Discrete Structures, Logic, and Computability

ISBN-10: 0763718432
ISBN-13: 9780763718435
Edition: 2nd 2001 (Revised)
Authors: James L. Hein
List price: $137.95
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Description: Discrete Structure, Logic, and Computability introduces the beginning computer science student to some of the fundamental ideas and techniques used by computer scientists today, focusing on discrete structures, logic and computability. The emphasis  More...

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Book details

List price: $137.95
Edition: 2nd
Copyright year: 2001
Publisher: Jones & Bartlett Learning, LLC
Publication date: 9/26/2001
Binding: Hardcover
Pages: 943
Size: 7.50" wide x 8.75" long x 1.50" tall
Weight: 3.278
Language: English

Discrete Structure, Logic, and Computability introduces the beginning computer science student to some of the fundamental ideas and techniques used by computer scientists today, focusing on discrete structures, logic and computability. The emphasis is on the computational aspects, so that the reader can see how the concepts are actually used. Because of logic's fundamental importance to computer science, the topic is examined extensively in three phases which cover: informal logic; the technique of inductive proof; and formal logic and its applications to computer science. Access the updated errata for this textClick to view/download: Errata.pdf(40 KBytes) The book contains a wide range of material that focuses on those parts of discrete mathematics, logic, and computability needed for computer science students. The topics can be taught in a variety of ways depending on the length of the course, the emphasis, and the background of students. For example, the preface includes syllabai for fourteen different courses in discrete mathematics ranging in length from ten weeks to a full year. It also includes syllabai for two courses in logic, and two courses in computability. The book is organized more along the lines of technique than on a subject-by-subject basis so that it doesn't fragment into a vast collection of seemingly unrelated ideas. The focus throughout the book is on the computation and construction of objects. So many traditional topics are dispersed throughout the text to places where they fit naturally with the techniques under discussion. Therefore there is a spiral approach to learning these topics. For example, to read about properties of-and techniques for processing-natural numbers, lists, strings, graphs, or trees, it's necessary to look in the index or scan the table of contents to find the several places where they are found. Every example has a title and, for ease of scanning, there are special marks to indicate the beginning and ending of each example. Titles and subtitles are used extensively to make it easy to find things. Important facts and results are boxed for easy reference. The second color also makes it easy to find things. There is a minimum of horizontal lines in the book so that scanning is easy. Exercises at the end of each section are categorized by topic and ordered within each topic by difficulty. There is also a collection of exercises at the end of each section labeled proofs and/or challenges. There are over 1700 exercises with answers provided for about half of them. The exercises with answers in the book are marked in color. The symbol glossary makes it easy to check definitions. The index is extensive so that topics can be easily found. The chapter guidelines and chapter summaries let students see what is to be covered and then to check whether they know the details about what was covered.

