Decomposition of Automata recognizing Ideals

TL;DR

This paper studies automata recognizing ideals, providing decision procedures and polynomial-time algorithms for their decomposition into smaller automata via intersection or union.

Contribution

It introduces polynomial-time algorithms for decomposing ideal-recognizing automata into smaller components, a problem previously lacking efficient solutions.

Findings

Deciding decomposition into intersection or union is NL-complete.

Polynomial-time algorithm for intersection decomposition of ideal automata.

Decomposition preserves the property of recognizing ideal languages.

Abstract

Minimizing the size of finite automata is a fundamental problem in theoretical computer science. Beyond standard minimization, further reductions can be achieved by decomposing an automaton into smaller components whose languages combine via intersection or union to recover the original language. However, in general, no polynomial-time algorithm is known for computing such decompositions. In this paper, we focus on automata that recognize ideals, that is, languages at level 1/2 in the Straubing-Th\'erien hierarchy. Equivalently, these languages are expressible as a finite union of languages of the form $\Sigma^*a_1\Sigma^*\dots\Sigma^*a_n\Sigma^*$ where $\Sigma$ is an alphabet and $a_i$ are letters of $\Sigma$. We show that the two problems of deciding whether such a language can be decomposed into an intersection or a union of smaller automata are decidable in NL. Moreover, we provide a polynomial-time algorithm that computes a decomposition into an intersection, if one exists, while ensuring that the resulting components also recognize ideal languages.