Distinguishability Operations and Closures on Regular Languages

TL;DR

This paper investigates the properties and complexities of distinguishability operations on regular languages, including their fixed points, bounds, and generalizations to various quotient types.

Contribution

It introduces a formal characterization of distinguishability operations, analyzes their fixed points, and provides tight bounds on their state complexity, extending results to different quotient types.

Findings

The distinguishability operation has a fixed point after iteration.

An upper bound for the state complexity of the operation is established and shown to be tight.

The set of minimal distinguishing words has at most n-1 elements, where n is the language's state complexity.

Abstract

Given a regular language $L$, we study the language of words $\mathsf{D}(L)$, that distinguish between pairs of different left-quotients of $L$. We characterize this distinguishability operation, show that its iteration has always a fixed point, and we generalize this result to operations derived from closure operators and Boolean operators. We give an upper bound for the state complexity of the distinguishability operation, and prove its tightness. We show that the set of minimal words that can be used to distinguish between different left-quotients of a language $L$ has at most $n-1$ elements, where $n$ is the state complexity of $L$, and we also study the properties of its iteration. We generalize the results for the languages of words that distinguish between pairs of different right-quotients and two-sided quotients of a language $L$.