Elementary Notions and Notations
A Proof Primer
Logical Statements
Something to Talk About
Proof Techniques
Exercises
Sets
Definition of a Set
Operations on Sets
Counting Finite Sets
Bags (Multisets)
Sets Should Not Be Too Complicated
Exercises
Ordered Structures
Tuples
Lists
Strings and Languages
Relations
Counting Tuples
Exercises
Graphs and Trees
Definition of a Graph
Paths and Graphs
Graph Traversals
Trees
Spanning Trees
Exercises
Chapter Summary
Facts about Functions
Definitions and Examples
Definition of a Function
Some Useful Functions
Partial Functions
Exercises
Constructing Functions
Composition of Functions
The Map Function
Exercise
Properties of Functions
Injections and Surjections
Bijections and Inverses
The Pigeonhole Principle
Simple Ciphers
Hash Functions
Exercises
Countability
Comparing the Size of Sets
Sets that Are Countable
Diagonalization
Limits on Computability
Exercises
Chapter Summary
Construction Techniques
Inductively Defined Sets
Numbers
Strings
Lists
Binary Trees
Cartesian Products of Sets
Exercises
Recursive Functions and Procedures
Numbers
Strings
Lists
Binary Trees
Two More Problems
Infinite Sequences
Exercises
Grammars
Recalling an English Grammar
Structure of Grammars
Derivations
Constructing Grammars
Meaning and Ambiguity
Exercises
Chapter Summary
Equivalence, Order, and Inductive Proof
Properties of Binary Relations
Composition of Relations
Closures
Path Problems
Exercises
Equivalence Relations
Definition and Examples
Equivalence Classes
Partitions
Generating Equivalence Relations
Exercises
Order Relations
Partial Orders
Topological Sorting
Well-Founded Orders
Ordinal Numbers
Exercises
Inductive Proof
Proof by Mathematical Induction
Proof by Well-Founded Induction
A Variety of Examples
Exercises
Chapter Summary
Analysis Techniques
Analyzing Algorithms
Worst-Case Running Time
Decision Trees
Exercises
Finding Closed Forms
Closed Forms for Sums
Exercises
Counting and Discrete Probability
Permutations (Order Is Important)
Combinations (Order Is Not Important)
Discrete Probability
Exercises
Solving Recurrences
Solving Simple Recurrences
Generating Functions
Exercises
Comparing Rates of Growth
Big Theta
Little Oh
Big Oh and Big Omega
Exercises
Chapter Summary
Elementary Logic
How Do We Reason?
What Is a Calculus?
How Can We Tell Whether Something Is a Proof?
Propositional Calculus
Well-Formed Formulas and Semantics
Equivalence
Truth Functions and Normal Forms
Complete Sets of Connectives
Exercises
Formal Reasoning
Inference Rules
Formal Proof
Proof Notes
Exercises
Formal Axiom Systems
An Example Axiom System
Other Axiom Systems
Exercises
Chapter Summary
Predicate Logic
First-Order Predicate Calculus
Predicates and Quantifiers
Well-Formed Formulas
Semantics and Interpretations
Validity
The Validity Problem
Exercises
Equivalent Formulas
Equivalence
Normal Forms
Formalizing English Sentences
Summary
Exercises
Formal Proofs in Predicate Calculus
Universal Instantiation
Existential Generalization (EG)
Existential Instantiation (EI)
Universal Generalization (UG)
Examples of Formal Proofs
Summary of Quantifier Proofs Rules
Exercises
Chapter Summary
Applied Logic
Equality
Describing Equality
Extending Equals for Equals
Exercises
Program Correctness
Imperative Program Correctness
Array Assignment
Termination
Exercises
Higher-Order Logics
Classifying Higher-Order Logics
Semantics
Higher-Order Reasoning
Exercises
Chapter Summary
Computational Logic
Automatic Reasoning
Clauses and Clausal Forms
Resolution for Propositions
Substitution and Unification
Resolution: The General Case
Theorem Proving with Resolution
Remarks
Exercises
Logic Programming
Family Trees
Definition of a Logic Program
Resolution and Logic Programming
Logic Programming Techniques
Exercises
Chapter Summary
Algebraic Structures and Techniques
What Is an Algebra?
Definition of an Algebra
Concrete Versus Abstract
Working in Algebras
Exercises
Boolean Algebra
Simplifying Boolean Expressions
Digital Circuits
Exercises
Abstract Data Types as Algebras
Natural Numbers
Lists and Strings
Stacks and Queues
Binary Trees and Priority Queues
Exercises
Computational Algebras
Relational Algebras
Functional Algebras
Exercises
Other Algebraic Ideas
Congruence
Cryptology: The RSA Algorithm
Subalgebras
Morphisms
Exercises
Chapter Summary
Regular Languages and Finite Automata
Regular Languages
Regular Expressions
The Algebra of Regular Expressions
Exercises
Finite Automata
Deterministic Finite Automata
Nondeterministic Finite Automata
Transforming Regular Expressions into Finite Automata
Transforming Finite Automata into Regular Expressions
Finite Automata as Output Devices
Representing and Executing Finite Automata
Exercises
Constructing Efficient Finite Automata
Another Regular Expression to NFA Algorithm
Transforming an NFA into a DFA
Minimum-State DFAs
Exercises
Regular Language Topics
Regular Grammars
Properties of Regular Languages
Exercises
Chapter Summary
Context-Free Languages and Pushdown Automata
Context-Free Languages
Exercises
Pushdown Automata
Equivalent Forms of Acceptance
Context-Free Grammars and Pushdown Automata
Representing and Executing Pushdown Automata
Exercises
Parsing Techniques
LL(k) Parsing
LR(k) Parsing
Exercises
Context-Free Language Topics
Transforming Grammars
Properties of Context-Free Languages
Exercises
Chapter Summary
Turing Machines and Equivalent Models
Turing Machines
Definition of a Turing Machine
Turing Machines with Output
Alternative Definitions
A Universal Turing Machine
Exercises
The Church-Turing Thesis
Equivalence of Computational Models
A Simple Programming Language
Recursive Functions
Machines That Transform Strings
Exercises
Chapter Summary
Computational Notions
Computability
Effective Enumerations
The Halting Problem
The Total Problem
Other Problems
Exercises
A Hierarchy of Languages
The Languages
Summary
Exercises
Complexity Classes
The Class P
The Class NP
The Class PSPACE
Intractable Problems
Completeness
Formal Complexity Theory
Exercises
Chapter Summary
Answers to Selected Exercises
Bibliography
Greek Alphabet
Symbol Glossary
Index

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