Document

**Published in:** LIPIcs, Volume 243, 33rd International Conference on Concurrency Theory (CONCUR 2022)

We prove two completeness results for Kleene algebra with a top element, with respect to languages and binary relations. While the equational theories of those two classes of models coincide over the signature of Kleene algebra, this is no longer the case when we consider an additional constant "top" for the full element. Indeed, the full relation satisfies more laws than the full language, and we show that those additional laws can all be derived from a single additional axiom. We recover that the two equational theories coincide if we slightly generalise the notion of relational model, allowing sub-algebras of relations where top is a greatest element but not necessarily the full relation.
We use models of closed languages and reductions in order to prove our completeness results, which are relative to any axiomatisation of the algebra of regular events.

Damien Pous and Jana Wagemaker. Completeness Theorems for Kleene Algebra with Top. In 33rd International Conference on Concurrency Theory (CONCUR 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 243, pp. 26:1-26:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{pous_et_al:LIPIcs.CONCUR.2022.26, author = {Pous, Damien and Wagemaker, Jana}, title = {{Completeness Theorems for Kleene Algebra with Top}}, booktitle = {33rd International Conference on Concurrency Theory (CONCUR 2022)}, pages = {26:1--26:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-246-4}, ISSN = {1868-8969}, year = {2022}, volume = {243}, editor = {Klin, Bartek and Lasota, S{\l}awomir and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2022.26}, URN = {urn:nbn:de:0030-drops-170890}, doi = {10.4230/LIPIcs.CONCUR.2022.26}, annote = {Keywords: Kleene algebra, Hypotheses, Completeness, Closed languages} }

Document

**Published in:** LIPIcs, Volume 171, 31st International Conference on Concurrency Theory (CONCUR 2020)

We study the equational theories of composition and intersection on binary relations, with or without their associated neutral elements (identity and full relation). Without these constants, the equational theory coincides with that of semilattice-ordered semigroups. We show that the equational theory is no longer finitely based when adding one or the other constant, refuting a conjecture from the literature. Our proofs exploit a characterisation in terms of graphs and homomorphisms, which we show how to adapt in order to capture standard equational theories over the considered signatures.

Amina Doumane and Damien Pous. Non Axiomatisability of Positive Relation Algebras with Constants, via Graph Homomorphisms. In 31st International Conference on Concurrency Theory (CONCUR 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 171, pp. 29:1-29:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{doumane_et_al:LIPIcs.CONCUR.2020.29, author = {Doumane, Amina and Pous, Damien}, title = {{Non Axiomatisability of Positive Relation Algebras with Constants, via Graph Homomorphisms}}, booktitle = {31st International Conference on Concurrency Theory (CONCUR 2020)}, pages = {29:1--29:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-160-3}, ISSN = {1868-8969}, year = {2020}, volume = {171}, editor = {Konnov, Igor and Kov\'{a}cs, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2020.29}, URN = {urn:nbn:de:0030-drops-128411}, doi = {10.4230/LIPIcs.CONCUR.2020.29}, annote = {Keywords: Relation algebra, graph homomorphisms, (in)equational theories} }

Document

**Published in:** LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

We consider a fragment of a cyclic sequent proof system for Kleene algebra, and we see it as a computational device for recognising languages of words. The starting proof system is linear and we show that it captures precisely the regular languages. When adding the standard contraction rule, the expressivity raises significantly; we characterise the corresponding class of languages using a new notion of multi-head finite automata, where heads can jump.

