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Kleene algebra with tests (KAT) is a foundational equational framework for reasoning about programs, which has found applications in program transformations, networking and compiler optimizations, among many other areas. In his seminal work, Kozen proved that KAT subsumes propositional Hoare logic, showing that one can reason about the (partial) correctness of while programs by means of the equational theory of KAT. In this work, we investigate the support that KAT provides for reasoning about incorrectness, instead, as embodied by O'Hearn's recently proposed incorrectness logic. We show that KAT cannot directly express incorrectness logic. The main reason for this limitation can be traced to the fact that KAT cannot express explicitly the notion of codomain, which is essential to express incorrectness triples. To address this issue, we study Kleene Algebra with Top and Tests (TopKAT), an extension of KAT with a top element. We show that TopKAT is powerful enough to express a codomain operation, to express incorrectness triples, and to prove all the rules of incorrectness logic sound. This shows that one can reason about the incorrectness of while-like programs by means of the equational theory of TopKAT.
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This content will become publicly available on January 1, 2026
Kleene Algebra with Commutativity Conditions Is Undecidable
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.
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- PAR ID:
- 10611964
- Editor(s):
- Endrullis, Jörg; Schmitz, Sylvain
- Publisher / Repository:
- Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- Date Published:
- Volume:
- 326
- ISSN:
- 1868-8969
- ISBN:
- 978-3-95977-362-1
- Page Range / eLocation ID:
- 36:1-36:25
- Subject(s) / Keyword(s):
- Kleene Algebra Hypotheses Complexity Theory of computation → Automated reasoning Theory of computation → Regular languages
- Format(s):
- Medium: X Size: 25 pages; 822868 bytes Other: application/pdf
- Size(s):
- 25 pages 822868 bytes
- Right(s):
- Creative Commons Attribution 4.0 International license; info:eu-repo/semantics/openAccess
- Sponsoring Org:
- National Science Foundation
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