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On the Complexity of Kleene Algebra with Domain

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Relational and Algebraic Methods in Computer Science (RAMiCS 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13896))

Abstract

We prove that the equational theory of Kleene algebra with domain is EXPTIME-complete. Our proof makes essential use of Hollenberg’s equational axiomatization of program equations valid in relational test algebra. We also show that the equational theory of Kleene algebra with domain coincides with the equational theory of *-continuous Kleene algebra with domain.

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Notes

  1. 1.

    Extensionality is closely related to the property of separability, well known from the literature on dynamic algebra [12, 22].

  2. 2.

    We note that the the main result of our original submission was an EXPTIME-completeness result on the equational theory of *-continuous KAD. The proof was essentially the same as the one given here, but we relied on the assumption of *-continuity in the proof of Theorem 4 (in particular, the argument showing that the translation of T8 is valid). An anonymous reviewer showed that the proof of Theorem 4 can be carried out without assuming *-continuity and that the resulting argument establishes in conjunction with Lemma 4 that the equational theory of KAD coincides with the equational theory of *-continuous KAD. We gratefully acknowledge the input of the reviewer and we suggest that the results of this paper be considered joint work of the official author and the anonymous reviewer.

  3. 3.

    In the original formulation [3], Kleene algebras with domain were expansions of Kleene algebras with test.

  4. 4.

    As before, we save letters by re-using p, q etc. as variables ranging over KAD terms, \(Tm_{KAD}\) (defined as expected), and letting the context disambiguate.

  5. 5.

    McLean [20] studies the equational theory of relational Kleene algebras expanded by the relational domain operator D seen as primitive. He shows that this theory is decidable, with a 3EXPTIME upper bound. The *-free fragment of \(Eq(\textsf{RKAD})\) is not finitely based [9], in contrast to the *-free fragment of \(Eq(\textsf{KAD})\).

  6. 6.

    Extensionality is closely related to the property of separability, well known from the literature on dynamic algebra [12, 22].

  7. 7.

    Strictly speaking, we should replace \(\textsf{a}\) (a total operation) by its restriction to \(\textsf{d}(\mathcal {D})\), but we will not bother with this detail.

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Acknowledgement

The author is grateful to three anonymous reviewers for valuable comments, and to Johann J. Wannenburg and Adam Přenosil for helpful discussions. Work on this paper was supported by the long-term strategic development financing of the Institute of Computer Science (RVO:67985807).

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  1. The Czech Academy of Sciences, Institute of Computer Science, Prague, Czech Republic

    Igor Sedlár

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  1. Igor Sedlár
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Correspondence to Igor Sedlár .

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  1. Deutsches Zentrum für Luft- und Raumfahrt, Augsburg, Germany

    Roland Glück

  2. Aix-Marseille University, Marseille, France

    Luigi Santocanale

  3. Brock University, St Catharines, ON, Canada

    Michael Winter

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Sedlár, I. (2023). On the Complexity of Kleene Algebra with Domain. In: Glück, R., Santocanale, L., Winter, M. (eds) Relational and Algebraic Methods in Computer Science. RAMiCS 2023. Lecture Notes in Computer Science, vol 13896. Springer, Cham. https://doi.org/10.1007/978-3-031-28083-2_13

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  • DOI: https://doi.org/10.1007/978-3-031-28083-2_13

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  • Online ISBN: 978-3-031-28083-2

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