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Decomposition of Automata Recognizing Ideals

Authors Mathias Berry , Pierre-Cyrille Héam , Ismaël Jecker

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LIPIcs.CONCUR.2026.16.pdf
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ACM Subject Classification
  • Theory of computation → Regular languages
Keywords
  • Finite state automata
  • decomposition
  • Shuffle ideals

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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^*a₁Σ^*… Σ^*a_nΣ^* where Σ is an alphabet and a_i are letters of Σ. We show that the two problems of deciding whether an automata recognizing an ideal 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.

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Mathias Berry, Pierre-Cyrille Héam, and Ismaël Jecker. Decomposition of Automata Recognizing Ideals. In 37th International Conference on Concurrency Theory (CONCUR 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 391, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CONCUR.2026.16

Author Details

Mathias Berry
  • Université Marie et Louis Pasteur, CNRS, institut FEMTO-ST, F-25000 Besançon, France
Pierre-Cyrille Héam
  • Université Marie et Louis Pasteur, CNRS, institut FEMTO-ST, F-25000 Besançon, France
Ismaël Jecker
  • Université Marie et Louis Pasteur, CNRS, institut FEMTO-ST, F-25000 Besançon, France

Funding

  • Jecker, Ismaël: This research was partially funded by the Agence Nationale de la Recherche (ANR) grant ANR-25-CE48-2447 FAVOR.

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