math library. discRt = math.sqrt(bb - 4ac) Here the value of bb - 4ac is the actual parameter of math.sqrt. This function call occurs on the right side of an assignment statement; that means it is an expression. The math.sqrt function must somehow produce a value that is then assigned to the variable discRt. Technically, we say that sqrt returns the square root of its argument. It’s very easy to write functions that return values. Here’s an example value-returning function that does the opposite of sqrt; it returns the square of its argument. def square(x): return x * x The body of this function consists of a single return statement. When Python encounters return, it exits the function and returns control to the point where the function was called. In addition, the value(s) provided in the return statement are sent back to the caller as an expression result. We can use our square function any place that an expression would be legal. Here are some interactive examples. >>> 9 >>> 16 >>> >>> >>> 25 >>> 34 square(3) print square(4) x=5 y = square(x) print y print square(x) + square(3) Let’s use the square function to write another function that ﬁnds the distance between two points. Given two points x1 y1 and x2 y2 , the distance between them is calculated from the Pythagorean Theorem as x2 x1 2 y2 y1 2 . Here is a Python function to compute the distance between two Point objects. def distance(p1, p2): dist = math.sqrt(square(p2.getX() - p1.getX()) + square(p2.getY() - p1.getY()) return dist Using the distance function, we can augment the interactive triangle program from last chapter to calculate the perimeter of the triangle. Here’s the complete program: ¡ ¤ ¡ ¢ ¡ ¤ ¢ ¡ ¢ £ Slide 102: 94 # Program: triangle2.py from graphics import * def square(x): return x * x CHAPTER 6. DEFINING FUNCTIONS def distance(p1, p2): dist = math.sqrt(square(p2.getX() - p1.getX()) + square(p2.getY() - p1.getY()) return dist def main(): win = GraphWin("Draw a Triangle") win.setCoords(0.0, 0.0, 10.0, 10.0) message = Text(Point(5, 0.5), "Click on three points") message.draw(win) # Get and draw three vertices of triangle p1 = win.getMouse() p1.draw(win) p2 = win.getMouse() p2.draw(win) p3 = win.getMouse() p3.draw(win) # Use Polygon object to draw the triangle triangle = Polygon(p1,p2,p3) triangle.setFill("peachpuff") triangle.setOutline("cyan") triangle.draw(win) # Calculate the perimeter of the triangle perim = distance(p1,p2) + distance(p2,p3) + distance(p3,p1) message.setText("The perimeter is: %0.2f" % perim) # Wait for another click to exit win.getMouse() You can see how distance is called three times in one line to compute the perimeter of the triangle. Using a function here saves quite a bit of tedious coding. Sometimes a function needs to return more than one value. This can be done by simply listing more than one expression in the return statement. As a silly example, here is a function that computes both the sum and the difference of two numbers. def sumDiff(x,y): sum = x + y diff = x - y return sum, diff As you can see, this return hands back two values. When calling this function, we would place it in a simultaneous assignment. num1, num2 = input("Please enter two numbers (num1, num2) ") s, d = sumDiff(num1, num2) print "The sum is", s, "and the difference is", d Slide 103: 6.6. FUNCTIONS AND PROGRAM STRUCTURE 95 As with parameters, when multiple values are returned from a function, they are assigned to variables by position. In this example, s will get the ﬁrst value listed in the return (sum), and d will get the second value (diff). That’s just about all there is to know about functions in Python. There is one “gotcha” to warn you about. Technically, all functions in Python return a value, regardless of whether or not the function actually contains a return statement. Functions without a return always hand back a special object, denoted None. This object is often used as a sort of default value for variables that don’t currently hold anything useful. A common mistake that new (and not-so-new) programmers make is writing what should be a value-returning function but forgetting to include a return statement at the end. Suppose we forget to include the return statement at the end of the distance function. def distance(p1, p2): dist = math.sqrt(square(p2.getX() - p1.getX()) + square(p2.getY() - p1.getY()) Running the revised triangle program with this version of distance generates this Python error message. Traceback (innermost last): File "<stdin>", line 1, in ? File "triangle2err.py", line 44, in ? main() File "triangle2err.py", line 37, in main perim = distance(p1,p2) + distance(p2,p3) + distance(p3,p1) TypeError: bad operand type(s) for + The problem here is that this version of distance does not return a number, but always hands back the value None. Addition is not deﬁned for None, and so Python complains. If your value-returning functions are producing strange error messages, check to make sure you remembered to include the return. 6.6 Functions and Program Structure So far, we have been discussing functions as a mechanism for reducing code duplication, thus shortening and simplifying our programs. Surprisingly, functions are often used even when doing so actually makes the program longer. A second reason for using functions is to make programs more modular. As the algorithms that you design get more complex, it gets more and more difﬁcult to make sense out of programs. Humans are pretty good at keeping track of eight to ten things at a time. When presented with an algorithm that is hundreds of lines long, even the best programmers will throw up their hands in bewilderment. One way to deal with this complexity is to break an algorithm into smaller subprograms, each of which makes sense on its own. I’ll have a lot more to say about this later when we discuss program design in Chapter 9. For now, we’ll just take a look at an example. Let’s return to the future value problem one more time. Here is the main program as we left it: def main(): # Introduction print "This program plots the growth of a 10-year investment." # Get principal and interest rate principal = input("Enter the initial principal: ") apr = input("Enter the annualized interest rate: ") # Create a graphics window with labels on left edge win = GraphWin("Investment Growth Chart", 320, 240) win.setBackground("white") win.setCoords(-1.75,-200, 11.5, 10400) Slide 104: 96 Text(Point(-1, Text(Point(-1, Text(Point(-1, Text(Point(-1, Text(Point(-1, CHAPTER 6. DEFINING FUNCTIONS 0), ’ 0.0K’).draw(win) 2500), ’ 2.5K’).draw(win) 5000), ’ 5.0K’).draw(win) 7500), ’ 7.5k’).draw(win) 10000), ’10.0K’).draw(win) # Draw bar for initial principal drawBar(win, 0, principal) # Draw a bar for each subsequent year for year in range(1, 11): principal = principal * (1 + apr) drawBar(win, year, principal) raw_input("Press <Enter> to quit.") Although we have already shortened this algorithm through the use of the drawBar function, it is still long enough to make reading through it awkward. The comments help to explain things, but, not to put too ﬁne a point on it, this function is just too long. One way to make the program more readable is to move some of the details into a separate function. For example, there are eight lines in the middle that simply create the window where the chart will be drawn. We could put these steps into a value returning function. def createLabeledWindow(): # Returns a GraphWin with title and labels drawn window = GraphWin("Investment Growth Chart", 320, 240) window.setBackground("white") window.setCoords(-1.75,-200, 11.5, 10400) Text(Point(-1, 0), ’ 0.0K’).draw(window) Text(Point(-1, 2500), ’ 2.5K’).draw(window) Text(Point(-1, 5000), ’ 5.0K’).draw(window) Text(Point(-1, 7500), ’ 7.5k’).draw(window) Text(Point(-1, 10000), ’10.0K’).draw(window) return window As its name implies, this function takes care of all the nitty-gritty details of drawing the initial window. It is a self-contained entity that performs this one well-deﬁned task. Using our new function, the main algorithm seems much simpler. def main(): print "This program plots the growth of a 10-year investment." principal = input("Enter the initial principal: ") apr = input("Enter the annualized interest rate: ") win = createLabeledWindow() drawBar(win, 0, principal) for year in range(1, 11): principal = principal * (1 + apr) drawBar(win, year, principal) raw_input("Press <Enter> to quit.") Notice that I have removed the comments; the intent of the algorithm is now clear. With suitably named functions, the code has become nearly self-documenting. Here is the ﬁnal version of our future value program: Slide 105: 6.7. EXERCISES # futval_graph4.py from graphics import * def createLabeledWindow(): window = GraphWin("Investment Growth Chart", 320, 240) window.setBackground("white") window.setCoords(-1.75,-200, 11.5, 10400) Text(Point(-1, 0), ’ 0.0K’).draw(window) Text(Point(-1, 2500), ’ 2.5K’).draw(window) Text(Point(-1, 5000), ’ 5.0K’).draw(window) Text(Point(-1, 7500), ’ 7.5k’).draw(window) Text(Point(-1, 10000), ’10.0K’).draw(window) return window def drawBar(window, year, height): bar = Rectangle(Point(year, 0), Point(year+1, height)) bar.setFill("green") bar.setWidth(2) bar.draw(window) def main(): print "This program plots the growth of a 10 year investment." principal = input("Enter the initial principal: ") apr = input("Enter the annualized interest rate: ") win = createLabeledWindow() drawBar(win, 0, principal) for year in range(1, 11): principal = principal * (1 + apr) drawBar(win, year, principal) raw_input("Press <Enter> to quit.") win.close() 97 Although this version is longer than the previous version, experienced programmers would ﬁnd it much easier to understand. As you get used to reading and writing functions, you too will learn to appreciate the elegance of more modular code. 6.7 Exercises 1. In your own words, describe the two motivations for deﬁning functions in your programs. 2. We have been thinking about computer programs as sequences of instructions where the computer methodically executes one instruction and then moves on to the next one. Do programs that contain functions ﬁt this model? Explain your answer. 3. Parameters are an important concept in deﬁning functions. (a) What is the purpose of parameters? (b) What is the difference between a formal parameter and an actual parameter? (c) In what ways are parameters similar to and different from ordinary variables? Slide 106: 98 CHAPTER 6. DEFINING FUNCTIONS 4. Functions can be thought of as miniature (sub)programs inside of other programs. Like any other program, we can think of functions as having input and output to communicate with the main program. (a) How does a program provide “input” to one of its functions? (b) How does a function provide “output” to the program? 5. Consider this very simple function: def cube(x): answer = x * x * x return answer (a) What does this function do? (b) Show how a program could use this function to print the value of y 3 , assuming y is a variable. (c) Here is a fragment of a program that uses this function: answer = 4 result = cube(3) print answer, result The output from this fragment is 4 27. Explain why the output is not 27 27, even though cube seems to change the value of answer to 27. 6. Write a program to print the lyrics of the song “Old MacDonald.” Your program should print the lyrics for ﬁve different animals, similar to the example verse below. Old MacDonald had a farm, Ee-igh, Ee-igh, Oh! And on that farm he had a cow, Ee-igh, Ee-igh, Oh! With a moo, moo here and a moo, moo there. Here a moo, there a moo, everywhere a moo, moo. Old MacDonald had a farm, Ee-igh, Ee-igh, Oh! 7. Write a program to print the lyrics for ten verses of “The Ants Go Marching.” A couple sample verses are given below. You may choose your own activity for the little one in each verse, but be sure to choose something that makes the rhyme work (or almost work). The ants go marching one by one, hurrah! hurrah! The ants go marching one by one, hurrah! hurrah! The ants go marching one by one. The little one stops to suck his thumb And they all go marching down... In the gound... To get out.... Of the rain. Boom! Boom! Boom! The ants go marching two by two, hurrah! hurrah! The ants go marching two by two, hurrah! hurrah! The ants go marching two by two. The little one stops to tie his shoe And they all go marching down... In the ground... To get out... Of the rain. Boom! Boom! Boom! Slide 107: 6.7. EXERCISES 99 8. Redo any of your favorite programming problems from previous chapters and use a function or two to encapsulate the calculations. For example, a program to compute the volume and surface area of a sphere could use functions sphereVol and sphereArea to do the calculations. 9. Redo some more problems.... Slide 108: 100 CHAPTER 6. DEFINING FUNCTIONS Slide 109: Chapter 7 Control Structures, Part 1 So far, we have viewed computer programs as sequences of instructions that are followed one after the other. Sequencing is a fundamental concept of programming, but alone, it is not sufﬁcient to solve every problem. Often it is necessary to alter the sequential ﬂow of a program to suit the needs of a particular situation. This is done with special statements known as control structures. In this chapter, we’ll take a look at decision structures, which are statements that allow a program to execute different sequences of instructions for different cases, effectively allowing the program to “choose” an appropriate course of action. 