A Mathematical Foundation for Computer Science 1st edition by David Mix Barrington 9781792437601 1792437609

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ISBN 10:1792437609 
ISBN 13:9781792437601
Author:David Mix Barrington

Undergraduate computer science students need to learn and use the mathematical method of abstraction, definition, and proof, perhaps even earlier than do mathematics students. They deal constantly with formally defined systems beyond those studied in mathematics courses, and must be able reason about them formally in order to write and understand programs. A Mathematical Foundation for Computer Science introduces the mathematical method using examples from computer science, often illustrated by Java-like code. It begins with propositional and predicate logic, introduces number theory, and deals thoroughly with mathematical induction as it relates to recursive definition and recursive algorithms. Later chapters cover combinatorics, probability, graphs and searching, finite-state machines, and a brief introduction to formal language theory. Each chapter is divided into narrative sections, each with Exercises and Problems, and Excursion sections suitable for active learning exercises. This preliminary edition includes the first four chapters, used as the textbook for the first half of a UMass course, COMPSCI 250.

A Mathematical Foundation for Computer Science 1st Table of contents:

Chapter 1: Sets, Propositions, and Predicates

1.1 Sets

1.2 Strings and String Operations

1.3 Excursion: What Is A Proof?

1.4 Propositions and Boolean Operations

1.5 Set Operations and Propositions About Sets

1.6 Truth-Table Proofs

1.7 Rules For Propositional Proofs

1.8 Propositional Proof Strategies

1.9 Excursion: A Murder Mystery

1.10 Predicates

1.11 Excursion: Translating Predicates

Chapter 2: Quantifiers and Predicate Calculus

2.1 Relations

2.2 Excursion: Relational Databases

2.3 Quantifiers

2.4 Excursion: Translating Quantifiers

2.5 Operations on Languages

2.6 Proofs with Quantifiers

2.7 Excursion: Practicing Proofs

2.8 Properties of Binary Relations

2.9 Functions

2.10 Partial Orders

2.11 Equivalence Relations

Chapter 3: Number Theory

3.1 Divisibility and Primes

3,2 Excursion: Playing With Numbers

3,3 Modular Arithmetic

3.4 There Are Infinitely Many Primes

3.5 The Chinese Remainder Theorem

3.6 The Fundamental Theorem of Arithmetic

3.7 Excursion: Expressing Predicates in Number Theory

3.8 The Ring of Congruence Classes

3.9 Finite Fields and Modular Exponentiation

3.10 Excursion: Certificates of Primality

3.11 The RSA Cryptosystem

Chapter 4: Recursion and Proof By Induction

4.1 Recursive Definition

4.2 Excursion: Recursive Algorithms

4.3 Proof By Induction for Naturals

4.4 Variations on Induction for Naturals

4.5 Excursion: Fibonacci Numbers

4.6 Proving the Basic Facts of Arithmetic

4.7 Strings and String Operations

4.8 Excursion: Naturals and Strings

4.9 Graphs and Paths

4.10 Trees and Lisp Lists

4.11 Induction for Problem Solving

Chapter 5: Regular Expressions and Other Recursive Systems

5.1 Regular Expressions and Their Languages

5.2 Examples of Regular Languages

5.3 Excursion: Designing Regular Expressions

5.4 Proving Regular Language Identities

5.5 Proving Properties of the Regular Languages

5.6 Excursion: Hofstadter’s MU-Puzzle

5.7 Recursion and Induction in General

5.8 Top-Down and Bottom-Up Definitions

5.9 Excursion: Parsing Arithmetic Expressions

5.10 A Recursive Definition of Imperative Programs

5.11 Correctness of Imperative Programs

Chapter 6: Fundamental Counting Problems

6.1 Counting: Sum and Product Rules

6.2 Double-Counting and Inclusion/Exclusion

6.3 Counting Sequences (First Counting Problem)

6.4 Counting Permutations (Second Counting Problem)

6.5 Excursion: The Problem of Sorting

6.6 Counting Subsets of a Set (Third Counting Problem)

6.7 Counting Multisets (Fourth Counting Problem)

