On $k$-piecewise testability (preliminary report)

TL;DR

This paper investigates the complexity of determining the minimal $k$ for which a language recognized by a DFA is $k$-piecewise testable, providing bounds and analysis for small values of $k$.

Contribution

It offers a complexity analysis and tight bounds on the depth of minimal DFAs recognizing $k$-piecewise testable languages, improving understanding of their structural properties.

Findings

Upper bound on $k$ can be exponentially larger than the minimal $k$

Provided tight bounds on DFA depth for $k$-PT languages

Analyzed complexity for small $k$ values

Abstract

For a non-negative integer $k$, a language is $k$-piecewise test\-able ($k$-PT) if it is a finite boolean combination of languages of the form $\Sigma^* a_1 \Sigma^* \cdots \Sigma^* a_n \Sigma^*$ for $a_i\in\Sigma$ and $0\le n \le k$. We study the following problem: Given a DFA recognizing a piecewise testable language, decide whether the language is $k$-PT. We provide a complexity bound and a detailed analysis for small $k$'s. The result can be used to find the minimal $k$ for which the language is $k$-PT. We show that the upper bound on $k$ given by the depth of the minimal DFA can be exponentially bigger than the minimal possible $k$, and provide a tight upper bound on the depth of the minimal DFA recognizing a $k$-PT language.