7.1 Simple Decisions 7.1.1 Example: Temperature Warnings Let’s start by getting the computer to make a simple decision. For an easy example, we’ll return to the Celsius to Fahrenheit temperature conversion program from Chapter 2. Remember, this was written by Suzie Programmer to help her ﬁgure out how to dress each morning in Europe. Here is the program as we left it: # convert.py # A program to convert Celsius temps to Fahrenheit # by: Suzie Programmer def main(): celsius = input("What is the Celsius temperature? ") fahrenheit = 9.0 / 5.0 * celsius + 32 print "The temperature is", fahrenheit, "degrees fahrenheit." main() This is a ﬁne program as far as it goes, but we want to enhance it. Suzie Programmer is not a morning person, and even though she has a program to convert the temperatures, sometimes she does not pay very close attention to the results. Our enhancement to the program will ensure that when the temperatures are extreme, the program prints out a suitable warning so that Suzie takes notice. The ﬁrst step is to fully specify the enhancement. An extreme temperature is either quite hot or quite cold. Let’s say that any temperature over 90 degrees Fahrenheit deserves a heat warning, and a temperature under 30 degrees warrants a cold warning. With this speciﬁcation in mind, we can design an enhanced algorithm. Input the temperature in degrees Celsius (call it celsius) Calculate fahrenheit as 9/5 celsius + 32 Output fahrenheit if fahrenheit > 90 print a heat warning if fahrenheit < 30 print a cold warning 101 Slide 110: 102 CHAPTER 7. CONTROL STRUCTURES, PART 1 This new design has two simple decisions at the end. The indentation indicates that a step should be performed only if the condition listed in the previous line is met. The idea here is that the decision introduces an alternative ﬂow of control through the program. The exact set of steps taken by the algorithm will depend on the value of fahrenheit. Figure 7.1 is a ﬂowchart showing the possible paths that can be taken through the algorithm. The diamond boxes show conditional decisions. If the condition is false, control passes to the next statement in the sequence (the one below). If the condition holds, however, control transfers to the instructions in the box to the right. Once these instructions are done, control then passes to the next statement. Input Celsius Temperature Farenheit = 9/5 * celsius + 32 Print Fahrenheit fahrenheit > 90? yes no Print a Heat Warning fahrenheit < 30? yes no Print a Cold Warning Figure 7.1: Flowchart of temperature conversion program with warnings. Here is how the new design translates into Python code: # convert2.py # A program to convert Celsius temps to Fahrenheit. # This version issues heat and cold warnings. def main(): celsius = input("What is the Celsius temperature? ") fahrenheit = 9.0 / 5.0 * celsius + 32 print "The temperature is", fahrenheit, "degrees fahrenheit." # Print warnings for extreme temps if fahrenheit > 90: print "It’s really hot out there, be careful!" if fahrenheit < 30: print "Brrrrr. Be sure to dress warmly!" main() You can see that the Python if statement is used to implement the decision. The form of the if is very similar to the pseudo-code in the algorithm. if <condition>: <body> Slide 111: 7.1. SIMPLE DECISIONS 103 The body is just a sequence of one or more statements indented under the if heading. In convert2.py there are two if statements, both of which have a single statement in the body. The semantics of the if should be clear from the example above. First, the condition in the heading is evaluated. If the condition is true, the sequence of statements in the body is executed, and then control passes to the next statement in the program. If the condition is false, the statements in the body are skipped. Figure 7.2 shows the semantics of the if as a ﬂowchart. Notice that the body of the if either executes or <condition> true? yes no <Body> Figure 7.2: Control ﬂow of simple if-statement not depending on the condition. In either case, control then passes to the next statement after the if. This is a one-way or simple decision. 7.1.2 Forming Simple Conditions One point that has not yet been discussed is exactly what a condition looks like. For the time being, our programs will use simple conditions that compare the values of two expressions. <expr> <relop> <expr> <relop> is short for relational operator. That’s just a fancy name for the mathematical concepts like “less than” or “equal to.” There are six relational operators in Python, shown in the following table. Python Mathematics ¢ ¡ Notice especially the use of for equality. Since Python uses the sign to indicate an assignment statement, a different symbol is required for the concept of equality. A common mistake in Python programs is using in conditions, where a is required. Conditions may compare either numbers or strings. When comparing strings, the ordering is lexicographic. Basically, this means that strings are put in alphabetic order according to the underlying ASCII codes. So all upper-case letters come before lower case letters (e.g., “Bbbb” comes before “aaaa”, since “B” precedes “a”). I should mention that conditions are actually a type of expression, called a Boolean expression, after George Boole, a 19th century English mathematician. When a Boolean expression is evaluated, it produces a value of either true (the condition holds) or false (it does not hold). In Python, conditions return int values. A true condition produces a 1, while a false condition produces a 0. Here are a few examples: ! Meaning Less than Less than or equal to Equal to Greater than or equal to Greater than Not equal to Slide 112: 104 >>> 1 >>> 0 >>> 1 >>> 0 >>> 1 3<4 34<3+4 "hello" == "hello" "hello" < "hello" "Hello" < "hello" CHAPTER 7. CONTROL STRUCTURES, PART 1 7.1.3 Example: Conditional Program Execution Back in Chapter 1, I mentioned that there are several different ways of running Python programs. Some Python module ﬁles are designed to be run directly. These are usually referred to as “programs” or “scripts.” Other Python modules are designed primarily to be imported and used by other programs, these are often called “libraries.” Sometimes we want to create a sort of hybrid module that can be used both as a standalone program and as a library that can be imported by other programs. So far, all of our programs have had a line at the bottom to invoke the main function. main() As you know, this is what actually starts a program running. These programs are suitable for running directly. In a windowing environment, you might run a ﬁle by (double-)clicking its icon. Or you might type a command like python <myfile>.py. Since Python evaluates the lines of a module during the import process, our current programs also run when they are imported into either an interactive Python session or into another Python program. Generally, it is nicer not to have modules run as they are imported. When testing a program interactively, the usual approach is to ﬁrst import the module and then call its main (or some other function) each time we want to run it. In a program that can be either imported (without running) or run directly, the call to main at the bottom must be made conditional. A simple decision should do the trick. if <condition>: main() We just need to ﬁgure out a suitable condition. Whenever a module is imported, Python sets a special variable in the module called name to be the name of the imported module. Here is an example interaction showing what happens with the math library. >>> import math >>> math.__name__ ’math’ You can see that, when imported, the name variable inside the math module is assigned the string ’math’. However, when Python code is being run directly (not imported), Python sets the value of name to be ’ main ’. To see this in action, you just need to start Python and look at the value. >>> __name__ ’__main__’ So, if a module is imported, the code in that module will see a variable called name whose value is the name of the module. When a ﬁle is run directly, the code will see that name has the value ’ main ’. A program can determine how it is being used by inspecting this variable. Putting the pieces together, we can change the ﬁnal lines of our programs to look like this: Slide 113: 7.2. TWO-WAY DECISIONS if __name__ == ’__main__’: main() 105 This guarantees that main will automatically run when the program is invoked directly, but it will not run if the module is imported. You will see a line of code similar to this at the bottom of virtually every Python program. 7.2 Two-Way Decisions Now that we have a way to selectively execute certain statements in a program using decisions, it’s time to go back and spruce up the quadratic equation solver from Chapter 3. Here is the program as we left it: # quadratic.py # A program that computes the real roots of a quadratic equation. # Illustrates use of the math library. # Note: this program crashes if the equation has no real roots. import math # Makes the math library available. def main(): print "This program finds the real solutions to a quadratic" print a, b, c = input("Please enter the coefficients (a, b, c): ") discRoot = math.sqrt(b b - 4 * a * c) root1 = (-b + discRoot) / (2 * a) root2 = (-b - discRoot) / (2 * a) print print "The solutions are:", root1, root2 main() As noted in the comments, this program crashes when it is given coefﬁcients of a quadratic equation that has no real roots. The problem with this code is that when b2 4ac is less than 0, the program attempts to take the square root of a negative number. Since negative numbers do not have real roots, the math library reports an error. Here’s an example. >>> import quadratic This program finds the real solutions to a quadratic Please enter the coefficients (a, b, c): 1,2,3 Traceback (innermost last): File "<stdin>", line 1, in ? File "quadratic.py", line 21, in ? main() File "quadratic.py", line 14, in main discRoot = math.sqrt(b * b - 4 * a * c) OverflowError: math range error We can use a decision to check for this situation and make sure that the program can’t crash. Here’s a ﬁrst attempt: # quadratic2.py import math ¢ Slide 114: 106 CHAPTER 7. CONTROL STRUCTURES, PART 1 def main(): print "This program finds the real solutions to a quadratic\n" a, b, c = input("Please enter the coefficients (a, b, c): ") dicrim = b * b - 4 * a * c if discrim >= 0: discRoot = math.sqrt(discrim) root1 = (-b + discRoot) / (2 * a) root2 = (-b - discRoot) / (2 * a) print "\nThe solutions are:", root1, root2 This version ﬁrst computes the value of the discriminant (b2 4ac) and then checks to make sure it is not negative. Only then does the program proceed to take the square root and calculate the solutions. This program will never attempt to call math.sqrt when discrim is negative. Incidently, you might also notice that I have replaced the bare print statements from the original version of the program with embedded newlines to put whitespace in the output; you hadn’t yet learned about n the ﬁrst time we encountered this program. Unfortunately, this updated version is not really a complete solution. Study the program for a moment. What happens when the equation has no real roots? According to the semantics for a simple if, when bb - 4ac is less than zero, the program will simply skip the calculations and go to the next statement. Since there is no next statement, the program just quits. Here’s what happens in an interactive session. >>> quadratic2.main() This program finds the real solutions to a quadratic Please enter the coefficients (a, b, c): 1,2,3 >>> This is almost worse than the previous version, because it does not give the user any indication of what went wrong; it just leaves them hanging. A better program would print a message telling the user that their particular equation has no real solutions. We could accomplish this by adding another simple decision at the end of the program. if discrim < 0: print "The equation has no real roots!" This will certainly solve our problem, but this solution just doesn’t feel right. We have programmed a sequence of two decisions, but the two outcomes are mutually exclusive. If discrim >= 0 is true then discrim < 0 must be false and vice versa. We have two conditions in the program, but there is really only one decision to make. Based on the value of discrim The program should either print that there are no real roots or it should calculate and display the roots. This is an example of a two-way decision. Figure 7.3 illustrates the situation. In Python, a two-way decision can be implemented by attaching an else clause onto an if. The result is called an if-else statement. if <condition>: <statements> else: <statements> When the Python interpreter encounters this structure, it will ﬁrst evaluate the condition. If the condition is true, the statements under the if are executed. If the condition is false, the statements under the else are executed. In either case, control then passes to the statement following the if-else. Using a two-way decision in the quadratic solver yields a more elegant solution. ¢ Slide 115: 7.3. MULTI-WAY DECISIONS 107 no discrim < 0 ? yes Calculate roots Print "no roots" Figure 7.3: Quadratic solver as a two-way decision. # quadratic3.py import math def main(): print "This program finds the real solutions to a quadratic\n" a, b, c = input("Please enter the coefficients (a, b, c): ") discrim = b b - 4 * a * c if discrim < 0: print "\nThe equation has no real roots!" else: discRoot = math.sqrt(b * b - 4 * a * c) root1 = (-b + discRoot) / (2 * a) root2 = (-b - discRoot) / (2 * a) print print "\nThe solutions are:", root1, root2 This program ﬁts the bill nicely. Here is a sample session that runs the new program twice. >>> quadratic3.main() This program finds the real solutions to a quadratic Please enter the coefficients (a, b, c): 1,2,3 The equation has no real roots! >>> quadratic3.main() This program finds the real solutions to a quadratic Please enter the coefficients (a, b, c): 2,4,1 The solutions are: -0.292893218813 -1.70710678119 7.3 Multi-Way Decisions The newest version of the quadratic solver is certainly a big improvement, but it still has some quirks. Here is another example run. Slide 116: 108 CHAPTER 7. CONTROL STRUCTURES, PART 1 >>> quadratic3.main() This program finds the real solutions to a quadratic Please enter the coefficients (a, b, c): 1, 2, 1 The solutions are: -1.0 -1.0 This is technically correct; the given coefﬁcients produce an equation that has a double root at -1. However, the output might be confusing to some users. It looks like the program has mistakenly printed the same number twice. Perhaps the program should be a bit more informative to avoid confusion. The double-root situation occurs when discrim is exactly 0. In this case, discRoot is also 0, and b both roots have the value 2a . If we want to catch this special case, it looks like our program actually needs a three-way decision. Here’s a quick sketch of the design. ... Check the value of discrim when < 0: handle the case of no roots when = 0: handle the case of a double root when > 0: handle the case of two distinct roots. One way to code this algorithm is to use two if-else statements. The body of an if or else clause can contain any legal Python statements, including other if or if-else statements. Putting one compound statement inside of another is called nesting. Here’s a fragment of code that uses nesting to achieve a three-way decision : if discrim < 0: print "Equation has no real roots" else: if discrim == 0: root = -b / (2 * a) print "There is a double root at", root else: # Do stuff for two roots If you trace through this code carefully, you will see that there are exactly three possible paths. The sequencing is determined by the value of discrim. A ﬂowchart of this solution is shown in Figure 7.4. You can see that the top-level structure is just an if-else. (Treat the dashed box as one big statement.) The dashed box contains the second if-else nested comfortably inside the else part of the top-level decision. Once again, we have a working solution, but the implementation doesn’t feel quite right. We have ﬁnessed a three-way decision by using two two-way decisions. The resulting code does not reﬂect the true three-fold decision of the original problem. Imagine if we needed to make a ﬁve-way decision using this technique. The if-else structures would nest four levels deep, and the Python code would march off the right-hand edge of the page. There is another way to write multi-way decisions in Python that preserves the semantics of the nested structures but gives it a more appealing look. The idea is to combine an else followed immediately by an if into a single clause called an elif. if <condition1>: <case1 