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Discrete Structures, Logic, and Computability

by: HEIN JAMES L.

Discrete Structures, Logic, and Computability

by: HEIN JAMES L.

ISBN 13:
9780763718435
ISBN 10:
0763718432
Edition: 2nd
Format: Hardcover
Copyright: 09/01/2001
Publisher: Jones & Bartlett
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Summary

This book 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.

Elementary Notions and Notations

(73)

A Proof Primer

(11)

Logical Statements

(3)

Something to Talk About

(1)

Proof Techniques

(6)

Exercises

(1)

Sets

(22)

Definition of a Set

(5)

Operations on Sets

(8)

Counting Finite Sets

(3)

Bags (Multisets)

(1)

Sets Should Not Be Too Complicated

(1)

Exercises

(4)

Ordered Structures

(20)

Tuples

(4)

Lists

(2)

Strings and Languages

(5)

Relations

(3)

Counting Tuples

(3)

Exercises

(3)

Graphs and Trees

(17)

Definition of a Graph

(4)

Paths and Graphs

(2)

Graph Traversals

(2)

Trees

(5)

Spanning Trees

(2)

Exercises

(2)

Chapter Summary

(1)

Facts about Functions

(54)

Definitions and Examples

(17)

Definition of a Function

(5)

Some Useful Functions

(8)

Partial Functions

(1)

Exercises

(3)

Constructing Functions

(9)

Composition of Functions

(5)

The Map Function

(2)

Exercises

(2)

Properties of Functions

100

(15)

Injections and Surjections

100

(2)

Bijections and Inverses

102

(3)

The Pigeonhole Principle

105

(1)

Simple Ciphers

106

(3)

Hash Functions

109

(2)

Exercises

111

(4)

Countability

115

(10)

Comparing the Size of Sets

115

(1)

Sets that Are Countable

116

(3)

Diagonalization

119

(2)

Limits on Computability

121

(3)

Exercises

124

(1)

Chapter Summary

125

(2)

Construction Techniques

127

(66)

Inductively Defined Sets

128

(17)

Numbers

129

(3)

Strings

132

(2)

Lists

134

(4)

Binary Trees

138

(2)

Cartesian Products of Sets

140

(2)

Exercises

142

(3)

Recursive Functions and Procedures

145

(28)

Numbers

146

(4)

Strings

150

(3)

Lists

153

(6)

Binary Trees

159

(4)

Two More Problems

163

(2)

Infinite Sequences

165

(3)

Exercises

168

(5)

Grammars

173

(18)

Recalling an English Grammar

173

(1)

Structure of Grammars

174

(3)

Derivations

177

(4)

Constructing Grammars

181

(5)

Meaning and Ambiguity

186

(2)

Exercises

188

(3)

Chapter Summary

191

(2)

Equivalence, Order, and Inductive Proof

193

(80)

Properties of Binary Relations

194

(19)

Composition of Relations

195

(4)

Closures

199

(5)

Path Problems

204

(5)

Exercises

209

(4)

Equivalence Relations

213

(19)

Definition and Examples

214

(4)

Equivalence Classes

218

(1)

Partitions

219

(6)

Generating Equivalence Relations

225

(4)

Exercises

229

(3)

Order Relations

232

(21)

Partial Orders

233

(6)

Topological Sorting

239

(3)

Well-Founded Orders

242

(8)

Ordinal Numbers

250

(1)

Exercises

251

(2)

Inductive Proof

253

(19)

Proof by Mathematical Induction

253

(6)

Proof by Well-Founded Induction

259

(2)

A Variety of Examples

261

(6)

Exercises

267

(5)

Chapter Summary

272

(1)

Analysis Techniques

273

(72)

Analyzing Algorithms

274

(7)

Worst-Case Running Time

274

(3)

Decision Trees

277

(4)

Exercises

281

(1)

Finding Closed Forms

281

(8)

Closed Forms for Sums

282

(5)

Exercises

287

(2)

Counting and Discrete Probability

289

(23)

Permutations (Order Is Important)

289

(4)

Combinations (Order Is Not Important)

293

(5)

Discrete Probability

298

(11)

Exercises

309

(3)

Solving Recurrences

312

(22)

Solving Simple Recurrences

313

(6)

Generating Functions

319

(13)

Exercises

332

(2)

Comparing Rates of Growth

334

(8)

Big Theta

334

(4)

Little Oh

338

(1)

Big Oh and Big Omega

339

(2)

Exercises

341

(1)

Chapter Summary

342

(3)

Elementary Logic

345

(52)

How Do We Reason?

346

(2)

What Is a Calculus?

347

(1)

How Can We Tell Whether Something Is a Proof?

348

(1)

Propositional Calculus

348

(21)

Well-Formed Formulas and Semantics

349

(4)

Equivalence

353

(5)

Truth Functions and Normal Forms

358

(7)

Complete Sets of Connectives

365

(2)

Exercises

367

(2)

Formal Reasoning

369

(15)

Inference Rules

370

(2)

Formal Proof

372

(8)

Proof Notes

380

(1)

Exercises

381

(3)

Formal Axiom Systems

384

(10)

An Example Axiom System

384

(7)

Other Axiom Systems

391

(1)

Exercises

392

(2)

Chapter Summary

394

(3)

Predicate Logic

397

(60)

First-Order Predicate Calculus

397

(19)

Predicates and Quantifiers

398

(4)

Well-Formed Formulas

402

(2)

Semantics and Interpretations

404

(5)

Validity

409

(4)

The Validity Problem

413

(1)

Exercises

413

(3)

Equivalent Formulas

416

(16)

