Introduction to the Theory of Computation

Introduction to the Theory of Computation

Michael Sipser

Language: English

Pages: 480

ISBN: 113318779X

Format: PDF / Kindle (mobi) / ePub

Gain a clear understanding of even the most complex, highly theoretical computational theory topics in the approachable presentation found only in the market-leading INTRODUCTION TO THE THEORY OF COMPUTATION, 3E. The number one choice for today's computational theory course, this revision continues the book's well-know, approachable style with timely revisions, additional practice, and more memorable examples in key areas. A new first-of-its-kind theoretical treatment of deterministic context-free languages is ideal for a better understanding of parsing and LR(k) grammars. You gain a solid understanding of the fundamental mathematical properties of computer hardware, software, and applications with a blend of practical and philosophical coverage and mathematical treatments, including advanced theorems and proofs. INTRODUCTION TO THE THEORY OF COMPUTATION, 3E's comprehensive coverage makes this a valuable reference for your continued studies in theoretical computing.

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union or concatenation of the languages Rl and R2 , or the star of the language R respectively.) {e}, 1, 0. Don'tconfuse the regular expressionse and The expressione represents the language containing a single string-namely,the empty string-whereas 0 that doesn't contain the any strings.))) represents language 1.3 REGULAR EXPRESSIONS 65) Seemingly,we are in danger of defining the notion of regular expressionin terms of itself. If true, we would have a circulardefinition, which would be

this sequencecannot be near each other. Nowconsiderthe two strings xy z and xy2 z. Thesestrings differ from each other by a single repetition of y, and consequently their lengths differ by the length of y. By condition 3 of the pumping lemma, Ixyl < p and thus Iyl < We have Ixyzl == p2 and so Ixy2 zl < p2 + Butp2 + p < p2 + 2p+ == (p+ Moreover, condition 2 implies that y is not the empty string and so Ixy2 zl p2. Therefore the length of xy2 z liesstrictly betweenthe consecutiveperfect squares

is a larger ED,but [\037] [\037] [\037] Considerthe be the same as in Problem 1.33. of Osand is and let E == {w E Show that E is not \037; I == I the bottom row is the bottom row}. D.Show that D is regular. rows to be and bottom top tt of w is the reverseof the top row of w }. regular. 1.36Let Bn {ak where k is a multiple Bn is regular. 1.37Let Cn [\037] number than of n}.Show that for eachn > 1,the language {xl x is a binary number that is a multiple of n}. Show that

the next symbol from the input and compare it to a. If they match, repeat. If they do not on this branch of the nondeterminism. match, reject c.If the top of stack is the symbol $, enter the acceptstate. Doingso acceptsthe input if it has all beenread.) b. PROOF We now give the formal detailsof the construction of the pushdown == automaton To make the construction clearer we use 6, ql, (Q, shorthand notation for the transition function. This notation provides a way to write an entire string on

the empty string. Otherwise the theorem would be trivially true. Condition 3))) 124 CHAPTER 2 / CONTEXT-FREE LANGUAGES) states that the piecesv, x, and y together have length at most p.This technical condition sometimes is useful in proving that certain languages are not context free.) PROOFIDEA Let A be a CFL and let G be a CFGthat generates it. We must show that any sufficiently long string s in A can be pumped and remain in A. The idea behind this approach is simple. Let s be a very long

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