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.

中文翻译：

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确定性有限自动机的局部阈值可测试性验证算法

局部阈值可测试语言L是具有以下属性的语言：对于某些非负整数k和l，单词u是否在语言L中取决于（1）长度为k>的单词u的前缀和后缀1和（2）单词u的长度为k的中间子串的集合，其中至少出现j次的子串的集合是相同的，对于j <=L。对于给定的k和L，该语言称为l阈值k可测试的 如果自动机接受l阈值k可测语言，则该有限自动机称为l阈值k可测。在本文中，找到了自动机局部可阈值测试的必要和充分条件。我们引入了第一个多项式时间算法，基于此特征来验证自动机的局部阈值可测试性。