On Separation by Locally Testable and Locally Threshold Testable Languages

TL;DR

This paper investigates the decidability of finding locally testable and locally threshold testable separators between regular languages, providing decision procedures and exploring extensions to context-free languages.

Contribution

It introduces decision algorithms for the separator problem and offers a characterization of possible separators, advancing understanding of language separation.

Findings

Decidable procedures for locally testable separators

Decidable procedures for locally threshold testable separators

Extension considerations to context-free languages

Abstract

A separator for two languages is a third language containing the first one and disjoint from the second one. We investigate the following decision problem: given two regular input languages, decide whether there exists a locally testable (resp. a locally threshold testable) separator. In both cases, we design a decision procedure based on the occurrence of special patterns in automata accepting the input languages. We prove that the problem is computationally harder than deciding membership. The correctness proof of the algorithm yields a stronger result, namely a description of a possible separator. Finally, we discuss the same problem for context-free input languages.