A Lower Bound for Primality of Finite Languages

TL;DR

This paper investigates the computational complexity of determining whether a finite language is prime, establishing NP lower bounds and Pi_2^P upper bounds for this decision problem.

Contribution

It provides the first complexity bounds for primality of finite languages, using techniques from prior work on deterministic finite automata.

Findings

NP lower bound for primality decision problem

Pi_2^P upper bound for primality decision problem

Complexity bounds for finite languages given as automata

Abstract

A regular language $L$ is said to be prime, if it is not the product of two non-trivial languages. Martens et al. settled the exact complexity of deciding primality for deterministic finite automata in 2010. For finite languages, Mateescu et al. and Wieczorek suspect the $\mathrm{NP}\text{ - }completeness$ of primality, but no actual bounds are given. Using techniques of Martens et al., we prove the $\mathrm{NP}$ lower bound and give a $\Pi_{2}^{\mathrm{P}}$ upper bound for deciding primality of finite languages given as deterministic finite automata.