statements> elif <condition2>: <case2 statements> elif <condition3>: <case3 statements> ... else: <default statements> ¡ Slide 117: 7.4. EXCEPTION HANDLING 109 yes discrim < 0 ? no Print "no roots" yes discrim == 0 ? no Do Double Root Do Unique Roots Figure 7.4: Three-way decision for quadratic solver using nested if-else. This form is used to set off any number of mutually exclusive code blocks. Python will evaluate each condition in turn looking for the ﬁrst one that is true. If a true condition is found, the statements indented under that condition are executed, and control passes to the next statement after the entire if-elif-else. If none of the conditions are true, the statements under the else are performed. The else clause is optional; if omitted, it is possible that no indented statement block will be executed. Using an if-elif-else to show the three-way decision in our quadratic solver yields a nicely ﬁnished program. # quadratic4.py import math def main(): print "This program finds the real solutions to a quadratic\n" a, b, c = input("Please enter the coefficients (a, b, c): ") discrim = b * b - 4 * a * c if discrim < 0: print "\nThe equation has no real roots!" elif discrim == 0: root = -b / (2 * a) print "\nThere is a double root at", root else: discRoot = math.sqrt(b * b - 4 * a * c) root1 = (-b + discRoot) / (2 * a) root2 = (-b - discRoot) / (2 * a) print "\nThe solutions are:", root1, root2 7.4 Exception Handling Our quadratic program uses decision structures to avoid taking the square root of a negative number and generating a run-time error. This is a common pattern in many programs: using decisions to protect against Slide 118: 110 CHAPTER 7. CONTROL STRUCTURES, PART 1 rare but possible errors. In the case of the quadratic solver, we checked the data before the call to the sqrt function. Sometimes functions themselves check for possible errors and return a special value to indicate that the operation was unsuccessful. For example, a different square root operation might return a negative number (say -1) to indicate an error. Since the square roots of real numbers are never negative, this value could be used to signal that an error had occurred. The program would check the result of the operation with a decision. discRt = otherSqrt(bb - 4ac) if discRt < 0: print "No real roots." else: ... Sometimes programs become so peppered with decisions to check for special cases that the main algorithm for handling the run-of-the-mill cases seems completely lost. Programming language designers have come up with mechanisms for exception handling that help to solve this design problem. The idea of an exception handling mechanism is that the programmer can write code that catches and deals with errors that arise when the program is running. Rather than explicitly checking that each step in the algorithm was successful, a program with exception handling can in essence say “Do these steps, and if any problem crops up, handle it this way.” We’re not going to discuss all the details of the Python exception handling mechanism here, but I do want to give you a concrete example so you can see how exception handling works and understand programs that use it. In Python, exception handling is done with a special control structure that is similar to a decision. Let’s start with a speciﬁc example and then take a look at the general approach. Here is a version of the quadratic program that uses Python’s exception mechanism to catch potential errors in the math.sqrt function. # quadratic5.py import math def main(): print "This program finds the real solutions to a quadratic\n" try: a, b, c = input("Please enter the coefficients (a, b, c): ") discRoot = math.sqrt(b b - 4 * a * c) root1 = (-b + discRoot) / (2 * a) root2 = (-b - discRoot) / (2 * a) print "\nThe solutions are:", root1, root2 except OverflowError: print "\nNo real roots" Notice that this is basically the very ﬁrst version of the quadratic program with the addition of a try...except around the heart of the program. A try statement has the general form: try: <body> except <ErrorType>: <handler> When Python encounters a try statement, it attempts to execute the statements inside the body. If these statements execute without error, control then passes to the next statement after the try...except. If an error occurs somewhere in the body, Python looks for an except clause with a matching error type. If a suitable except is found, the handler code is executed. The original program without the exception-handling produced the following error. Slide 119: 7.4. EXCEPTION HANDLING Traceback (innermost last): File "<stdin>", line 1, in ? File "quadratic.py", line 13, in ? discRoot = math.sqrt(b * b - 4 * a * c) OverflowError: math range error 111 The last line of this error message indicates the type of error that was generated, namely an OverflowError. The updated version of the program provides an except clause to catch the OverflowError. Here is the error handling version in action: This program finds the real solutions to a quadratic Please enter the coefficients (a, b, c): 1,2,3 No real roots Instead of crashing, the exception handler catches the error and prints a message indicating that the equation does not have real roots. The nice thing about the try...except statement is that it can be used to catch any kind of error, even ones that might be difﬁcult to test for, and hopefully, provide a graceful exit. For example, in the quadratic solver, there are lots of other things that could go wrong besides having a bad set of coefﬁcients. If the user fails to type the correct number of inputs, the program generates a ValueError. If the user accidently types an identiﬁer instead of a number, the program generates a NameError. If the user types in a valid Python expression that produces non-numeric results, the program generates a TypeError. A single try statement can have multiple except clauses to catch various possible classes of errors. Here’s one last version of the program designed to robustly handle any possible errors in the input. # quadratic6.py import math def main(): print "This program finds the real solutions to a quadratic\n" try: a, b, c = input("Please enter the coefficients (a, b, c): ") discRoot = math.sqrt(b * b - 4 * a * c) root1 = (-b + discRoot) / (2 * a) root2 = (-b - discRoot) / (2 * a) print "\nThe solutions are:", root1, root2 except OverflowError: print "\nNo real roots" except ValueError: print "\nYou didn’t give me three coefficients." except NameError: print "\nYou didn’t enter three numbers" except TypeError: print "\nYour inputs were not all numbers" except: print "\nSomething went wrong, sorry!" The multiple excepts are similar to elifs. If an error occurs, Python will try each except in turn looking for one that matches the type of error. The bare except at the bottom acts like an else and will be used if none of the others match. If there is no default at the bottom and none of the except types match the error, then the program crashes and Python reports the error. You can see how the try...except statement allows us to write really bullet-proof programs. Whether you need to go to this much trouble depends on the type of program that you are writing. In your beginning Slide 120: 112 CHAPTER 7. CONTROL STRUCTURES, PART 1 programs, you might not worry too much about bad input; however, professional quality software should do whatever is feasible to shield users from unexpected results. 7.5 Study in Design: Max of Three Now that we have decisions that can alter the control ﬂow of a program, our algorithms are liberated from the monotony of step-by-step, strictly sequential processing. This is both a blessing and a curse. The positive side is that we can now develop more sophisticated algorithms, as we did for our quadratic solver. The negative side is that designing these more sophisticated algorithms is much harder. In this section, we’ll step through the design of a more difﬁcult decision problem to illustrate some of the challenge and excitement of the design process. Suppose we need an algorithm to ﬁnd the largest of three numbers. This algorithm could be part of a larger problem such as determining grades or computing taxes, but we are not interested in the ﬁnal details, just the crux of the problem. That is, how can a computer determine which of three user inputs is the largest? Here is a program outline. We need to ﬁll in the missing part. def main(): x1, x2, x3 = input("Please enter three values: ") # missing code sets max to the value of the largest print "The largest value is", max Before reading the following analysis, you might want to try your hand at solving this problem. 7.5.1 Strategy 1: Compare Each to All Obviously, this program presents us with a decision problem. We need a sequence of statements that sets the value of max to the largest of the three inputs x1, x2, and x3. At ﬁrst glance, this looks like a three-way decision; we need to execute one of the following assignments. max = x1 max = x2 max = x3 It would seem we just need to preface each one with the appropriate condition(s), so that it is executed only in the proper situation. Let’s consider the ﬁrst possibility, that x1 is the largest. To see that x1 is actually the largest, we just need to check that it is at least as large as the other two. Here is a ﬁrst attempt: if x1 >= x2 >= x3: max = x1 Your ﬁrst concern here should be whether this statement is syntactically correct. The condition x1 x2 x3 does not match the template for conditions shown above. Most computer languages would not accept this as a valid expression. It turns out that Python does allow this compound condition, and it behaves exactly like the mathematical relations x1 x2 x3. That is, the condition is true when x1 is at least as large as x2 and x2 is at least as large as x3. So, Python has no problem with this condition. Whenever you write a decision, you should ask yourself two crucial questions. First, when the condition is true, are you absolutely certain that executing the body of the decision is the right action to take? In this case, the condition clearly states that x1 is at least as large as x2 and x3, so assigning its value to max should be correct. Always pay particular attention to borderline values. Notice that our condition includes equal as well as greater. We should convince ourselves that this is correct. Suppose that x1, x2, and x3 are all the same; this condition will return true. That’s OK because it doesn’t matter which we choose, the ﬁrst is at least as big as the others, and hence, the max. ¡ ¡ Slide 121: 7.5. STUDY IN DESIGN: MAX OF THREE 113 The second question to ask is the converse of the ﬁrst. Are we certain that this condition is true in all cases where x1 is the max? Unfortunately, our condition does not meet this test. Suppose the values are 5, 2, and 4. Clearly, x1 is the largest, but our condition returns false since the relationship 5 2 4 does not hold. We need to ﬁx this. We want to ensure that x1 is the largest, but we don’t care about the relative ordering of x2 and x3. What we really need is two separate tests to determine that x1 >= x2 and that x2 >= x3. Python allows us to test multiple conditions like this by combining them with the keyword and. We’ll discuss the exact semantics of and in Chapter 8. Intuitively, the following condition seems to be what we are looking for: if x1 >= x2 and x1 >= x3: max = x1 # x1 is greater than each of the others ¡ ¡ To complete the program, we just need to implement analogous tests for the other possibilities. if x1 >= x2 and x1 >= x3: max = x1 elif x2 >= x1 and x2 >= x3: max = x2 else: max = x3 Summing up this approach, our algorithm is basically checking each possible value against all the others to determine if it is the largest. With just three values the result is quite simple. But how would this solution look if we were trying to ﬁnd the max of ﬁve values? Then we would need four Boolean expressions, each consisting of four conditions anded together. The complex expressions result from the fact that each decision is designed to stand on its own; information from one test is ignored in the following. To see what I mean, look back at our simple max of three code. Suppose the ﬁrst decision discovers that x1 is greater than x2, but not greater than x3. At this point, we know that x3 must be the max. Unfortunately, our code ignores this; Python will go ahead and evaluate the next expression, discover it to be false and ﬁnally execute the else. 7.5.2 Strategy 2: Decision Tree One way to avoid the redundant tests of the previous algorithm is to use a decision tree approach. Suppose we start with a simple test x1 x2. This knocks either x1 or x2 out of contention to be the max. If the condition is true, we just need to see which is larger, x1 or x3. Should the initial condition be false, the result boils down to a choice between x2 and x3. As you can see, the ﬁrst decision “branches” into two possibilities, each of which is another decision. Hence the name, decision tree. Figure 7.5 shows the situation in a ﬂowchart. This ﬂowchart translates easily into nested if-else statements. if x1 >= x2: if x1 >= x3: max = x1 else: max = x3 else: if x2 >= x3: max = x2 else: max = x3 The strength of this approach is its efﬁciency. No matter what the ordering of the three values, this algorithm will make exactly two comparisons and assign the correct value to max. However, the structure of this approach is more complicated than the ﬁrst, and it suffers a similar complexity explosion, should we try this design with more than three values. As a challenge, you might see if you can design a decision tree to ﬁnd the max of four values. (You will need if-elses nested three levels deep leading to eight assignment statements.) Slide 122: 114 CHAPTER 7. CONTROL STRUCTURES, PART 1 yes x1 >= x2 no yes x1 >= x3 no yes x2 > =x3 no max = x1 max = x3 max = x2 max = x3 Figure 7.5: Flowchart of the decision tree approach to max of three 7.5.3 Strategy 3: Sequential Processing So far, we have designed two very different algorithms, but neither one seems particularly elegant. Perhaps there is yet a third way. When designing an algorithm, a good starting place is to ask yourself how you would solve the problem if you were asked to do the job. For ﬁnding the max of three numbers, you probably don’t have a very good intuition about the steps you go through. You’d just look at the numbers and know which is the largest. But what if you were handed a book containing hundreds of numbers in no particular order. How would you ﬁnd the largest in this collection? When confronted with the larger problem, most people develop a simple strategy. Scan through the numbers until you ﬁnd a big one, and put your ﬁnger on it. Continue scanning; if you ﬁnd a number bigger than the one your ﬁnger is on, move your ﬁnger to the new one. When you get to the end of the list, your ﬁnger will remain on the largest value. In a nutshell, this strategy has us look through the list sequentially, keeping track of the largest number seen so far. A computer doesn’t have ﬁngers, but we can use a variable to keep