A necessary and sufficient pumping lemma for regular languages

A necessary and sufficient pumping lemma for regular languages SIGACT News Summer 197 8 A Necessary and Sufficient Pumping Lemma for Regular Language s Jeffrey Jaffe, MI T 1. Introductio n A well known necessary and sufficient condition to guarantee that a language is regular i s provided by Neroce ' s Theorem [3] . However, the "pumping lemma" is often a more useful tool i n demonstrating properties of regular languages . The question has been raised [1] whether there is a necessary and sufficient pumping lemma . In this note we answer the question in the affirmative . 2. Pumping Lemma Theorem . Let E be a finite alphabet . A language LcE * is regular if f (*) ]k.EIN such that VyEE * with Iyl=k 3u,v,wEE * satisfying y=uvw, ivl>O and dzEE * ,diErN yzELu.v t rvzEL and yzsLuv t wzi$L. Let M-(Q,E,S,g0,F) be a deterministic finite state recognizer (fsr) for L wher e Q is the state set, 6 :Q X E Q is the state transition function, g 0 EQ is the initial state and Fc Q is the set of final states . We let b ' denote the extension of b to domain Q X E * in the usual http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ACM SIGACT News Association for Computing Machinery

A necessary and sufficient pumping lemma for regular languages

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Publisher
Association for Computing Machinery
Copyright
Copyright © 1978 by ACM Inc.
ISSN
0163-5700
D.O.I.
10.1145/990524.990528
Publisher site
See Article on Publisher Site

Abstract

SIGACT News Summer 197 8 A Necessary and Sufficient Pumping Lemma for Regular Language s Jeffrey Jaffe, MI T 1. Introductio n A well known necessary and sufficient condition to guarantee that a language is regular i s provided by Neroce ' s Theorem [3] . However, the "pumping lemma" is often a more useful tool i n demonstrating properties of regular languages . The question has been raised [1] whether there is a necessary and sufficient pumping lemma . In this note we answer the question in the affirmative . 2. Pumping Lemma Theorem . Let E be a finite alphabet . A language LcE * is regular if f (*) ]k.EIN such that VyEE * with Iyl=k 3u,v,wEE * satisfying y=uvw, ivl>O and dzEE * ,diErN yzELu.v t rvzEL and yzsLuv t wzi$L. Let M-(Q,E,S,g0,F) be a deterministic finite state recognizer (fsr) for L wher e Q is the state set, 6 :Q X E Q is the state transition function, g 0 EQ is the initial state and Fc Q is the set of final states . We let b ' denote the extension of b to domain Q X E * in the usual

Journal

ACM SIGACT NewsAssociation for Computing Machinery

Published: Jul 1, 1978

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