Equivalence

416

(8)

Normal Forms

424

(3)

Formalizing English Sentences

427

(2)

Summary

429

(1)

Exercises

430

(2)

Formal Proofs in Predicate Calculus

432

(24)

Universal Instantiation

433

(4)

Existential Generalization (EG)

437

(1)

Existential Instantiation (EI)

438

(2)

Universal Generalization (UG)

440

(3)

Examples of Formal Proofs

443

(7)

Summary of Quantifier Proof Rules

450

(1)

Exercises

451

(5)

Chapter Summary

456

(1)

Applied Logic

457

(48)

Equality

458

(8)

Describing Equality

458

(6)

Extending Equals for Equals

464

(1)

Exercises

465

(1)

Program Correctness

466

(25)

Imperative Program Correctness

467

(11)

Array Assignment

478

(4)

Termination

482

(4)

Exercises

486

(5)

Higher-Order Logics

491

(12)

Classifying Higher-Order Logics

492

(4)

Semantics

496

(2)

Higher-Order Reasoning

498

(3)

Exercises

501

(2)

Chapter Summary

503

(2)

Computational Logic

505

(52)

Automatic Reasoning

505

(28)

Clauses and Clausal Forms

506

(6)

Resolution for Propositions

512

(2)

Substitution and Unification

514

(7)

Resolution: The General Case

521

(5)

Theorem Proving with Resolution

526

(3)

Remarks

529

(1)

Exercises

530

(3)

Logic Programming

533

(22)

Family Trees

534

(2)

Definition of a Logic Program

536

(1)

Resolution and Logic Programming

537

(12)

Logic Programming Techniques

549

(4)

Exercises

553

(2)

Chapter Summary

555

(2)

Algebraic Structures and Techniques

557

(76)

What Is an Algebra?

558

(14)

Definition of an Algebra

560

(2)

Concrete Versus Abstract

562

(2)

Working in Algebras

564

(6)

Exercises

570

(2)

Boolean Algebra

572

(13)

Simplifying Boolean Expressions

574

(4)

Digital Circuits

578

(5)

Exercises

583

(2)

Abstract Data Types as Algebras

585

(16)

Natural Numbers

585

(4)

Lists and Strings

589

(3)

Stacks and Queues

592

(4)

Binary Trees and Priority Queues

596

(3)

Exercises

599

(2)

Computational Algebras

601

(12)

Relational Algebras

601

(6)

Functional Algebras

607

(4)

Exercises

611

(2)

Other Algebraic Ideas

613

(19)

Congruence

613

(3)

Cryptology: The RSA Algorithm

616

(5)

Subalgebras

621

(2)

Morphisms

623

(6)

Exercises

629

(3)

Chapter Summary

632

(1)

Regular Languages and Finite Automata

633

(64)

Regular Languages

634

(8)

Regular Expressions

635

(3)

The Algebra of Regular Expressions

638

(2)

Exercises

640

(2)

Finite Automata

642

(24)

Deterministic Finite Automata

642

(4)

Nondeterministic Finite Automata

646

(2)

Transforming Regular Expressions into Finite Automata

648

(2)

Transforming Finite Automata into Regular Expressions

650

(5)

Finite Automata as Output Devices

655

(3)

Representing and Executing Finite Automata

658

(6)

Exercises

664

(2)

Constructing Efficient Finite Automata

666

(17)

Another Regular Expression to NFA Algorithm

667

(2)

Transforming an NFA into a DFA

669

(6)

Minimum-State DFAs

675

(6)

Exercises

681

(2)

Regular Language Topics

683

(12)

Regular Grammars

684

(5)

Properties of Regular Languages

689

(4)

Exercises

693

(2)

Chapter Summary

695

(2)

Context-Free Languages and Pushdown Automata

697

(60)

Context-Free Languages

697

(3)

Exercises

700

(1)

Pushdown Automata

700

(17)

Equivalent Forms of Acceptance

703

(4)

Context-Free Grammars and Pushdown Automata

707

(5)

Representing and Executing Pushdown Automata

712

(3)

Exercises

715

(2)

Parsing Techniques

717

(29)

LL(k) Parsing

717

(14)

LR(k) Parsing

731

(13)

Exercises

744

(2)

Context-Free Language Topics

746

(10)

Transforming Grammars

746

(5)

Properties of Context-Free Languages

751

(4)

Exercises

755

(1)

Chapter Summary

756

(1)

Turing Machines and Equivalent Models

757

(34)

Turing Machines

757

(17)

Definition of a Turing Machine

758

(4)

Turing Machines with Output

762

(3)

Alternative Definitions

765

(4)

A Universal Turing Machine

769

(4)

Exercises

773

(1)

The Church-Turing Thesis

774

(15)

Equivalence of Computational Models

775

(1)

A Simple Programming Language

776

(2)

Recursive Functions

778

(3)

Machines That Transform Strings

781

(6)

Exercises

787

(2)

Chapter Summary

789

(2)

Computational Notions

791

(36)

Computability

791

(12)

Effective Enumerations

792

(3)

The Halting Problem

795

(1)

The Total Problem

796

(2)

Amazon no longer offers textbook rentals. We do!

Discrete Structures, Logic, and Computability

Discrete Structures, Logic, and Computability

9780763718435

0763718432

Summary

Table of Contents

Supplemental Materials

Write a Review

Amazon no longer offers textbook rentals. We do!

Discrete Structures, Logic, and Computability

Discrete Structures, Logic, and Computability

9780763718435

0763718432

Summary

Table of Contents

Supplemental Materials

Write a Review

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