track of the max so far. In fact, the easiest approach is just to use max to do this job. That way, when we get to the end, max automatically contains the value of the largest. A ﬂowchart depicting this strategy for the max of three problem is shown in Figure 7.6. Here is the translation into Python code: max = x1 if x2 > max: max = x2 if x3 > max: max = x3 Clearly, the sequential approach is the best of our three algorithms. The code itself is quite simple, containing only two simple decisions, and the sequencing is easier to understand than the nesting used in the previous algorithm. Furthermore, the idea scales well to larger problems; adding a fourth item adds only one more statement. max = x1 if x2 > max: max = x2 if x3 > max: max = x3 if x4 > max: max = x4 Slide 123: 7.5. STUDY IN DESIGN: MAX OF THREE 115 max = x1 x2 > max max = x2 x3 > max max = x3 Figure 7.6: Flowchart of a sequential approach to the max of three problem. It should not be surprising that the last solution scales to larger problems; we invented the algorithm by explicitly considering how to solve a more complex problem. In fact, you can see that the code is very repetitive. We can easily write a program that allows the user to ﬁnd the largest of n numbers by folding our algorithm into a loop. Rather than having separate variables for x1, x2, x3, etc., we can just get the values one at a time and keep reusing a single variable x. Each time, we compare the newest x against the current value of max to see if it is larger. # program: maxn.py # Finds the maximum of a series of numbers def main(): n = input("How many numbers are there? ") # Set max to be the first value max = input("Enter a number >> ") # Now compare the n-1 successive values for i in range(n-1): x = input("Enter a number >> ") if x > max: max = x print "The largest value is", max main() This code uses a decision nested inside of a loop to get the job done. On each iteration of the loop, max contains the largest value seen so far. Slide 124: 116 CHAPTER 7. CONTROL STRUCTURES, PART 1 7.5.4 Strategy 4: Use Python Before leaving this problem, I really should mention that none of the algorithm development we have so painstakingly pursued was necessary. Python actually has a built-in function called max that returns the largest of its parameters. Here is the simplest version of our program. def main(): x1, x2, x3 = input("Please enter three values: ") print "The largest value is", max(x1, x2, x3) Of course, this version didn’t require any algorithm development at all, which rather defeats the point of the exercise! Sometimes Python is just too simple for our own good.... 7.5.5 Some Lessons The max of three problem is not particularly earth shattering, but the attempt to solve this problem has illustrated some important ideas in algorithm and program design. There is more than one way to do it. For any non-trivial computing problem, there are many ways to approach the problem. While this may seem obvious, many beginning programmers do not really take this point to heart. What does this mean for you? Don’t rush to code up the ﬁrst idea that pops into your head. Think about your design, ask yourself if there is a better way to approach the problem. Once you have written the code, ask yourself again if there might be a better way. Your ﬁrst task is to ﬁnd a correct algorithm. After that, strive for clarity, simplicity, efﬁciency, scalability and elegance. Good algorithms and programs are like poems of logic. They are a pleasure to read and maintain. Be the computer. Especially for beginning programmers, one of the best ways to formulate an algorithm is to simply ask yourself how you would solve the problem. There are other techniques for designing good algorithms (see Chapter 13); however, the straightforward approach is often simple, clear and efﬁcient enough. Generality is good. We arrived at the best solution to the max of three problem by considering the more general max of n problem. It is not unusual that consideration of a more general problem can lead to a better solution for some special case. Don’t be afraid to step back and think about the overarching problem. Similarly, when designing programs, you should always have an eye toward making your program more generally useful. If the max of n program is just as easy to write as max of three, you may as well write the more general program because it is more likely to be useful in other situations. That way you get the maximum utility from your programming effort. Don’t reinvent the wheel. Our fourth solution was to use Python’s max function. You may think that was cheating, but this example illustrates an important point. A lot of very smart programmers have designed countless good algorithms and programs. If the problem you are trying to solve seems to be one that lots of others must have encountered, you might begin by ﬁnding out if the problem has already been solved for you. As you are learning to program, designing from scratch is great experience. Truly expert programmers, however, know when to borrow. 7.6 Exercises 1. Explain the following patterns in your own words. (a) simple decision (b) two-way decision (c) multi-way decision 2. The following is a (silly) decision structure. Slide 125: 7.6. EXERCISES a, b, c = input(’Enter three numbers: ’) if a > b: if b > c: print "Spam Please!" else: print "It’s a late parrot!" elif b > c: print "Cheese Shoppe" if a >= c: print "Cheddar" elif a < b: print "Gouda" elif c == b: print "Swiss" else: print "Trees" if a == b: print "Chestnut" else: print "Larch" print "Done" Show the output that would result from each of the following possible inputs. (a) 3, 4, 5 (b) 3, 3, 3 (c) 5, 4, 3 (d) 3, 5, 2 (e) 5, 4, 7 (f) 3, 3, 2 117 3. Many companies pay time-and-a-half for any hours worked above 40 in a given week. Write a program to input the number of hours worked and the hourly rate and calculate the total wages for the week. 4. A certain CS professor gives 5-point quizzes that are graded on the scale 5-A, 4-B, 3-C, 2-D, 1-F, 0-F. Write a program that accepts a quiz score as an input and uses a decision structure to calculate the corresponding grade. 5. A certain CS professor gives 100-point exams that are graded on the scale 90–100:A, 80–89:B, 70– 79:C, 60–69:D, 60:F. Write a program that accepts an exam score as input and uses a decision structure to calculate the corresponding grade. 6. A certain college classiﬁes students according to credits earned. A student with less than 7 credits is a Freshman. At least 7 credits are required to be a Sophomore, 16 to be a Junior and 26 to be classiﬁed as a Senior. Write a program that calculates class standing from the number of credits earned. 7. The body mass index (BMI) is calculated as a person’s weight (in pounds) times 720, divided by the square of the person’s height (in inches). A BMI in the range 19–25, inclusive, is considered healthy. Write a program that calculates a person’s BMI and prints a message telling whether they are above, within or below the healthy range. 8. The speeding ticket ﬁne policy in Podunksville is $50 plus $5 for each mph over the limit plus a penalty of $200 for any speed over 90 mph. Write a program that accepts a speed limit and a clocked speed and either prints a message indicating the speed was legal or prints the amount of the ﬁne, if the speed is illegal. Slide 126: 118 CHAPTER 7. CONTROL STRUCTURES, PART 1 9. A babysitter charges $2.50 an hour until 9:00 PM when the rate drops to $1.75 an hour (the children are in bed). Write a program that accepts a starting time and ending time in hours and minutes and calculates the total babysitting bill. You may assume that the starting and ending times are in a single 24 hour period. Partial hours should be appropriately prorated. 10. A person is eligible to be a US senator if they are at least 30 years old and have been a US citizen for at least 9 years. To be a US representative these numbers are 25 and 7, respectively. Write a program that accepts a person’s age and years of citizenship as input and outputs their eligibility for the Senate and House. 11. A formula for computing Easter in the years 1982–2048, inclusive, is as follows: let a year%19, b year%4, c year%7, d 19a 24¡ %30, e 2b 4c 6d 5¡ %7. The date of Easter is March 22 d e (which could be in April). Write a program that inputs a year, veriﬁes that it is in the proper range and then prints out the date of Easter that year. 12. The formula for Easter in the previous problem works for every year in the range 1900–2099 except for 1954, 1981, 2049 and 2076. For these 4 years it produces a date that is one week too late. Modify the above program to work for the entire range 1900–2099. 13. A year is a leap year if it is divisible by 4, unless it is a century year that is not divisible by 400. (1800 and 1900 are not leap years while 1600 and 2000 are.) Write a program that calculates whether a year is a leap year. 14. Write a program that accepts a date in the form month/day/year and outputs whether or not the date is valid. For example 5/24/1962 is valid, but 9/31/2000 is not. (September has only 30 days.) 15. The days of the year are often numbered from 1 to through 365 (or 366). This number can be computed in three steps using int arithmetic: ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ (a) dayNum 31 month (c) if it’s a leap year and after February 29, add 1 Write a program that accepts a date as month/day/year, veriﬁes that it is a valid date (see previous problem) and then calculates the corresponding day number. 16. Take a favorite programming problem from a previous chapter and add decisions and/or exception handling as required to make it truly robust (will not crash on any inputs). 17. Archery Scorer. Write a program that draws an archery target (see exercises from Chapter 5) and allows the user to click 5 times to represent arrows shot at the target. Using 5-band scoring, a bulls-eye (yellow) is worth 9 points and each successive ring is worth 2 fewer points down to 1 for white. The program should output a score for each click and keep track of a running sum for the entire series. ¢ (b) if the month is after February subtract 4month 23¡ 10 ¡ 1¡ day Slide 127: Chapter 8 Control Structures, Part 2 In Chapter 7, we looked in detail at the Python if statement and its use in implementing design patterns such as one-way, two-way and multi-way decisions. In this chapter, we’ll wrap up our tour of control structures with a detailed look at loops and Boolean expressions. 8.1 For Loops: A Quick Review You already know that the Python for statement provides a kind of loop. It allows us to iterate through a sequence of values. for <var> in <sequence>: <body> The loop index variable var takes on each successive value in the sequence, and the statements in the body of the loop are executed once for each value. Suppose we want to write a program that can compute the average of a series of numbers entered by the user. To make the program general, it should work for any size set of numbers. You know that an average is calculated by summing up the numbers and dividing by the count of how many numbers there are. We don’t need to keep track of all the numbers that have been entered; we just need a running sum so that we can calculate the average at the end. This problem description should start some bells ringing in your head. It suggests the use of some design patterns you have seen before. We are dealing with a series of numbers—that will be handled by some form of loop. If there are n numbers, the loop should execute n times; we can use the counted loop pattern. We also need a running sum; that calls for a loop accumulator. Putting the two ideas together, we can generate a design for this problem. Input the count of the numbers, n Initialize sum to 0 Loop n times Input a number, x Add x to sum Output average as sum / n Hopefully, you see both the counted loop and accumulator patterns integrated into this design. We can translate this design almost directly into a Python implementation. # average1.py def main(): n = input("How many numbers do you have? ") sum = 0.0 for i in range(n): 119 Slide 128: 120 CHAPTER 8. CONTROL STRUCTURES, PART 2 x = input("Enter a number >> ") sum = sum + x print "\nThe average of the numbers is", sum / n The running sum starts at 0, and each number is added in turn. Notice that sum is initialized to a ﬂoat 0.0. This ensures that the division sum / n on the last line returns a ﬂoat even if all the input values were ints. Here is the program in action. How many numbers do you have? 5 Enter a number >> 32 Enter a number >> 45 Enter a number >> 34 Enter a number >> 76 Enter a number >> 45 The average of the numbers is 46.4 Well, that wasn’t too bad. Knowing a couple of common patterns, counted loop and accumulator, got us to a working program with minimal difﬁculty in design and implementation. Hopefully, you can see the worth of committing these sorts of programming clich´ s to memory. e 8.2 Indeﬁnite Loops Our averaging program is certainly functional, but it doesn’t have the best user interface. It begins by asking the user how many numbers there are. For a handful of numbers this is OK, but what if I have a whole page of numbers to average? It might be a signiﬁcant burden to go through and count them up. It would be much nicer if the computer could take care of counting the numbers for us. Unfortunately, as you no doubt recall, the for loop is a deﬁnite loop, and that means the number of iterations is determined when the loop starts. We can’t use a deﬁnite loop unless we know the number of iterations ahead of time, and we can’t know how many iterations this loop needs until all of the numbers have been entered. We seem to be stuck. The solution to this dilemma lies in another kind of loop, the indeﬁnite or conditional loop. An indeﬁnite loop keeps iterating until certain conditions are met. There is no guarantee ahead of time regarding how many times the loop will go around. In Python, an indeﬁnite loop is implemented using a while statement. Syntactically, the while is very simple. while <condition>: <body> Here condition is a Boolean expression, just like in if statements. The body is, as usual, a sequence of one or more statements. The semantics of while is straightforward. The body of the loop executes repeatedly as long as the condition remains true. When the condition is false, the loop terminates. Figure 8.1 shows a ﬂowchart for the while. Notice that the condition is always tested at the top of the loop, before the loop body is executed. This kind of structure is called a pre-test loop. If the loop condition is initially false, the loop body will not execute at all. Here is an example of a simple while loop that counts from 0 to 10: i=0 while i <= 10: print i i=i+1 This code will have the same output as if we had written a for loop like this: Slide 129: 8.3. COMMON LOOP PATTERNS 121 <condition>? no yes <body> Figure 8.1: Flowchart of a while loop. for i in range(11): print i Notice that the while version requires us to take care of initializing i before the loop and incrementing i at the bottom of the loop body. In the for loop, the loop variable is handled automatically. The simplicity of the while statement makes it both powerful and dangerous. Because it is less rigid, it is more versatile; it can do more than just iterate through sequences. But it is also a common source of errors. Suppose we forget to increment i at the bottom of the loop body in the counting example. i=0 while i <= 10: print i What will the output from this program be? When Python gets to the loop, i will be 0, which is less than 10, so the loop body executes, printing a 0. Now control returns to the condition; i is still 0, so the loop body executes again, printing a 0. Now control returns to the condition; i is still 0, so the loop body executes again, printing a 0.... You get the picture. This is an example of an inﬁnite loop. Usually, inﬁnite loops are a bad thing. Clearly this version of the program does nothing useful. That reminds me, did you hear about the computer scientist who died of exhaustion while washing his hair? The instructions on the bottle said: “Lather. Rinse. Repeat.” As a beginning programmer, it would surprising if you did not accidently write a few programs with inﬁnite loops—it’s a rite of passage for programmers. Even more experienced programmers have been known to do this from time to time. Usually, you can break out of a loop by pressing Ctrl -c (holding down the Ctrl key and pressing “c”). If your loop is really tight, this might not work, and you’ll have to resort to more drastic means (such as Ctrl - Alt - Delete on a PC). If all else fails, there is always the trusty reset button on your computer. The best idea is to avoid writing inﬁnite loops in the ﬁrst place. 