A Mathematical Foundation for Computer Science

731 Pages·2019·223.438 MB·English

by  David Mix Barrington

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A Mathematical Foundation
for Computer Science
REVISED PRELIMINARY EDITION
David Mix Barrington
Kendall Hunt
publishing co mpany
All chapter heading quotes are taken from Monty Python's Flying Circus: All the Words
(Volumes I and 2) (New York: Pantheon Books, 1989) and Monty Python and The Holy Grail
(Book) [Monti Python ik den Helie Grailen (Bok)} (New York, Methuen Inc., 1979).
Excursion 1.11 uses text from Through the Looking Glass, and What Alice Found There by
Lewis Carroll (London, McMillan, 1871) and Fox in Socks by Dr. Seuss (New York, Random
House, 1965).
Problems 2.6.2 and 2.6.3 use text from The Number oft he Beast by Robert A. Heinlein (New
York: Fawcett, 1980).
There are many references in the text to Godel, Escher, Bach: An Eternal Golden Braid by
Douglas R. Hofstadter (New York: Basic Books, 1979).
Cover image of Stalker Castle, Scotland, by Frank Parolek © Shutterstock, Inc.
Kendall Hunt
publishing company
www.kendallhunt.com
Send all inquiries to:
4050 Westmark Drive
Dubuque, IA 52004-1840
Copyright© 2020 by Kendall Hunt Publishing Company
Revised 2020
ISBN 978-1-7924-3760-l
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 the prior written permission of the copyright owner.
Published in the United States of America
REVISED PRELIMINARY EDITION
CONTENTS
Chapter 1: Sets, Propositions, and Predicates 1-1
1.1: Sets 1-2
1.2: Strings and String Operations 1-11
1.3: Excursion: What is a Proof? 1-20
1.4: Propositions and Boolean Operations 1-24
1.5: Set Operations and Propositions About Sets 1-34
1.6: Truth-Table Proofs 1-45
1.7: Rules for Propositional Proofs 1-52
1.8: Propositional Proof Strategies 1-59
1.9: Excursion: A Murder Mystery 1-65
1. 10: Predicates 1-68
1.11: Excursion: Translating Predicates 1-75
Glossary for Chapter 1 1-78
Chapter 2: Quantifiers and Predicate Calculus 2-1
2.1: Relations 2-2
2.2: Excursion: Relational Databases 2-8
2.3: Quantifiers 2-10
2.4: Excursion: Translating Quantifiers 2-17
2.5: Operations on Languages 2-19
2.6: Proofs With Quantifiers 2-25
2.7: Excursion: Practicing Proofs 2-32
2.8: Properties of Binary Relations 2-34
2.9: Functions 2-41
2.10: Partial Orders 2-48
2.11: Equivalence Relations 2-55
Glossary for Chapter 2 2-62
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Chapter 3: Number Theory 3-1
3.1: Divisibility and Primes 3-2
3.2: Excursion: Playing With Numbers 3-11
3.3: Modular Arithmetic 3-14
3.4: There are Infinitely Many Primes 3-23
3.5: The Chinese Remainder Theorem 3-28
3.6: The Fundamental Theorem of Arithmetic 3-35
3.7: Excursion: Expressing Predicates in Number Theory 3-43
3.8: The Ring of Congruence Classes 3-46
3.9: Finite Fields and Modular Exponentiation 3-52
3.10: Excursion: Certificates of Primality 3-58
3.11: The RSA Cryptosystem 3-61
Glossary for Chapter 3 3-71
Chapter 4: Recursion and Proof by Induction 4-1
4.1: Recursive Definition 4-2
4.2: Excursion: Recursive Algorithms 4-10
4.3: Proof By Induction for Naturals 4-13
4.4: Variations on Induction for Naturals 4-20
4.5: Excursion: Fibonacci Numbers 4-27
4.6: Proving the Basic Facts of Arithmetic 4-30
4.7: Recursive Definition for Strings 4-37
4.8: Excursion: Naturals and Strings 4-45
4.9: Graphs and Paths 4-47
4.10: Trees and Lisp Lists 4-56
4.11: Induction for Problem Solving 4-66
Glossary for Chapter 4 4-75
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Chapter 5: Regular Expressions and Other Recursive Systems
5.1: Regular Expressions and Their Languages 5-2
5.2: Examples of Regular Languages 5-7
5.3: Excursion: Designing Regular Expressions 5-13
5.4: Proving Regular Language Identities 5-15
5 .5: Proving Properties of the Regular Languages 5-19
5.6: Excursion: Hofstadter's MU-Puzzle 5-28
5.7: Recursion and Induction in General 5-30
5.8: Top-Down and Bottom-Up Definitions 5-38
5.9: Excursion: Parsing Arithmetic Expressions 5-45
5.10: A Recursive Definition oflmperative Programs 5-47
5.11: Correctness oflmperative Programs 5-56
Glossary for Chapter 5 5-61
Chapter 9: Trees and Searching
