Closures in Formal Languages: Concatenation, Separation, and Algorithms

TL;DR

This paper explores the properties of open and closed languages in formal language theory, focusing on their behavior under concatenation, topological analogues, and computational complexity of related decision problems.

Contribution

It introduces new results on how openness and closedness are preserved under concatenation and establishes complexity bounds for deciding closure in DFA and NFA languages.

Findings

Closure under concatenation depends on language properties.

A clopen partition exists if and only if words commute.

Deciding closure is quadratic time for DFA, PSPACE-complete for NFA.

Abstract

We continue our study of open and closed languages. We investigate how the properties of being open and closed are preserved under concatenation. We investigate analogues, in formal languages, of the separation axioms in topological spaces; one of our main results is that there is a clopen partition separating two words if and only if the words commute. We show that we can decide in quadratic time if the language specified by a DFA is closed, but if the language is specified by an NFA, the problem is PSPACE-complete.