8.3 Common Loop Patterns 8.3.1 Interactive Loops One good use of the indeﬁnite loop is to write interactive loops. The idea behind an interactive loop is that it allows the user to repeat certain portions of a program on demand. Let’s take a look at this loop pattern in the context of our number averaging problem. Slide 130: 122 CHAPTER 8. CONTROL STRUCTURES, PART 2 Recall that the previous version of the program forced the user to count up how many numbers there were to be averaged. We want to modify the program so that it keeps track of how many numbers there are. We can do this with another accumulator, call it count, that starts at zero and increases by 1 each time through the loop. To allow the user to stop at any time, each iteration of the loop will ask whether there is more data to process. The general pattern for an interactive loop looks like this: set moredata to "yes" while moredata is "yes" get the next data item process the item ask user if there is moredata Combining the interactive loop pattern with accumulators for the sum and count yields this algorithm for the averaging program. initialize sum to 0.0 initialize count to 0 set moredata to "yes" while moredata is "yes" input a number, x add x to sum add 1 to count ask user if there is moredata output sum / count Notice how the two accumulators are interleaved into the basic structure of the interactive loop. Here is the corresponding Python program: # average2.py def main(): sum = 0.0 count = 0 moredata = "yes" while moredata[0] == "y": x = input("Enter a number >> ") sum = sum + x count = count + 1 moredata = raw_input("Do you have more numbers (yes or no)? ") print "\nThe average of the numbers is", sum / count Notice this program uses string indexing (moredata[0]) to look just at the ﬁrst letter of the user’s input. This allows for varied responses such as “yes,” “y,” “yeah,” etc. All that matters is that the ﬁrst letter is a “y.” Also note the use of raw input to get this value. Remember you should use raw input to get string data. Here is sample output from this program. Enter a number >> 32 Do you have more numbers Enter a number >> 45 Do you have more numbers Enter a number >> 34 Do you have more numbers Enter a number >> 76 Do you have more numbers Enter a number >> 45 Do you have more numbers (yes or no)? yes (yes or no)? y (yes or no)? y (yes or no)? y (yes or no)? nope Slide 131: 8.3. COMMON LOOP PATTERNS 123 The average of the numbers is 46.5 In this version, the user doesn’t have to count the data values, but the interface is still not good. The user will almost certainly be annoyed by the constant prodding for more data. The interactive loop has many good applications; this is not one of them. 8.3.2 Sentinel Loops A better solution to the number averaging problem is to employ a pattern commonly known as a sentinel loop. A sentinel loop continues to process data until reaching a special value that signals the end. The special value is called the sentinel. Any value may be chosen for the sentinel. The only restriction is that it be distinguishable from actual data values. The sentinel is not processed as part of the data. Here is a general pattern for designing sentinel loops: get the first data item while item is not the sentinel process the item get the next data item Notice how this pattern avoids processing the sentinel item. The ﬁrst item is retrieved before the loop starts. This is sometimes called the priming read, as it gets the process started. If the ﬁrst item is the sentinel, the loop immediately terminates and no data is processed. Otherwise, the item is processed and the next one is read. The loop test at the top ensures this next item is not the sentinel before processing it. When the sentinel is reached, the loop terminates. We can apply the sentinel pattern to our number averaging problem. The ﬁrst step is to pick a sentinel. Suppose we are using the program to average exam scores. In that case, we can safely assume that no score will be below 0. The user can enter a negative number to signal the end of the data. Combining the sentinel loop with the two accumulators from the interactive loop version yields this program. # average3.py def main(): sum = 0.0 count = 0 x = input("Enter a number (negative to quit) >> ") while x >= 0: sum = sum + x count = count + 1 x = input("Enter a number (negative to quit) >> ") print "\nThe average of the numbers is", sum / count I have changed the prompt so that the user knows how to signal the end of the data. Notice that the prompt is identical at the priming read and the bottom of the loop body. Now we have a useful form of the program. Here it is in action: Enter Enter Enter Enter Enter Enter a a a a a a number number number number number number (negative (negative (negative (negative (negative (negative to to to to to to quit) quit) quit) quit) quit) quit) >> >> >> >> >> >> 32 45 34 76 45 -1 The average of the numbers is 46.4 This version provides the ease of use of the interactive loop without the hassle of having to type “yes” all the time. The sentinel loop is a very handy pattern for solving all sorts of data processing problems. It’s another clich´ that you should commit to memory. e Slide 132: 124 CHAPTER 8. CONTROL STRUCTURES, PART 2 This sentinel loop solution is quite good, but there is still a limitation. The program can’t be used to average a set of numbers containing negative as well as positive values. Let’s see if we can’t generalize the program a bit. What we need is a sentinel value that is distinct from any possible valid number, positive or negative. Of course, this is impossible as long as we restrict ourselves to working with numbers. No matter what number or range of numbers we pick as a sentinel, it is always possible that some data set may contain such a number. In order to have a truly unique sentinel, we need to broaden the possible inputs. Suppose that we get the input from the user as a string. We can have a distinctive, non-numeric string that indicates the end of the input; all others would be converted into numbers and treated as data. One simple solution is to have the sentinel value be an empty string. Remember, an empty string is represented in Python as "" (quotes with no space between). If the user types a blank line in response to a raw input (just hits Enter ), Python returns an empty string. We can use this as a simple way to terminate input. The design looks like this: Initialize sum to 0.0 Initialize count to 0 Input data item as a string, xStr while xStr is not empty Convert xStr to a number, x Add x to sum Add 1 to count Input next data item as a string, xStr Output sum / count Comparing this to the previous algorithm, you can see that converting the string to a number has been added to the processing section of the sentinel loop. Translating into Python yields this program: # average4.py def main(): sum = 0.0 count = 0 xStr = raw_input("Enter a number (<Enter> to quit) >> ") while xStr != "": x = eval(xStr) sum = sum + x count = count + 1 xStr = raw_input("Enter a number (<Enter> to quit) >> ") print "\nThe average of the numbers is", sum / count This code makes use of eval (from Chapter 4) to convert the input string into a number. Here is an example run, showing that it is now possible to average arbitrary sets of numbers: Enter Enter Enter Enter Enter Enter Enter a a a a a a a number number number number number number number (<Enter> (<Enter> (<Enter> (<Enter> (<Enter> (<Enter> (<Enter> to to to to to to to quit) quit) quit) quit) quit) quit) quit) >> >> >> >> >> >> >> 34 23 0 -25 -34.4 22.7 The average of the numbers is 3.38333333333 We ﬁnally have an excellent solution to our original problem. You should study this solution so that you can incorporate these techniques into your own programs. Slide 133: 8.3. COMMON LOOP PATTERNS 125 8.3.3 File Loops One disadvantage of all the averaging programs presented so far is that they are interactive. Imagine you are trying to average 87 numbers and you happen to make a typo near the end. With our interactive program, you will need to start all over again. A better approach to the problem might be to type all of the numbers into a ﬁle. The data in the ﬁle can be perused and edited before sending it to a program that generates a report. This ﬁle-oriented approach is typically used for data processing applications. Back in Chapter 4, we looked at reading data from ﬁles using the Python readlines method and a for loop. We can apply this technique directly to the number averaging problem. Assuming that the numbers are typed into a ﬁle one per line, we can compute the average with this program. # average5.py def main(): fileName = raw_input("What file are the numbers in? ") infile = open(fileName,’r’) sum = 0.0 count = 0 for line in infile.readlines(): sum = sum + eval(line) count = count + 1 print "\nThe average of the numbers is", sum / count In this code, readlines reads the entire ﬁle into memory as a sequence of strings. The loop variable line then iterates through this sequence; each line is converted to a number and added to the running sum. One potential problem with this kind of ﬁle processing loop is that the entire contents of the ﬁle are ﬁrst read into main memory via the readlines method. As you know from Chapter 1, secondary memory where ﬁles reside is usually much larger than the primary memory. It’s possible that a large data ﬁle may not ﬁt into memory all at one time. In that case, this approach to ﬁle processing will be very inefﬁcient and, perhaps, not work at all. With very large data ﬁles, it is better to read and process small sections of the ﬁle at a time. In the case of text ﬁles, a simple approach is to process the ﬁle one line at a time. This is easily accomplished by a sentintel loop and the Python readline method on ﬁles. Recall, the readline method gets the next line from a ﬁle as a string. When we come to the end of the ﬁle, readline returns an empty string, which we can use as a sentinel value. Here is a general pattern for an end-of-ﬁle loop in Python. line = infile.readline() while line != "": # process line line = infile.readline() At ﬁrst glance, you may be concerned that this loop stops prematurely if it encounters an empty line in the ﬁle. This is not the case. Remember, a blank line in a text ﬁle contains a single newline character (" n"), and the readline method includes the newline character in its return value. Since " n" != "", the loop will continue. Here is the code that results from applying the end-of-ﬁle sentinel loop to our number averaging problem. # average6.py def main(): fileName = raw_input("What file are the numbers in? ") infile = open(fileName,’r’) sum = 0.0 count = 0 Slide 134: 126 CHAPTER 8. CONTROL STRUCTURES, PART 2 line = infile.readline() while line != "": sum = sum + eval(line) count = count + 1 line = infile.readline() print "\nThe average of the numbers is", sum / count Obviously, this version is not quite as concise as the version using readlines and a for loop. If the ﬁle sizes are known to be relatively modest, that approach is probably better. When ﬁle sizes are quite large, however, an end-of-ﬁle loop is invaluable. 8.3.4 Nested Loops In the last chapter, you saw how control structures such as decisions and loops could be nested inside one another to produce sophisticated algorithms. One particularly useful, but somewhat tricky technique is the nesting of loops. Let’s take a look at an example program. How about one last version of our number averaging problem? Honest, I promise this is the last time I’ll use this example.1 Suppose we modify the speciﬁcation of our ﬁle averaging problem slightly. This time, instead of typing the numbers into the ﬁle one-per-line, we’ll allow any number of values on a line. When multiple values appear on a line, they will be separated by commas. At the top level, the basic algorithm will be some sort of ﬁle-processing loop that computes a running sum and count. For practice, let’s use an end-of-ﬁle loop. Here is the code comprising the top-level loop. sum = 0.0 count = 0 line = infile.readline() while line != "": # update sum and count for values in line line = infile.readline() print "\nThe average of the numbers is", sum / count Now we need to ﬁgure out how to update the sum and count in the body of the loop. Since each individual line of the ﬁle contains one or more numbers separated by commas, we can split the line into substrings, each of which represents a number. Then we need to loop through these substrings, convert each to a number, and add it to sum. We also need to add 1 to count for each number. Here is a code fragment that processes a line: for xStr in string.split(line,","): sum = sum + eval(xStr) count = count +1 Notice that the iteration of the for loop in this fragment is controlled by the value of line, which just happens to be the loop-control variable for the ﬁle-processing loop we outlined above. Knitting these two loops together, here is our program. # average7.py import string def main(): fileName = raw_input("What file are the numbers in? ") infile = open(fileName,’r’) sum = 0.0 count = 0 line = infile.readline() while line != "": 1 until Chapter 11... Slide 135: 8.4. COMPUTING WITH BOOLEANS # update sum and count for values in line for xStr in string.split(line): sum = sum + eval(xStr) count = count + 1 line = infile.readline() print "\nThe average of the numbers is", sum / count 127 As you can see, the loop that processes the numbers in a line is indented inside of the ﬁle processing loop. The outer while loop iterates once for each line of the ﬁle. On each iteration of the outer loop, the inner for loop iterates as many times as there are numbers on that line. When the inner loop ﬁnishes, the next line of the ﬁle is read, and the outer loop goes through its next iteration. The individual fragments of this problem are not complex when taken separately, but the ﬁnal result is fairly intricate. The best way to design nested loops is to follow the process we did here. First design the outer loop without worrying about what goes inside. Then design what goes inside, ignoring the outer loop(s). Finally, put the pieces together, taking care to preserve the nesting. If the individual loops are correct, the nested result will work just ﬁne; trust it. With a little practice, you’ll be implementing double-, even triplenested loops with ease. 8.4 Computing with Booleans We now have two control structures, if and while, that use conditions, which are Boolean expressions. Conceptually, a Boolean expression evaluates to one of two values: false or true. In Python, these values are represented by the ints 0 and 1. So far, we have used simple Boolean expressions that compare two values (e.g., while x >= 0). 