9.1: Graphs, Trees, and Recursion 9-2
9 .2: Excursion: Boolean Expressions 9-10
9.3: Expressions and Recursive Algorithms 9-12
9.4: General Search Algorithms 9-20
9.5: General Depth-First and Breadth-First Search 9-28
9.6: Depth-First and Breadth-First Search on Graphs 9-36
9.7: Excursion: Middle-First Search 9-44
9.8: Uniform-Cost Search 9-46
9.9: A* Search 9-54
9.10: Adversary Search 9-60
9.11: Excursion: Hexapawn 9-69
Glossary for Chapter 9 9-71
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Chapter 14: Finite-State Machines
14.1: Deterministic Finite Automata 14-2
14.2: Proving that DFA's Can't Do Things 14-9
14.3: The Myhill-Nerode Theorem 14-16
14.4: Excursion: Syntactic Monoids 14-26
14.5: Nondeterministic Finite Automata 14-28
14.6: The Subset Construction: NFA's into DFA's 14-36
14.7: Killing A -Moves: A -NFA's into NFA's 14-43
14.8: Constructing A -NFA's From Regular Expressions 14-53
14.9: Excursion: Practicing Multiple Constructions 14-62
14.10: State Elimination: NFA's into Regular Expresions 14-65
14.11: Excursion: Another Way From NFA's to Regular Expressions 14-73
Glossary for Chapter 14 14-76
Chapter 15: A Brief Tour of Formal Language Theory
15.1: Two-Way Deterministic Finite Automata 15-3
15.2: Grammars, Regular and Otherwise 15-14
15.3: Context-Free Languages 15-22
15.4: Excursion: Chomsky Normal Form 15-31
15.5: CFL's and Pushdown Automata 15-34
15.6: Turing Machine Definitions 15-43
15.7: Excursion: Unrestricted Grammars 15-51
15.8: Turing Machine Semantics 15-54
15.9: Excursion: Turing-Hangable Languages 15-62
15.10: The Halting Problem and Unsolvability 15-63
15.11: A Brief Look at Complexity Theory 15-72
Glossary for Chapter 15 15-83
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S.1: Solutions to Exercises from Chapter 1 S-1
S.2: Solutions to Exercises from Chapter 2 S-16
S.3: Solutions to Exercises from Chapter 3 S-32
S.4: Solutions to Exercises from Chapter 4 S-48
S.5: Solutions to Exercises from Chapter 5 S-64
S.9: Solutions to Exercises from Chapter 9 S-80
S.14: Solutions to Exercises from Chapter 14 S-96
S.15: Solutions to Exercises from Chapter 15 S-113
FULL VERSION CONTENTS
Chapter 1: Sets, Propositions and Predicates
Chapter 2: Quantifiers and Predicate Calculus
Chapter 3: Number Theory
Chapter 4: Recursion and Proof By Induction
Chapter 5: Regular Expressions and Other Recursive Systems
Chapter 6: Fundamental Counting Problems
Chapter 7: Further Topics in Combinatorics
Chapter 8: Graphs
Chapter 9: Trees and Searching
Chapter 10: Discrete Probability
Chapter 11: Reasoning About Uncertainty
Chapter 12: Markov Processes and Classical Games
Chapter 13: Information Theory
Chapter 14: Finite-State Machines
Chapter 15: A Brief Tour of Formal Language Theory
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AUTHOR'S NOTE
• This Revised Preliminary Edition contains eight of the entire
book's fifteen chapters, which form the text for COMPSCI 250
at UMass Amherst in Fall 2020. The final version will include
these chapters and seven others which could be used in
COMPSCI 240.
• Each chapter has eight ordinary sections and three Excursions.
In COMPSCI 250 the 50-minute lectures cover one or
sometimes two ordinary sections. Excursions are used for team
problem-solving sessions in the weekly discussions. Each
ordinary section contains ten Exercises (w ith solutions in the
back) and ten Problems (suitable for homework assignments).
Grateful Thanks to:
• First and foremost, my wife Jessica and daughter Julia.
• The many teachers who helped form me as a mathematician,
including David Cox, the late Jim Mauldon, and especially
Mark Kidwell at Amherst College, Adrian Mathias at
Cambridge University, and Mike Sipser at M.I.T.
• Colleagues at UMass who helped form me as a teacher and
COMPSCI 250 as a course, including Amy Rosenberg, Neil
Immerman, Hava Siegelmann, Marius Minea, and dozens of
graduate and undergraduate teaching assistants.
• My students in this and other courses.
• Emma Strubell, who made most of the diagrams in Chapters 1
and 2 and many of those in Chapter 14.
• Bev Kraus and Lenell Wyman at Kendall Hunt.
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