TY - JOUR

T1 - Optimal estimation on the order of local testability of finite automata

AU - Trahtman, A. N.

PY - 2000/1/17

Y1 - 2000/1/17

N2 - A locally testable language L is a language with the property that for some nonnegative integer k, called the order of local testability, whether or not a word u is in the language L depends on (1) the prefix and suffix of the word u of length k - 1 and (2) the set of subwords of length k of the word u. For given k the language is called k-testable. We improve the upper bound on the order of local testability of a locally testable deterministic finite automaton with n states to 1/2(n2 - n) + 1. This bound is the best possible. We give an answer to the following conjecture of Kim, McNaughton and McCloskey for deterministic finite locally testable automata with n states: "Is the order of local testability no greater than O(n1.5) when the alphabet size is two?" Our answer is negative. In the case of size two the situation is the same as in general case: the order of local testability is Ω(n2). The necessary and sufficient conditions for the language of an automaton to be k-testable are given in terms of the length of paths of a related graph. Some estimates of the bounds on the order of local testability follow from these results.

AB - A locally testable language L is a language with the property that for some nonnegative integer k, called the order of local testability, whether or not a word u is in the language L depends on (1) the prefix and suffix of the word u of length k - 1 and (2) the set of subwords of length k of the word u. For given k the language is called k-testable. We improve the upper bound on the order of local testability of a locally testable deterministic finite automaton with n states to 1/2(n2 - n) + 1. This bound is the best possible. We give an answer to the following conjecture of Kim, McNaughton and McCloskey for deterministic finite locally testable automata with n states: "Is the order of local testability no greater than O(n1.5) when the alphabet size is two?" Our answer is negative. In the case of size two the situation is the same as in general case: the order of local testability is Ω(n2). The necessary and sufficient conditions for the language of an automaton to be k-testable are given in terms of the length of paths of a related graph. Some estimates of the bounds on the order of local testability follow from these results.

KW - Finite automaton

KW - Language

KW - Locally testable

KW - Order of local testability

KW - Semigroup

UR - http://www.scopus.com/inward/record.url?scp=0347998643&partnerID=8YFLogxK

U2 - 10.1016/s0304-3975(99)00017-1

DO - 10.1016/s0304-3975(99)00017-1

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AN - SCOPUS:0347998643

SN - 0304-3975

VL - 231

SP - 59

EP - 74

JO - Theoretical Computer Science

JF - Theoretical Computer Science

IS - 1

ER -