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A Unifying Categorical View of Nondeterministic Iteration and Tests

Authors Sergey Goncharov , Tarmo Uustalu

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Author Details

Sergey Goncharov
  • Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany
Tarmo Uustalu
  • Reykjavik University, Iceland, and Tallinn University of Technology, Estonia

Funding

  • Goncharov, Sergey: German Research Foundation (DFG) project 501369690, Icelandic Research Fund project 228684-052
  • Uustalu, Tarmo: Icelandic Research Fund project 228684-052

Cite AsGet BibTex

Sergey Goncharov and Tarmo Uustalu. A Unifying Categorical View of Nondeterministic Iteration and Tests. In 35th International Conference on Concurrency Theory (CONCUR 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 311, pp. 25:1-25:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.CONCUR.2024.25

Abstract

We study Kleene iteration in the categorical context. A celebrated completeness result by Kozen introduced Kleene algebra (with tests) as a ubiquitous tool for lightweight reasoning about program equivalence, and yet, numerous variants of it came along afterwards to answer the demand for more refined flavors of semantics, such as stateful, concurrent, exceptional, hybrid, branching time, etc. We detach Kleene iteration from Kleene algebra and analyze it from the categorical perspective. The notion, we arrive at is that of Kleene-iteration category (with coproducts and tests), which we show to be general and robust in the sense of compatibility with programming language features, such as exceptions, store, concurrent behaviour, etc. We attest the proposed notion w.r.t. various yardsticks, most importantly, by characterizing the free model as a certain category of (nondeterministic) rational trees.

Subject Classification

ACM Subject Classification
  • Theory of computation → Categorical semantics
  • Theory of computation → Axiomatic semantics
Keywords
  • Kleene iteration
  • Elgot iteration
  • Kleene algebra
  • coalgebraic resumptions

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