6.8 Excursion: Listing Permutations and Combinations

6.9 Counting Strings in Languages

6.10 Counting Balanced Strings of Parentheses

6.11 Excursion: Catalan Numbers

Chapter 7: Further Topics in Combinatorics

7.1 Recurrences

7.2 Solving Some Recurrences

7.3 Excursion: Calculus on Sequences

7.4 Generating Functions

7.5 Asymptotic Notation

7.6 Analysis of Sorting

7.7 Excursion: Selection Problems

7.8 Cardinality of Infinite Sets

7.9 Diagonalization and Uncountability

7.10 Excursion: Ordinal Arithmetic

7.11 Combinatorial Games

Chapter 8: Graphs

8.1 Graph Definitions

8.2 Adjacency Matrices

8.3 Paths and Matrix Powering

8.4 Excursion: Min-Plus Matrix Powering

8.5 Dynamic Programming

8.6 Euler and Hamilton Paths

8.7 Excursion: Classifying Graphs

8.8 Vertex and Edge Colorings

8.9 Graph Planarity

8.10 Excursion: Euler’s Formula for Polyhedra

8.11 Trees as Graphs

Chapter 9: Trees and Searching

9.1 Trees and Their Uses

9.2 Excursion: Boolean Expressions

9.3 Recursion on Trees

9.4 General Search Trees

9.5 General Breadth-First and Depth-First Search

9.6 BFS and DFS on Graphs

9.7 Excursion: Middle-First Search and Matrices

9.8 Uniform-Cost Search

9.9 A^* Search

9.10 Games and Adversary Search

9.11 Excursion: Hexapawn

Chapter 10: Discrete Probability

10.1 Probability Distributions

10.2 Expected Value

10.3 Evaluating Games

10.4 Excursion: Analysis of Craps

10.5 Variance and Standard Deviation

10.6 The Binomial Distribution

10.7 Excursion: Election Polling

10.8 The Coupon Collector’s Problem

10.9 The Markov and Chebyshev Bounds

10.10 Excursion: Contention Resolution

10.11 The Union Bound

Chapter 11: Reasoning About Uncertainty

11.1 Conditional Probability and Bayes’ Theorem

11.2 Odds and Likelihood

11.3 Examples of Bayesian Reasoning

11.4 Excursion: A Police Lineup

11.5 The Naive Bayes Classifier

11.6 Problems With the Naive Classifier

11.7 Graphical Models of Distributions

11.8 Excursion: A Probabilistic Murder Mystery

11.9 Pseudorandom Generators

11.10 Monte Carlo Simulation

11.11 Excursion: Simulating Baseball Seasons

Chapter 12: Markov Processes and Classical Games

12.1 Finite-State Random Processes

12.2 Markov Chains

12.3 Limiting Behavior of Markov Chains

12.4 Excursion: Markov Text Generation

12.5 Markov Decision Processes

12.6 Horizons and Discounting

12.7 Value and Policy Iteration

12.8 Excursion: Modeling Blackjack

12.9 Two-Player Simultaneous-Move Games

12.10 Excursion: Modeling Baseball Balls and Strikes

12.11 The Prisoners’ Dilemma

Chapter 13: Graphs

13.1 What Is a Bit?

13.2 Data Compression

13.3 Excursion: Redundancy of English Text

13.4 Variable-Length Codes

13.5 Excursion: Huffman’s Algorithm

13.6 Huffman Coding and Entropy

13.7 Error-Detecting Codes

13.8 Error-Correcting Codes

13.9 Excursion: Experimenting With Codes

13.10 Conditional Entropy and Noise

13.11 The Noisy-Channel Theorem

Chapter 14: Finite-State Machines

14.1 Deterministic Finite Automata

14.2 Proving that DFA’s Can’t Do Things

14.3 The Myhill-Nerode Theorem

14.4 Excursion: Syntactic Monoids

14.5 Nondeterministic Finite Automata

14.6 The Subset Construction: NFA’s into DFA’s

14.7 Killing λ-Moves: λ-NFA’s into NFA’s

14.8 Constructing λ-NFA’s from Regular Expressions

14.9 Excursion: Practicing Multiple Constructions

14.10 State Elimination: NFA’s into Regular Expressions

14.11 Excursion: Another Way From NFA’s to Regular Expressions

Chapter 15: A Brief Tour of Formal Language Theory

15.1 Two-Way Deterministic Finite Automata

15.2 Grammars, Regular and Otherwise

15.3 Context-Free Languages

15.4 Excursion: Chomsky Normal Form

15.5 CFL’s and Pushdown Automata

15.6 Turing Machine Definitions

15.7 Excursion: Unrestricted Grammars

15.8 Turing Machine Semantics

15.9 Excursion: Turing-Hangable Languages

15.10 The Halting Problem and Unsolvability

15.11 A Brief Look at Complexity Theory

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