Decomposition of Automata recognizing Ideals

2026-04-28

AI Summary

This study addresses the decomposability of automata recognizing ideal languages in level 1/2 of the Straubing–Thérien hierarchy, specifically investigating whether such automata can be expressed as intersections or unions of smaller automata. For the first time, we establish that the corresponding decomposition decision problem lies within the complexity class NL. Furthermore, we propose a polynomial-time algorithm that effectively computes intersection decompositions while preserving the ideality of the language. By integrating techniques from formal language theory with structural analysis of automata, our approach enables efficient and property-preserving decomposition of ideal-language automata, significantly enhancing their tractability and modularity.

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-Thérien hierarchy. Equivalently, these languages are expressible as a finite union of languages of the form $Σ^*a_1Σ^*\dotsΣ^*a_nΣ^*$ where $Σ$ is an alphabet and $a_i$ are letters of $Σ$. 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.

Problem

• automata decomposition
• ideal languages
• Straubing-Thérien hierarchy
• finite automata
• language decomposition

Innovation

• automata decomposition
• ideal languages
• polynomial-time algorithm
• NL complexity
• Straubing-Thérien hierarchy