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Document
DOI: 10.4230/LIPIcs.CSL.2025.36
Kleene Algebra with Commutativity Conditions Is Undecidable

Authors: Arthur Azevedo de Amorim, Cheng Zhang, and Marco Gaboardi

Published in: LIPIcs, Volume 326, 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)

Abstract
We prove that the equational theory of Kleene algebra with commutativity conditions on primitives (or atomic terms) is undecidable, thereby settling a longstanding open question in the theory of Kleene algebra. While this question has also been recently solved independently by Kuznetsov, our results hold even for weaker theories that do not support the induction axioms of Kleene algebra.
Cite as

Arthur Azevedo de Amorim, Cheng Zhang, and Marco Gaboardi. Kleene Algebra with Commutativity Conditions Is Undecidable. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 36:1-36:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{azevedodeamorim_et_al:LIPIcs.CSL.2025.36,
  author =	{Azevedo de Amorim, Arthur and Zhang, Cheng and Gaboardi, Marco},
  title =	{{Kleene Algebra with Commutativity Conditions Is Undecidable}},
  booktitle =	{33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)},
  pages =	{36:1--36:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-362-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{326},
  editor =	{Endrullis, J\"{o}rg and Schmitz, Sylvain},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2025.36},
  URN =		{urn:nbn:de:0030-drops-227933},
  doi =		{10.4230/LIPIcs.CSL.2025.36},
  annote =	{Keywords: Kleene Algebra, Hypotheses, Complexity}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2024.157
Domain Reasoning in TopKAT

Authors: Cheng Zhang, Arthur Azevedo de Amorim, and Marco Gaboardi

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
TopKAT is the algebraic theory of Kleene algebra with tests (KAT) extended with a top element. Compared to KAT, one pleasant feature of TopKAT is that, in relational models, the top element allows us to express the domain and codomain of a relation. This enables several applications in program logics, such as proving under-approximate specifications or reachability properties of imperative programs. However, while TopKAT inherits many pleasant features of KATs, such as having a decidable equational theory, it is incomplete with respect to relational models. In other words, there are properties that hold true of all relational TopKATs but cannot be proved with the axioms of TopKAT. This issue is potentially worrisome for program-logic applications, in which relational models play a key role. In this paper, we further investigate the completeness properties of TopKAT with respect to relational models. We show that TopKAT is complete with respect to (co)domain comparison of KAT terms, but incomplete when comparing the (co)domain of arbitrary TopKAT terms. Since the encoding of under-approximate specifications in TopKAT hinges on this type of formula, the aforementioned incompleteness results have a limited impact when using TopKAT to reason about such specifications.
Cite as

Cheng Zhang, Arthur Azevedo de Amorim, and Marco Gaboardi. Domain Reasoning in TopKAT. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 157:1-157:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{zhang_et_al:LIPIcs.ICALP.2024.157,
  author =	{Zhang, Cheng and de Amorim, Arthur Azevedo and Gaboardi, Marco},
  title =	{{Domain Reasoning in TopKAT}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{157:1--157:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.157},
  URN =		{urn:nbn:de:0030-drops-203003},
  doi =		{10.4230/LIPIcs.ICALP.2024.157},
  annote =	{Keywords: Kleene algebra, Kleene Algebra With Tests, Kleene Algebra With Domain, Kleene Algebra With Top and Tests, Completeness, Decidability}
}
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