Denis Kuperberg, Laureline Pinault, and Damien Pous. Cyclic Proofs and Jumping Automata. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 45:1-45:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kuperberg_et_al:LIPIcs.FSTTCS.2019.45, author = {Kuperberg, Denis and Pinault, Laureline and Pous, Damien}, title = {{Cyclic Proofs and Jumping Automata}}, booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)}, pages = {45:1--45:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-131-3}, ISSN = {1868-8969}, year = {2019}, volume = {150}, editor = {Chattopadhyay, Arkadev and Gastin, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.45}, URN = {urn:nbn:de:0030-drops-116071}, doi = {10.4230/LIPIcs.FSTTCS.2019.45}, annote = {Keywords: Cyclic proofs, regular languages, multi-head automata, transducers} }

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Invited Paper

**Published in:** LIPIcs, Volume 139, 8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019)

Coinduction is a mathematical tool that is used pervasively in computer science: to program and reason about infinite data-structures, to give semantics to concurrent systems, to obtain automata algorithms. We present some of these applications in automata theory and in formalised mathematics. Then we discuss recent developments on the abstract theory of coinduction and its enhancements.

Damien Pous. Coinduction: Automata, Formal Proof, Companions (Invited Paper). In 8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 139, pp. 4:1-4:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{pous:LIPIcs.CALCO.2019.4, author = {Pous, Damien}, title = {{Coinduction: Automata, Formal Proof, Companions}}, booktitle = {8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019)}, pages = {4:1--4:4}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-120-7}, ISSN = {1868-8969}, year = {2019}, volume = {139}, editor = {Roggenbach, Markus and Sokolova, Ana}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2019.4}, URN = {urn:nbn:de:0030-drops-114323}, doi = {10.4230/LIPIcs.CALCO.2019.4}, annote = {Keywords: Coinduction, Automata, Coalgebra, Formal proofs} }

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**Published in:** LIPIcs, Volume 141, 10th International Conference on Interactive Theorem Proving (ITP 2019)

We present a library to verify rigorous approximations of univariate functions on real numbers, with the Coq proof assistant. Based on interval arithmetic, this library also implements a technique of validation a posteriori based on the Banach fixed-point theorem. We illustrate this technique on the case of operations of division and square root. This library features a collection of abstract structures that organise the specfication of rigorous approximations, and modularise the related proofs. Finally, we provide an implementation of verified Chebyshev approximations, and we discuss a few examples of computations.

Florent Bréhard, Assia Mahboubi, and Damien Pous. A Certificate-Based Approach to Formally Verified Approximations. In 10th International Conference on Interactive Theorem Proving (ITP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 141, pp. 8:1-8:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{brehard_et_al:LIPIcs.ITP.2019.8, author = {Br\'{e}hard, Florent and Mahboubi, Assia and Pous, Damien}, title = {{A Certificate-Based Approach to Formally Verified Approximations}}, booktitle = {10th International Conference on Interactive Theorem Proving (ITP 2019)}, pages = {8:1--8:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-122-1}, ISSN = {1868-8969}, year = {2019}, volume = {141}, editor = {Harrison, John and O'Leary, John and Tolmach, Andrew}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2019.8}, URN = {urn:nbn:de:0030-drops-110638}, doi = {10.4230/LIPIcs.ITP.2019.8}, annote = {Keywords: approximation theory, Chebyshev polynomials, Banach fixed-point theorem, interval arithmetic, Coq} }

Document

**Published in:** LIPIcs, Volume 118, 29th International Conference on Concurrency Theory (CONCUR 2018)

We provide a finite set of axioms for identity-free Kleene lattices, which we prove sound and complete for the equational theory of their relational models. Our proof builds on the completeness theorem for Kleene algebra, and on a novel automata construction that makes it possible to extract axiomatic proofs using a Kleene-like algorithm.

Amina Doumane and Damien Pous. Completeness for Identity-free Kleene Lattices. In 29th International Conference on Concurrency Theory (CONCUR 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 118, pp. 18:1-18:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{doumane_et_al:LIPIcs.CONCUR.2018.18, author = {Doumane, Amina and Pous, Damien}, title = {{Completeness for Identity-free Kleene Lattices}}, booktitle = {29th International Conference on Concurrency Theory (CONCUR 2018)}, pages = {18:1--18:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-087-3}, ISSN = {1868-8969}, year = {2018}, volume = {118}, editor = {Schewe, Sven and Zhang, Lijun}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2018.18}, URN = {urn:nbn:de:0030-drops-95564}, doi = {10.4230/LIPIcs.CONCUR.2018.18}, annote = {Keywords: Kleene algebra, Graph languages, Petri Automata, Kleene theorem} }

