# Separability by Piecewise Testable Languages is PTime-Complete

## TL;DR

This paper investigates the computational complexity of determining whether two regular languages can be separated by piecewise testable languages, establishing PTime-completeness for DFAs and NFAs.

## Contribution

It proves that the separability problem by piecewise testable languages is PTime-complete for both minimal DFAs and NFAs, filling a gap in complexity understanding.

## Key findings

- Separability by piecewise testable languages is PTime-hard for DFAs.
- The problem remains PTime-hard even for minimal DFAs.
- Membership testing for NFAs is PSpace-complete.

## Abstract

Piecewise testable languages form the first level of the Straubing-Th\'erien hierarchy. The membership problem for this level is decidable and testing if the language of a DFA is piecewise testable is NL-complete. The question has not yet been addressed for NFAs. We fill in this gap by showing that it is PSpace-complete. The main result is then the lower-bound complexity of separability of regular languages by piecewise testable languages. Two regular languages are separable by a piecewise testable language if the piecewise testable language includes one of them and is disjoint from the other. For languages represented by NFAs, separability by piecewise testable languages is known to be decidable in PTime. We show that it is PTime-hard and that it remains PTime-hard even for minimal DFAs.