# On boolean combinations forming piecewise testable languages

From International Center for Computational Logic

# On boolean combinations forming piecewise testable languages

##### Tomáš MasopustTomáš Masopust, Michaël ThomazoMichaël Thomazo

Tomáš Masopust, Michaël Thomazo

Theoretical Computer Science, 682:165-179, June 2017

**On boolean combinations forming piecewise testable languages**Theoretical Computer Science, 682:165-179, June 2017

**Kurzfassung****Abstract**

A regular language is k-piecewise testable (k -PT) if it is a Boolean combination of languages of the form L_{a_1a_2…a_n} = Σ*a_1Σ*a_2Σ*⋯Σ*a_nΣ*, where a_i ∈ Σ and 0≤n≤k. Given a finite automaton A, if the language L(A) is piecewise testable, we want to express it as a Boolean combination of languages of the above form. The idea is as follows. If the language is k-PT, then there exists a congruence ∼_k of finite index such that L(A) is a finite union of ∼_k-classes. Every such class is characterized by an intersection of languages of the from L_u, for the length of u at most k, and their complements. To represent the ∼_k-classes, we make use of the ∼_k-canonical DFA. We identify the states of the ∼_k-canonical DFA whose union forms the language L(A) and use them to construct the required Boolean combination. We study the computational and descriptional complexity of related problems.**Projekt:****Project:**Cfaed, DIAMOND**Forschungsgruppe:****Research Group:**Wissensbasierte Systeme

```
@article{MT2017,
author = {Tom{\'{a}}{\v{s}} Masopust and Micha{\"{e}}l Thomazo},
title = {On boolean combinations forming piecewise testable languages},
journal = {Theoretical Computer Science},
volume = {682},
publisher = {Elsevier},
year = {2017},
month = {June},
pages = {165-179},
doi = {10.1016/j.tcs.2017.01.017}
}
```