An algorithm to verify local threshold testability of deterministic finite automata

TL;DR

This paper introduces a polynomial-time algorithm to verify whether a deterministic finite automaton recognizes a locally threshold testable language, based on new necessary and sufficient conditions.

Contribution

It provides the first polynomial-time algorithm for verifying local threshold testability of automata, along with improved methods for testing local testability.

Findings

Established necessary and sufficient conditions for local threshold testability.

Developed the first polynomial-time verification algorithm.

Presented an improved algorithm for local testability verification.

Abstract

A locally threshold testable language L is a language with the property that for some non negative integers k and l, 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 intermediate substrings of length k of the word u where the sets of substrings occurring at least j times are the same, for j <= L. For given k and L the language is called l-threshold k-testable. A finite deterministic automaton is called l-threshold k-testable if the automaton accepts a l-threshold k-testable language. In this paper, the necessary and sufficient conditions for an automaton to be locally threshold testable are found. We introduce the first polynomial time algorithm to verify local threshold testability of the automaton based on this characterization. New version of polynomial time algorithm to verify the local testability will be presented too.