# AUTOMATA AND COMPUTABILITY

Language: English

Pages: 413

ISBN: 0387949070

Format: PDF / Kindle (mobi) / ePub

This textbook provides undergraduate students with an introduction to the basic theoretical models of computability, and develops some of the model's rich and varied structure. The first part of the book is devoted to finite automata and their properties. Pushdown automata provide a broader class of models and enable the analysis of context-free languages. In the remaining chapters, Turing machines are introduced and the book culminates in analyses of effective computability, decidability, and Gödel's incompleteness theorems. Students who already have some experience with elementary discrete mathematics will find this a well-paced first course, and a number of supplementary chapters introduce more advanced concepts.

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Social Issues in the Information Age Kozen, Automata and Computability Merritt and Stix, Migrating from Pascal to C++ Pearce, Programming and Meta-Programming in Scheme Zeigler, Objects and Systems Dexter C. Kozen Automata and Computability ~ Springer Dexter C. Kozen Department of Computer Science Cornell Uni versity Ithaca, NY 14853-7501 USA Series Edirors DavidGries Department of Computer Science 415 Boyd Studies Research Center The University of Georgia Athens, Georgia 30602 USA Fred

0, {p}, {q}, {r}, {p,q}, {p,r}, {q,r}, {p,q,r}. Here is the deterministic automaton M: - 0 {p} {q} {r}F {p,q} {p,r}F {q,r}F {p,q,r}F 0 1 0 0 {P} {p,q} {r} {r} 0 0 {p,r} {p,q,r} {p,q} {r} {p,q,r} {P} {r} {p,r} For example, if we have pebbles on p and q (the fifth row of the table), and if we see input symbol 0 (first column), then in the next step there will be pebbles on p and r. This is because in the automaton N, p is reachable from p under input 0 and r is reachable from q

E F 6(q, ay) E F <==} 6(6(q,a),y) E F. <==} <==} since p ';:!, q Since y was arbitrary, 6(p, a) ';:!, 6(q, a) by definition of >:::. Lemma 13.6 PEF <==} IP] o E F'. Proof. The direction =? is immediate from the definition of F'. For the direction ¢::, we need to show that if p ';:!, q and p E F, then q E F. In other words, every ';:!,-equivalence class is either a subset of F or disjoint from F. This follows immediately by taking x = € in the definition of p ';:!, q. 0 Lemma 13.7 For

unmarked pairs and find out that {1,2} -+ {3,4} and {3,4} -+ {5,5} under both a and b, and neither {3,4} nor {S,S} is marked, so there are no new marks. We are left with unmarked pairs 0 {1,2} and {3, 4}, indicating that 1 ~ 2 and 3 ~ 4. Example 14.2 Now let's do Example 13.4 of Lecture 13. -+ a 0 1 1F 2 3 4F 5 2 3 4 5 0 Here is the table after step 2. o ,f 1 ,f ,f ,f 2 3 ,f ,f ,f 4 ,f 5 Then: • {O,2} -+ {1,3}, which is marked, so mark {O,2}. A Minimization Algorithm 87

called the carrier of A, along with a map that assigns a function IA or relation RA of the appropriate arity to each function symbol leE or relation symbol R e E. If I is an n-ary function symbol, then the function associated with I must be an n-ary function IA : An -+ A. If R is an n-ary relation symbol, then the relation RA must be an n-ary relation RA ~ An. Constant function symbols c are interpreted as O-ary functions (functions with no inputs), which are just elements cA of A. A unary