Decision Problems For Convex Languages

TL;DR

This paper investigates the computational complexity of decision problems related to convex languages, establishing polynomial-time algorithms for DFA representations and PSPACE-hardness for NFA, along with bounds on shortest non-convex words.

Contribution

It provides the first complexity classifications for convex language decision problems and tight bounds on minimal non-convex words for various subclasses.

Findings

Polynomial-time decidability for DFA representations.

PSPACE-hardness for NFA representations.

Tight bounds on shortest non-convex words.

Abstract

In this paper we examine decision problems associated with various classes of convex languages, studied by Ang and Brzozowski (under the name "continuous languages"). We show that we can decide whether a given language L is prefix-, suffix-, factor-, or subword-convex in polynomial time if L is represented by a DFA, but that the problem is PSPACE-hard if L is represented by an NFA. In the case that a regular language is not convex, we prove tight upper bounds on the length of the shortest words demonstrating this fact, in terms of the number of states of an accepting DFA. Similar results are proved for some subclasses of convex languages: the prefix-, suffix-, factor-, and subword-closed languages, and the prefix-, suffix-, factor-, and subword-free languages.