8.4.1 Boolean Operators Sometimes the simple conditions that we have been using do not seem expressive enough. For example, suppose you need to determine whether two point objects are in the same position—that is, they have equal x coordinates and equal y coordinates. One way of handling this would be a nested decision. if p1.getX() == p2.getX(): if p1.getY() == p2.getY(): # points are the same else: # points are different else: # points are different You can see how awkward this is. Instead of working around this problem with a decision structure, another approach would be to construct a more complex expression using Boolean operations. Like most programming languages, Python provides three Boolean operators: and, or and not. Let’s take a look at these three operators and then see how they can be used to simplify our problem. The Boolean operators and and or are used to combine two Boolean expressions and produce a Boolean result. <expr> and <expr> <expr> or <expr> The and of two expressions is true exactly when both of the expressions are true. We can represent this deﬁnition in a truth table. Slide 136: 128 P T T F F Q T F T F CHAPTER 8. CONTROL STRUCTURES, PART 2 P and Q T F F F In this table, P and Q represent smaller Boolean expressions. Since each expression has two possible values, there are four possible combinations of values, each shown as one row in the table. The last column gives the value of P and Q for each possible combination. By deﬁnition, the and is true only in the case where both P and Q are true. The or of two expressions is true when either expression is true. Here is the truth table deﬁning or: P T T F F Q T F T F P or Q T T T F The only time the or is false is when both expressions are false. Notice especially that or is true when both expressions are true. This is the mathematical deﬁnition of or, but the word “or” is sometimes used in an exclusive sense in everyday English. If your mom said that you could have cake or cookies for dessert, she would probably scold you for taking both. The not operator computes the opposite of a Boolean expression. It is a unary operator, meaning that it operates on a single expression. The truth table is very simple. P T F not P F T Using Boolean operators, it is possible to build arbitrarily complex Boolean expressions. As with arithmetic operators, the exact meaning of a complex expression depends on the precedence rules for the operators. Consider this expression. a or not b and c How should this be evaluated? Python follows a standard convention that the order of precedence is not, followed by and, followed by or. So the expression would be equivalent to this parenthesized version. (a or ((not b) and c)) Unlike arithmetic, however, most people don’t tend to know or remember the precedence rules for Booleans. I suggest that you always parenthesize your complex expressions to prevent confusion. Now that we have some Boolean operators, we are ready to return to our example problem. To test for the co-location of two points, we could use an and operation. if p1.getX() == p2.getX() and p2.getY() == p1.getY(): # points are the same else: # points are different Here the entire expression will only be true when both of the simple conditions are true. This ensures that both the x and y coordinates have to match for the points to be the same. Obviously, this is much simpler and clearer than the nested ifs from the previous version. Let’s look at a slightly more complex example. In the next chapter, we will develop a simulation for the game of racquetball. Part of the simulation will need to determine when a game has ended. Suppose that scoreA and scoreB represent the scores of two racquetball players. The game is over as soon as either of the players has reached 15 points. Here is a Boolean expression that is true when the game is over: Slide 137: 8.4. COMPUTING WITH BOOLEANS scoreA == 15 or scoreB == 15 129 When either score reaches 15, one of the two simple conditions becomes true, and, by deﬁnition of or, the entire Boolean expression is true. As long as both conditions remain false (neither player has reached 15) the entire expression is false. Our simulation will need a loop that continues as long as the game is not over. We can construct an appropriate loop condition by taking the negation of the game-over condition: while not (scoreA == 15 or scoreB == 15): # continue playing We can also construct more complex Boolean expressions that reﬂect different possible stopping conditions. Some racquetball players play shutouts (sometimes called a skunk). For these players, a game also ends when one of the players reaches 7 and the other has not yet scored a point. For brevity, I’ll use a for scoreA and b for scoreB. Here is an expression for game-over when shutouts are included: a == 15 or b == 15 or (a == 7 and b == 0) or (b == 7 and a == 0) Do you see how I have added two more situations to the original condition? The new parts reﬂect the two possible ways a shutout can occur, and each requires checking both scores. The result is a fairly complex expression. While we’re at it, let’s try one more example. Suppose we were writing a simulation for volleyball, rather than racquetball. Volleyball does not have shutouts, but it requires a team to win by at least two points. If the score is 15 to 14, or even 21 to 20, the game continues. Let’s write a condition that computes when a volleyball game is over. Here’s one approach. (a >= 15 and a - b >= 2) or (b >= 15 and b - a >= 2) Do you see how this expression works? It basically says the game is over when team A has won (scored at least 15 and leading by at least 2) or when team B has won. Here is another way to do it. (a >= 15 or b >= 15) and abs(a - b) >= 2 This version is a bit more succinct. It states that the game is over when one of the teams has reached a winning total and the difference in the scores is at least 2. Remember that abs returns the absolute value of an expression. 8.4.2 Boolean Algebra All decisions in computer programs boil down to appropriate Boolean expressions. The ability to formulate, manipulate and reason with these expressions is an important skill for programmers and computer scientists. Boolean expressions obey certain algebraic laws similar to those that apply to numeric operations. These laws are called Boolean logic or Boolean algebra. Let’s look at a few examples. The following table shows some rules of algebra with their correlates in Boolean algebra. Algebra a0 0 a1 a a0a Boolean algebra a and false == false a and true == a a or false == a From these examples, you can see that and has similarities to multiplication, and or has similarities to addition; while 0 and 1 correspond to false and true. Here are some other interesting properties of Boolean operations. Anything ored with true is just true. a or true == true Both and and or distribute over each other. ¢ Slide 138: 130 CHAPTER 8. CONTROL STRUCTURES, PART 2 a or (b and c) == (a or b) and (a or c) a and (b or c) == (a and b) or (a and c) A double negative cancels out. not(not a) == a The next two identities are known as DeMorgan’s laws. not(a or b) == (not a) and (not b) not(a and b) == (not a) or (not b) Notice how the operator changes between and and or when the not is pushed into an expression. One application of Boolean algebra is the analysis and simpliﬁcation of Boolean expressions inside of programs. For example, let’s go back to the racquetball game one more time. Above, we developed a loop condition for continuing the game that looked like this: while not (scoreA == 15 or scoreB == 15): # continue playing You can read this condition as something like: While it is not the case that player A has 15 or player B has 15, continue playing. We’re pretty sure that’s correct, but negating complex conditions like this can be somewhat awkward, to say the least. Using a little Boolean algebra, we can transform this result. Applying DeMorgan’s law, we know that the expression is equivalent to this. (not scoreA == 15) and (not scoreB == 15) Remember, we have to change the or to and when “distributing” the not. This condition is no better than the ﬁrst, but we can go one step farther by pushing the nots into the conditions themselves. while scoreA != 15 and scoreB != 15: # continue playing Now we have a version that is much easier to understand. This reads simply as while player A has not reached 15 and player B has not reached 15, continue playing. This particular example illustrates a generally useful approach to loop conditions. Sometimes it’s easier to ﬁgure out when a loop should stop, rather than when the loop should continue. In that case, simply write the loop termination condition and then put a not in front of it. An application or two of DeMorgan’s laws can then get you to a simpler but equivalent version suitable for use in a while statement. 8.5 Other Common Structures Taken together, the decision structure (if) along with a pre-test loop (while) provide a complete set of control structures. This means that every algorithm can be expressed using just these. Once you’ve mastered the while and the if, there is no algorithm that you cannot write, in principle. However, for certain kinds of problems, alternative structures can sometimes be convenient. This section outlines some of those alternatives. 8.5.1 Post-Test Loop Suppose you are writing an input algorithm that is supposed to get a nonnegative number from the user. If the user types an incorrect input, the program asks for another value. It continues to reprompt until the user enters a valid value. This process is called input validation. Well-engineered programs validate inputs whenever possible. Here is a simple algorithm. repeat get a number from the user until number is >= 0 Slide 139: 8.5. OTHER COMMON STRUCTURES 131 The idea here is that the loop keeps getting inputs until the value is acceptable. The ﬂowchart depicting this design in shown in Figure 8.2. Notice how this algorithm contains a loop where the condition test comes after the loop body. This is a post-test loop. A post-test loop must always execute the body of the loop at least once. Get a number number < 0 ? yes no Figure 8.2: Flowchart of a post-test loop. Unlike some other languages, Python does not have a statement that directly implements a post-test loop. However, this algorithm can be implemented with a while by “seeding” the loop condition for the ﬁrst iteration. number = -1 # Start with an illegal value to get into the loop. while number < 0: number = input("Enter a positive number: ") This forces the loop body to execute at least once and is equivalent to the post-test algorithm. You might notice that this is similar to the structure given earlier for the interactive loop pattern. Interactive loops are naturally suited to a post-test implementation. Some programmers prefer to simulate a post-test loop more directly by using a Python break statement. Executing break causes Python to immediately exit the enclosing loop. Often a break statement is used to leave what looks syntactically like an inﬁnite loop. Here is the same algorithm implemented with a break. while 1: number = input("Enter a positive number: ") if x >= 0: break # Exit loop if number is valid. The ﬁrst line may look a bit strange to you. Remember that conditions in Python evaluate to either a 0 for false or a 1 for true. The heading while 1 appears to be an inﬁnite loop, since the expression always evaluates to 1 (i.e., it is always true). However, when the value of x is nonnegative, the break statement executes, which terminates the loop. Notice the break is placed on the same line as the if. This is legal when the body of the if only contains one statement. It’s common to see a one-line if-break combination used as a loop exit. Even this small example can be improved. It would be nice if the program issued a warning explaining why the input was invalid. In the while version of the post-test loop, this is a bit awkward. We need to add an if so that the warning is not displayed for valid inputs. Slide 140: 132 CHAPTER 8. CONTROL STRUCTURES, PART 2 number = -1 # Start with an illegal value to get into the loop. while number < 0: number = input("Enter a positive number: ") if number < 0: print "The number you entered was not positive" See how the validity check gets repeated in two places? Adding a warning to the version using break only requires adding an else to the existing if. while 1: number = input("Enter a positive number: ") if x >= 0: break # Exit loop if number is valid. else: print "The number you entered was not positive" 8.5.2 Loop and a Half Some programmers would solve the warning problem from the previous section using a slightly different style. while 1: number = input("Enter a positive number: ") if x >= 0: break # Loop exit print "The number you entered was not positive" Here the loop exit is actually in the middle of the loop body. This is called a loop and a half. Some purists frown on exits in the midst of a loop like this, but the pattern can be quite handy. The loop and a half is an elegant way to avoid the priming read in a sentinel loop. Here is the general pattern of a sentinel loop implemented as a loop and a half. while 1: Get next data item if the item is the sentinel: break process the item Figure 8.3 shows a ﬂowchart of this approach to sentinel loops. You can see that this implementation is faithful to the ﬁrst rule of sentinel loops: avoid processing the sentinel value. The choice of whether to use break statements or not is largely a matter of taste. Either style is acceptable. One temptation that should generally be avoided is peppering the body of a loop with multiple break statements. The logic of a loop is easily lost when there are multiple exits. However, there are times when even this rule should be broken to provide the most elegant solution to a problem. 