A characterization of strictly locally testable languages and its application to subsemigroups of a free semigroup

Aldo De Luca, Antonio Restivo
<span title="">1980</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/fbkexx2qg5anljkxraobuswdq4" style="color: black;">Information and Control</a> </i> &nbsp;
A syntactic characterization of strictly locally testable languages is given by means of the concept of constant. If S is a semigroup and X a subset of S, an element c of S is called constant for X if fi)r all p, q, r, s ~ S ~ [pcq, rcs eX =. pcs c X]. The main result of the paper states that a recognizable subset X of a free semigroup A ~ is strictly locally testable if and only if all the idempotents of the syntactic semigroup S(X) of X are constants for X' ~ X~, where ,6 : A +-~ S(X) is the
more &raquo; ... yntactic morphism. By this result some remarkable consequences are derived for recognizable subsemigroups of A". In particular we prove that if X is a recognizable free subsemigroup of A ~ and Y ~ X/X ~ its base then the following conditions are equivalent : (1) X is strictly locally testable. (2) X is locally testable. (3) X is locally parsable and Y is strictly locally testable. (4) X has a bounded synchronization delay and Y is strictly locally testable (5) A positive integer k exists such that all the elements of A + whose length is greater than or equal to k, are constants for X. (6) For all the idempotents e of the syntactic semigroup S(X) of X, eS(X)e _C {e, 0}. 0. INTRODUCTION In this paper a characterization of strictly locally testable languages in the sense of MeNaughton and Papert (1971) is given bv means of the concept of constant introduced by Schiitzenberger (1975). If S is a semigroup and X a subset of S, an element c c-S is a constant for X if for allp, q,r,s~S a pcq, rcs ~ X ~> pcs ~ X. Let S be the free semigroup ,4 ~ generated by a finite alphabet A. The characterization of strictly locally subsets X of A + is that X is strictly locally testable if and only if all the sufficiently long words of A-are constants for X. From this result one derives for recognizable languages a syntactic characterization of strictly local testability: All the idempotents of the syntactic semigroup S(X) have to be constants for X' = X¢, where 4~: A : ~ S(X) is the syntactic morphism. This condition gives then an algebraic decision procedure for strictly local 300
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