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Automata Classes Accepting Languages Whose Commutative Closure is Regular

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SOFSEM 2024: Theory and Practice of Computer Science (SOFSEM 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14519))

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Abstract

The commutative closure operation, which corresponds to the Parikh image, is a natural operation on formal languages occurring in verification and model-checking. Commutative closures of regular languages correspond to semilinear sets and, by Parikh’s theorem, to the commutative closures of context-free languages. The commutative closure is not regularity-preserving on the class of regular languages, for example already the commutative closure of the simple language \((ab)^*\) is not regular. Here, we show that the commutative closure of a binary regular language accepted by a circular automaton yields a regular language. Then, we deduce a sufficient condition on the cycles in automata for regularity of the commutative closure. This yields this property, for example, for the following classes of automata: automata with threshold one transformation semigroups, automata with simple idempotents and almost-group automata. The fact that the commutative closure on group languages and polynomials of group languages is regularity-preserving is known in the literature. Polynomials of group languages correspond to level one-half of the group hierarchy. We also show that on the next level in this hierarchy, i.e., level one, this property is lost and the commutative closure is no longer regularity-preserving. Lastly, we give a binary circular automaton not contained in the largest proper positive variety \(\mathcal W\) closed under shuffle and commutative closure.

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Notes

  1. 1.

    A language is cofinite if its complement is finite.

  2. 2.

    We note that this result can be stated a little more general using so-called chains of simple semigroups and well-quasi order arguments due to Kunc [26] but which we leave out due to space.

  3. 3.

    See Sect. 8 for a simpler example due to an anonymous reviewer.

  4. 4.

    We note that it follows easily from results in the literature that languages accepted by binary circular automata are not in the other classes mentioned in this work closed for commutation.

  5. 5.

    For the definition of monoids and semigroups and their relation to automata and formal language theory we refer to the literature, for example [32].

  6. 6.

    An element \(y \in M\) is idempotent if \(yy = y\).

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Acknowledgement

I thank the anonymous reviewers for careful reading and spotting typos. I also thank one reviewer for mentioning the simpler argument for the result from Sect. 6 (see the conclusion, Sect. 8).

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  1. Universität Trier, Behringstraße 21, 54296, Trier, Germany

    Stefan Hoffmann

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  1. University of Trier, Trier, Germany

    Henning Fernau

  2. UNSW Sydney, Sydney, NSW, Australia

    Serge Gaspers

  3. CNRS and University of Bordeaux, Talence, France

    Ralf Klasing

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Hoffmann, S. (2024). Automata Classes Accepting Languages Whose Commutative Closure is Regular. In: Fernau, H., Gaspers, S., Klasing, R. (eds) SOFSEM 2024: Theory and Practice of Computer Science. SOFSEM 2024. Lecture Notes in Computer Science, vol 14519. Springer, Cham. https://doi.org/10.1007/978-3-031-52113-3_22

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