8.5.3 Boolean Expressions as Decisions So far, we have talked about Boolean expressions only within the context of other control structures. Sometimes, Boolean expressions themselves can act as control structures. In fact, Boolean expressions are so ﬂexible in Python that they can sometimes lead to subtle programming errors. Consider writing an interactive loop that keeps going as long as the user response starts with a “y.” To allow the user to type either an upper or lower case response, you could use a loop like this: while response[0] == "y" or response[0] == "Y": You must be be careful not to abbreviate this condition as you might think of it in English: “While the ﬁrst letter is ’y’ or ’Y”’. The following form does not work. while response[0] == "y" or "Y": Slide 141: 8.5. OTHER COMMON STRUCTURES 133 Get next Data item Item is the sentinel ? yes no Process the item Figure 8.3: Loop-and-a-half implementation of sentinel loop pattern In fact, this is an inﬁnite loop. Understanding why this condition is always true requires digging into some idiosyncrasies of Python Boolean expressions. You already know that Python does not have any special Boolean type. We have been using the int values 0 and 1 to represent the Boolean values false and true, respectively. The Python condition operators (i.e., ==) always evaluate to either 0 or 1. However, Python is actually very ﬂexible about what can be a Boolean expression. Any built-in type can be interpreted as a Boolean. For numbers (ints, ﬂoats and long ints) a zero value is considered as false, anything other than zero is taken as true. Other types can also be used in Boolean expressions. For example, Python interprets an empty string as false and any nonempty string as true. The ﬂexibility of Python Booleans extends to the Boolean operators. Although the main use of these operators is forming Boolean expressions, they have operational deﬁnitions that make them useful for other purposes as well. This table summarizes the behavior of these operators. operator x and y x or y not x operational deﬁnition If x is false, return x. Otherwise, return y. If x is false, return y. Otherwise, return x. If x is false, return 1. Otherwise, return 0. The deﬁnition of not is straightforward. It might take a bit of thinking to convince yourself that these descriptions of and and or faithfully reﬂect the truth tables you saw at the beginning of the chapter. Consider the expression x and y. In order for this to be true, both expressions x and y must be true. As soon as one of them is discovered to be false, the party is over. Python looks at the expressions left-to-right. If x is false, Python should return a false result. Whatever the false value of x was, that is what is returned. If x turns out to be true, then the truth or falsity of the whole expression turns on the result of y. Simply returning y guarantees that if y is true, the whole result is true, and if y is false, the whole result is false. Similar reasoning can be used to show that the description of or is faithful to the logical deﬁnition of or given in the truth table. These operational deﬁnitions show that Python’s Boolean operators are short-circuit operators. That means that a true or false value is returned as soon as the result is known. In an and where the ﬁrst expression is false and in an or where the ﬁrst expression is true, Python will not even evaluate the second expression. Now let’s take a look at our inﬁnite loop problem. Slide 142: 134 response[0] == "y" or "Y" CHAPTER 8. CONTROL STRUCTURES, PART 2 Treated as a Boolean expression, this will always evaluate to true. The ﬁrst thing to notice is that the Boolean operator is combining two expressions; the ﬁrst is a simple condition, and the second is a string. Here is an equivalent parenthesized version: (response[0] == "y") or ("Y"): By the operational description of or, this expression returns either 1 (returned by == when response[0] is “y”) or "Y" (when response[0] is not a “y”). Either of these results is interpreted by Python as true. A more logic-oriented way to think about this is to simply look at the second expression. It is a nonempty string, so Python will always interpret it as true. Since at least one of the two expressions is always true, the or of the expressions must always be true as well. So, the strange behavior of this example is due to some quirks in the deﬁnitions of the Boolean operators. This is one of the few places where the design of Python has a potential pitfall for the beginning programmer. You may wonder about the wisdom of this design, yet the ﬂexibility of Python allows for certain succinct programming idioms that many programmers ﬁnd useful. Let’s look at an example. Frequently, programs prompt users for information but offer a default value for the response. The default value, sometimes listed in square brackets, is used if the user simply hits the Enter key. Here is an example code fragment: ans = raw_input("What flavor do you want [vanilla]: ") if ans != "": flavor = ans else: flavor = "vanilla" Exploiting the fact that the string in ans can be treated as a Boolean, the condition in this code can be simpliﬁed as follows. ans = raw_input("What flavor do you want [vanilla]: ") if ans: flavor = ans else: flavor = "vanilla" Here a Boolean condition is being used to decide how to set a string variable. If the user just hits Enter , ans will be an empty string, which Python interprets as false. In this case, the empty string will be replaced by "vanilla" in the else clause. The same idea can be succinctly coded by treating the strings themselves as Booleans and using an or. ans = raw_input("What flavor do you want [vanilla]: ") flavor = ans or "vanilla" The operational deﬁnition of or guarantees that this is equivalent to the if-else version. Remember, any nonempty answer is interpreted as “true.” In fact, this task can easily be accomplished in a single line of code. flavor = raw_input("What flavor do you want [vanilla]: ") or "vanilla" I don’t know whether it’s really worthwhile to save a few lines of code using Boolean operators this way. If you like this style, by all means, feel free to use it. Just make sure that your code doesn’t get so tricky that others (or you) have trouble understanding it. 8.6 Exercises 1. Compare and contrast the following pairs of terms. Slide 143: 8.6. EXERCISES (a) Deﬁnite loop vs. Indeﬁnite loop (b) For loop vs. While loop (c) Interactive loop vs. Sentinel loop (d) Sentinel loop vs. End-of-ﬁle loop 135 2. Give a truth table that shows the (Boolean) value of each of the following Boolean expressions, for every possible combination of “input” values. Hint: including columns for “intermediate” expressions is helpful. (a) not (P and Q) (b) (not P) and Q (c) (not P) or (not Q) (d) (P and Q) or R (e) (P or R) and (Q or R) 3. Write a while loop fragment that calculates the following values. ¢ ¢ ¢ ¡ ¢ ¢ ¢ ¢ ¢ (a) Sum of the ﬁrst n counting numbers: 1 ¢ ¢ 2 ¢ 3 n 1 (b) Sum of the ﬁrst n odd numbers: 1 3 5 2n (c) Sum of a series of numbers entered by the user until the value 999 is entered. Note: 999 should not be part of the sum. (d) The number of times a whole number n can be divided by 2 (using integer division) before reaching 1 (i.e., log2 n). 4. The Fibonacci sequence starts 1¤ 1¤ 2¤ 3¤ 5¤ 8¤ . Each number in the sequence (after the ﬁrst two) is the sum of the previous two. Write a program that computes and outputs the nth Fibonacci number, where n is a value entered by the user. 5. The National Weather Service computes the windchill index using the following formula. ¡ ¢ ¢ 35 74 0 6215T Where T is the temperature in degrees Fahrenheit, and V is the wind speed in miles per hour. Write a program that prints a nicely formatted table of windchill values. Rows should represent wind speed for 0 to 50 in 5 mph increments, and the columns represent temperatures from -20 to +60 in 10-degree increments. 6. Write a program that uses a while loop to determine how long it takes for an investment to double at a given interest rate. The input will be an annualized interest rate, and the output is the number of years it takes an investment to double. Note: the amount of the initial investment does not matter; you can use $1. 7. The Syracuse (also called Collatz or Hailstone) sequence is generated by starting with a natural number and repeatedly applying the following function until reaching 1. syr x¡ x2 if x is even 3x 1 if x is odd For example, the Syracuse sequence starting with 5 is: 5¤ 16¤ 8¤ 4¤ 2¤ 1. It is an open question in mathematics whether this sequence will always go to 1 for every possible starting value. Write a program that gets a starting value from the user and then prints the Syracuse sequence for that starting value. 35 75 V 0 16 0 4275T V 0 16 ¡ ¡ Slide 144: 136 CHAPTER 8. CONTROL STRUCTURES, PART 2 8. A postive whole number n 2 is prime if no number between 2 and n (inclusive) evenly divides n. Write a program that accepts a value of n as input and determines if the value is prime. If n is not prime, your program should quit as soon as it ﬁnds a value that evenly divides n. 9. Modify the previous program to ﬁnd every prime number less than or equal to n. 10. The greatest common divisor (GCD) of two values can be computed using Euclid’s algorithm. Starting with the values m and n, we repeatedly apply the formula: n, m = m, n%m until m is 0. At that point, n is the GCD of the original m and n. Write a program that ﬁnds the GCD of two numbers using this algorithm. 11. Write a program that computes the fuel efﬁciency of a multi-leg journey. The program will ﬁrst prompt for the starting odometer reading and then get information about a series of legs. For each leg, the user enters the current odometer reading and the amount of gas used (separated by a space). The user signals the end of the trip with a blank line. The program should print out the miles per gallon achieved on each leg and the total MPG for the trip. 12. Modify the previous program to get its input from a ﬁle. 13. Heating and cooling degree-days are measures used by utility companies to estimate energy requirements. If the average temperature for a day is below 60, then the number of degrees below 60 is added to the heating degree-days. If the temperature is above 80, the amount over 80 is added to the cooling degree-days. Write a program that accepts a sequence of average daily temps and computes the running total of cooling and heating degree-days. The program should print these two totals after all the data has been processed. 14. Modify the previous program to get its input from a ﬁle. 15. Write a program that graphically plots a regression line, that is, the line with the best ﬁt through a collection of points. First ask the user to specify the data points by clicking on them in a graphics window. To ﬁnd the end of input, place a small rectangle labelled “Done” in the lower left corner of the window; the program will stop gathering points when the user clicks inside that rectangle. The regression line is the line with the following equation: ¢ ¢ y where ¯ y mx ¯ x¡ ¯ is the mean of the x-values and ¯ is the mean of the y-values. x y As the user clicks on points, the program should draw them in the graphics window and keep track of the count of input values and the running sum of x, y, x2 and xy values. When the user clicks inside the “Done” rectangle, the program then computes value of y (using the equations above) correponding to the x values at the left and right edges of the window to compute the endpoints of the regression line spanning the window. After the line is drawn, the program will pause for another mouse click before closing the window and quitting. ¢ m xy ∑ xi yi n ¯ ¯ 2 2 x ∑ xi n ¯ ¢ ¡ Slide 145: Chapter 9 Simulation and Design You may not realize it, but you have reached a signiﬁcant milestone in the journey to becoming a computer scientist. You now have all the tools to write programs that solve interesting problems. By interesting, I mean problems that would be difﬁcult or impossible to solve without the ability to write and implement computer algorithms. You are probably not yet ready to write the next great killer application, but you can do some nontrivial computing. One particularly powerful technique for solving real-world problems is simulation. Computers can model real-world processes to provide otherwise unobtainable information. Computer simulation is used every day to perform myriad tasks such as predicting the weather, designing aircraft, creating special effects for movies, and entertaining video game players, to name just a few. Most of these applications require extremely complex programs, but even relatively modest simulations can sometimes shed light on knotty problems. In this chapter we are going to develop a simple simulation of the game of racquetball. Along the way, you will learn some important design and implementation strategies that will help you in tackling your own problems. 9.1 Simulating Racquetball 9.1.1 A Simulation Problem Suzie Programmer’s friend, Denny Dibblebit, plays racquetball. Over years of playing, he has noticed a strange quirk in the game. He often competes with players who are just a little bit better than he is. In the process, he always seems to get thumped, losing the vast majority of matches. This has led him to question what is going on. On the surface, one would think that players who are slightly better should win slightly more often, but against Denny, they seem to win the lion’s share. One obvious possibility is that Denny Dibblebit’s problem is in his head. Maybe his mental game isn’t up to par with his physical skills. Or perhaps the other players are really much better than he is, and he just refuses to see it. One day, Denny was discussing racquetball with Suzie, when she suggested another possibility. Maybe it is the nature of the game itself that small differences in ability lead to lopsided matches on the court. Denny was intrigued by the idea; he didn’t want to to waste money on an expensive sports psychologist if it wasn’t going to help. But how could he ﬁgure out if the problem was mental or just part of the game? Suzie suggested she could write a computer program to simulate certain aspects of racquetball. Using the simulation, they could let the computer model thousands of games between players of differing skill levels. Since there would not be any mental aspects involved, the simulation would show whether Denny is losing more than his share of matches. Let’s write our own racquetball simulation and see what Suzie and Denny discovered. 