Computability, Complexity and Languages: Fundamentals of Theoretical Computer Science (Computer Science and Applied Mathematics)

Computability, Complexity and Languages: Fundamentals of Theoretical Computer Science (Computer Science and Applied Mathematics)

Martin Davis

Language: English

Pages: 425

ISBN: 0122063805

Format: PDF / Kindle (mobi) / ePub

This introductory text covers the key areas of computer science, including recursive function theory, formal languages, and automata. It assumes a minimal background in formal mathematics. The book is divided into five parts: Computability, Grammars and Automata, Logic, Complexity, and Unsolvability.

* Computability theory is introduced in a manner that makes maximum use of previous programming experience, including a "universal" program that takes up less than a page.
* The number of exercises included has more than tripled.
* Automata theory, computational logic, and complexity theory are presented in a flexible manner, and can be covered in a variety of different arrangements.

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(Vt) \&(Vt)

if / = 0 1)} otherwise An easy induction argument on / shows that $>e is a total, and therefore computable, function. Now, e satisfies the equations O,(0) = k e is obtained from g by primitive recursion of the form (2.1) in Chapter 3, so the recursion theorem gives us another proof of Theorem 2.1 in Chapter 3. In fact, the recursion theorem can be used to justify definitions based on much more general forms of recursion, which explains how it came by

we shall also write as (f>t(x), that will enumerate the unary primitive recursive functions: (x + 1 0 Kx) r(x) if if if if if t / í / / = = = = = 0 1 2 3 4, n > 0 (l(n)(x),(f>r(n)(x)) if / = 5, « > 0 0 4>Kn)((x - if / = 6, n > 0 and .v = 0 if t = 6, A7 > 0 and A is odd 0/2) 0 and x is even Here 0(x), (fr^ix), 2(x), ^>3(JC) are the four initial functions. For t > 3, t is represented as 3« + / where n > 0 and / = 4, 5, or 6, the three operations

each symbol a in the alphabet A and each label L in the n symbol alphabet {sl9...,sn} ordered as shown is to replace the numerical value x by / * AZ|A| + x. Just as the instructions of S? are natural as basic numerical operations, but complex as string operations, so the instructions of J/^ are natural as basic string operations, but complex as numerical operations. We now give some macros for use in S?n with the corresponding expansions. 1. The macro IF V ¥= 0 GOTO L has the expansion IF V ENDS

computer science / Martin D. Davis, Ron Sigal, Elaine J. Weyuker. -2nded. p. cm. -(Computer science and applied mathematics) Includes bibliographical references and index. ISBN 0-12-206382-1 1. Machine theory. 2. Computational complexity. 3. Formal Languages. I. Sigal, Ron. II. Weyuker, Elaine J. III. Title. IV. Series. QA267.D38 1994 511.3-dc20 93-26807 CIP PRINTED IN THE UNITED STATES OF AMERICA 02 03 04 05 06 SB 9 8 7 6 To the memory of Helen and Harry Davis and to Hannah and Herman Sigal

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