137 Slide 146: 138 CHAPTER 9. SIMULATION AND DESIGN 9.1.2 Program Speciﬁcation Racquetball is a sport played between two players using racquets to strike a ball in a four-walled court. It has aspects similar to many other ball and racquet games such as tennis, volleyball, badminton, squash, table tennis, etc. We don’t need to understand all the rules of racquetball to write the program, just the basic outline of the game. To start the game, one of the players puts the ball into play—this is called serving. The players then alternate hitting the ball to keep it in play; this is a rally. The rally ends when one of the players fails to hit a legal shot. The player who misses the shot loses the rally. If the loser is the player who served, then service passes to the other player. If the server wins the rally, a point is awarded. Players can only score points during their service. The ﬁrst player to reach 15 points wins the game. In our simulation, the ability-level of the players will be represented by the probability that the player wins the rally when he or she serves. Thus, players with a 0.6 probability win a point on 60% of their serves. The program will prompt the user to enter the service probability for both players and then simulate multiple games of racquetball using those probabilities. The program will then print a summary of the results. Here is a detailed speciﬁcation: Input The program ﬁrst prompts for and gets the service probabilities of the two players (called “Player A” and “Player B”). Then the program prompts for and gets the number of games to be simulated. Output The program will provide a series of initial prompts such as the following: What is the prob. player A wins a serve? What is the prob. player B wins a serve? How many games to simulate? The program will print out a nicely formatted report showing the number of games simulated and the number of wins and winning percentage for each player. Here is an example: Games Simulated: 500 Wins for A: 268 (53.6%) Wins for B: 232 (46.4%) Notes: All inputs are assumed to be legal numeric values, no error or validity checking is required. In each simulated game, player A serves ﬁrst. 9.2 Random Numbers Our simulation program will have to deal with uncertain events. When we say that a player wins 50% of the serves, that does not mean that every other serve is a winner. It’s more like a coin toss. Overall, we expect that half the time the coin will come up heads and half the time it will come up tails, but there is nothing to prevent a run of ﬁve tails in a row. Similarly, our racquetball player should win or lose rallies randomly. The service probability provides a likelihood that a given serve will be won, but there is no set pattern. Many simulations share this property of requiring events to occur with a certain likelihood. A driving simulation must model the unpredictability of other drivers; a bank simulation has to deal with the random arrival of customers. These sorts of simulations are sometimes called Monte Carlo algorithms, because the results depend on “chance” probabilities. Of course, you know that there is nothing random about computers; they are instruction-following machines. How can computer programs model seemingly random happenings? Simulating randomness is a well-studied problem in computer science. Remember the chaos program from Chapter 1? The numbers produced by that program seemed to jump around randomly between zero and one. This apparent randomness came from repeatedly applying a function to generate a sequence of numbers. A similar approach can be used to generate random (actually pseudorandom) numbers. A pseudorandom number generator works by starting with some seed value. This value is fed to a function to produce a “random” number. The next time a random number is needed, the current value is fed back into Slide 147: 9.2. RANDOM NUMBERS 139 the function to produce a new number. With a carefully chosen function, the resulting sequence of values looks essentially random. Of course, if you start the process over again with the same seed value, you end up with exactly the same sequence of numbers. It’s all determined by the generating function and the value of the seed. Python provides a library module that contains a number of useful functions for generating pseudorandom numbers. The functions in this module derive an initial seed value from the date and time when the module is loaded, so you get a different seed value each time the program is run. This means that you will also get a unique sequence of pseudorandom values. The two functions of greatest interest to us are randrange and random. The randrange function is used to select a pseudorandom int from a given range. It can be used with one, two or three parameters to specify a range exactly as with the range function. For example, randrange(1,6) returns some number from the range [1,2,3,4,5], and randrange(5,105,5) returns a multiple of 5 between 5 and 100, inclusive. (Remember, ranges go up to, but not including, the stopping value.) Each call to randrange generates a new pseudorandom int. Here is an interactive session that shows randrange in action. >>> >>> 3 >>> 3 >>> 5 >>> 5 >>> 5 >>> 1 >>> 5 >>> 4 >>> 2 from random import randrange randrange(1,6) randrange(1,6) randrange(1,6) randrange(1,6) randrange(1,6) randrange(1,6) randrange(1,6) randrange(1,6) randrange(1,6) Notice it took ten calls to randrange to eventually generate every number in the range 1–5. The value 5 came up almost half of the time. This shows the probabilistic nature of random numbers. Over the long haul, this function produces a uniform distribution, which means that all values will appear an (approximately) equal number of times. The random function can be used to generate pseudorandom ﬂoating point values. It requires no parameters and returns values uniformly distributed between 0 and 1 (including 0, but excluding 1). Here are some interactive examples. >>> from random import random >>> random() 0.545146406725 >>> random() 0.221621655814 >>> random() 0.928877335157 >>> random() 0.258660828538 >>> random() 0.859346793436 Slide 148: 140 CHAPTER 9. SIMULATION AND DESIGN The name of the module (random) is the same as the name of the function, which gives rise to the funnylooking import line. Our racquetball simulation can make use of the random function to determine whether or not a player wins a serve. Let’s look at a speciﬁc example. Suppose a player’s service probability is 0.70. This means that they should win 70% of their serves. You can imagine a decision in the program something like this: if <player wins serve>: score = score + 1 We need to insert a probabilistic condition that will succeed 70% of the time. Suppose we generate a random value between 0 and 1. Exactly 70% of the interval 0 1 is to the left of 0.7. So 70% of the time the random number will be 0.7, and it will be 0.7 the other 30% of the time. (The goes on the upper end, because the random generator can produce a 0, but never a 1.) In general, if prob represents the probability that the player wins a serve, the condition random() < prob will succeed with just the right probability. Here is how the decision will look: if random() < prob: score = score + 1 ¡ 9.3 Top-Down Design Now you have the complete speciﬁcation for our simulation and the necessary knowledge of random numbers to get the job done. Go ahead and take a few minutes to write up the program; I’ll wait. OK, seriously, this is a more complicated program than you’ve probably attempted so far. You may not even know where to begin. If you’re going to make it through with minimal frustration, you’ll need a systematic approach. One proven technique for tackling complex problems is called top-down design. The basic idea is to start with the general problem and try to express a solution in terms of smaller problems. Then each of the smaller problems is attacked in turn using the same technique. Eventually the problems get so small that they are trivial to solve. Then you just put all the pieces back together and, voil` , you’ve got a program. a 9.3.1 Top-Level Design Top-down design is easier to illustrate than it is to deﬁne. Let’s give it a try on our racquetball simulation and see where it takes us. As always, a good start is to study the program speciﬁcation. In very broad brush strokes, this program follows the basic Input-Process-Output pattern. We need to get the simulation inputs from the user, simulate a bunch of games, and print out a report. Here is a basic algorithm. Print an Introduction Get the inputs: probA, probB, n Simulate n games of racquetball using probA and probB Print a report on the wins for playerA and playerB Now that we’ve got an algorithm, we’re ready to write a program. I know what you’re thinking: this design is too high-level; you don’t have any idea yet how it’s all going to work. That’s OK. Whatever we don’t know how to do, we’ll just ignore for now. Imagine that all of the components you need to implement the algorithm have already been written for you. Your job is to ﬁnish this top-level algorithm using those components. First we have to print an introduction. I think I know how to do this. It just requires a few print statements, but I don’t really want to bother with it right now. It seems an unimportant part of the algorithm. I’ll procrastinate and pretend that someone else will do it for me. Here’s the beginning of the program. def main(): printInstructions() Slide 149: 9.3. TOP-DOWN DESIGN 141 Do you see how this works. I’m just assuming there is a printInstructions function that takes care of printing the instructions. That step was easy! Let’s move on. Next, I need to get some inputs from the user. I also know how to do that—I just need a few input statements. Again, that doesn’t seem very interesting, and I feel like putting off the details. Let’s assume that a component already exists to solve that problem. We’ll call the function getInputs. The point of this function is to get values for variables probA, probB and n. The function must return these values for the main program to use. Here is our program so far: def main(): printInstructions() probA, probB, n = getInputs() We’re making progress, let’s move on to the next line. Here we’ve hit the crux of the problem. We need to simulate n games of racquetball using the values of probA, and probB. This time, I really don’t have a very good idea how that will even be accomplished. Let’s procrastinate again and push the details off into a function. (Maybe we can get someone else to write that part for us later.) But what should we put into main? Let’s call our function simNGames. We need to ﬁgure out what the call of this function looks like. Suppose you were asking a friend to actually carry out a simulation of n games. What information would you have to give him? Your friend would need to know how many games he was supposed to simulate and what the values of probA and probB should be for those simulations. These three values will, in a sense, be inputs to the function. What information do you need to get back from your friend? Well, in order to ﬁnish out the program (print a report) you need to know how many games were won by player A and how many games were won by Player B. These must be outputs from the simNGames function. Remember in the discussion of functions in Chapter 6, I said that parameters were used as function inputs, and return values serve as function outputs. Given this analysis, we now know how the next step of the algorithm can be coded. def main(): printInstructions() probA, probB, n = getInputs() winsA, winsB = simNGames(n, probA, probB) Are you getting the hang of this? The last step is to print a report. If you told your friend to type up the report, you would have to tell him how many wins there were for each player; these values are inputs to the function. Here’s the complete program. def main(): printInstructions() probA, probB, n = getInputs() winsA, winsB = simNGames(n, probA, probB) printSummary(winsA, winsB) That wasn’t very hard. The main function is only ﬁve lines long, and the program looks like a more precise formulation of the rough algorithm. 9.3.2 Separation of Concerns Of course, the main function alone won’t do very much; we’ve put off all of the interesting details. In fact, you may think that we have not yet accomplished anything at all, but that is far from true. We have broken the original problem into four independent tasks: printInstructions, getInputs, simNGames and printSummary. Further, we have speciﬁed the name, parameters and expected return values of the functions that perform these tasks. This information is called the interface or signature of a function. Having signatures allows us to tackle pieces independently. For the purposes of main, we don’t care how simNGames does its job. The only concern is that, when given the number of games to simulate and the two Slide 150: 142 CHAPTER 9. SIMULATION AND DESIGN probabilities, it must hand back the correct number of wins for each player. The main function only cares what each (sub-)function does. Our work so far can be represented as a structure chart (also called a module hierarchy chart). Figure 9.1 illustrates. Each component in the design is a rectangle. A line connecting two rectangles indicates that the one above uses the one below. The arrows and annotations show the interfaces between the components in terms of information ﬂow. main winsA winsB probA probB n probA probB n winsA winsB printIntro getInputs simNGames printSummary Figure 9.1: First-level structure chart for racquetball simulation. At each level of a design, the interface tells us which details of the lower level are important. Anything else can be ignored (for the moment). The general process of determining the important characteristics of something and ignoring other details is called abstraction. Abstraction is the fundamental tool of design. You might view the entire process of top-down design as a systematic method for discovering useful abstractions. 9.3.3 Second-Level Design Now all we need to do is repeat the design process for each of the remaining components. Let’s take them in order. The printIntro function should print an introduction to the program. Let’s compose a suitable sequence of print statements. def printIntro(): print "This program simulates a game of racquetball between two" print ’players called "A" and "B". The abilities of each player is’ print "indicated by a probability (a number between 0 and 1) that" print "the player wins the point when serving. Player A always" print "has the first serve." Notice the second line. I wanted to put double quotes around “A” and “B,” so the entire string is enclosed in apostrophes. This function comprises only primitive Python instructions. Since we didn’t introduce any new functions, there is no change to our structure chart. Now let’s tackle getInputs. We need to prompt for and get three values, which are returned to the main program. Again, this is simple to code. def getInputs(): # RETURNS the three simulation parameters probA, probB and n a = input("What is the prob. player A wins a serve? ") b = input("What is the prob. player B wins a serve? ") n = input("How many games to simulate? ") return a, b, n Notice that I have taken some shortcuts with the variable names. Remember, variables inside of a function are local to that function. This function is so short, it’s very easy to see what the three values represent. The main concern here is to make sure the values are returned in the correct order to match with the interface we established between getInputs and main.