Document

**Published in:** LIPIcs, Volume 119, 27th EACSL Annual Conference on Computer Science Logic (CSL 2018)

We prove cut-elimination for a sequent-style proof system which is sound and complete for the equational theory of Kleene algebra, and where proofs are (potentially) non-wellfounded infinite trees. We extend these results to systems with meets and residuals, capturing `star-continuous' action lattices in a similar way. We recover the equational theory of all action lattices by restricting to regular proofs (with cut) - those proofs that are unfoldings of finite graphs.

Anupam Das and Damien Pous. Non-Wellfounded Proof Theory For (Kleene+Action)(Algebras+Lattices). In 27th EACSL Annual Conference on Computer Science Logic (CSL 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 119, pp. 19:1-19:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{das_et_al:LIPIcs.CSL.2018.19, author = {Das, Anupam and Pous, Damien}, title = {{Non-Wellfounded Proof Theory For (Kleene+Action)(Algebras+Lattices)}}, booktitle = {27th EACSL Annual Conference on Computer Science Logic (CSL 2018)}, pages = {19:1--19:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-088-0}, ISSN = {1868-8969}, year = {2018}, volume = {119}, editor = {Ghica, Dan R. and Jung, Achim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2018.19}, URN = {urn:nbn:de:0030-drops-96869}, doi = {10.4230/LIPIcs.CSL.2018.19}, annote = {Keywords: Kleene algebra, proof theory, sequent system, non-wellfounded proofs} }

Document

**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

We give a new and elementary proof that the graphs of treewidth at most two can be seen as a free algebra. This result was originally established through an elaborate analysis of the structure of K_4-free graphs, ultimately reproving the well-known fact that the graphs of treewidth at most two are precisely those excluding K_4 as a minor. Our new proof is based on a confluent and terminating rewriting system for term-labeled graphs and does not involve graph minors anymore. The new strategy is simpler and robust in the sense that it can be adapted to subclasses of treewidth-two graphs, e.g., graphs without self-loops.

Christian Doczkal and Damien Pous. Treewidth-Two Graphs as a Free Algebra. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 60:1-60:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{doczkal_et_al:LIPIcs.MFCS.2018.60, author = {Doczkal, Christian and Pous, Damien}, title = {{Treewidth-Two Graphs as a Free Algebra}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {60:1--60:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.60}, URN = {urn:nbn:de:0030-drops-96429}, doi = {10.4230/LIPIcs.MFCS.2018.60}, annote = {Keywords: Treewidth, Universal Algebra, Rewriting} }

Document

**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

Binary relations are such a basic object that they appear in many
places in mathematics and computer science. For instance, when
dealing with graphs, program semantics, or termination guarantees,
binary relations are always used at some point.
In this survey, we focus on the relations themselves, and we
consider algebraic and algorithmic questions. On the algebraic side, we want to understand and characterise the laws governing the behaviour of the following standard operations on relations: union, intersection, composition, converse, and reflexive-transitive closure. On the algorithmic side, we look for decision procedures for equality or inequality of relations.
After having formally defined the calculus of relations, we recall
the existing results about two well-studied fragments of particular importance: Kleene algebras and allegories. Unifying those fragments yields a decidable theory whose axiomatisability remains an open problem.

Damien Pous. On the Positive Calculus of Relations with Transitive Closure. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 3:1-3:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{pous:LIPIcs.STACS.2018.3, author = {Pous, Damien}, title = {{On the Positive Calculus of Relations with Transitive Closure}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {3:1--3:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.3}, URN = {urn:nbn:de:0030-drops-85382}, doi = {10.4230/LIPIcs.STACS.2018.3}, annote = {Keywords: Relation Algebra, Kleene Algebra, Allegories, Automata, Graphs} }

Document

**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

Graphs of treewidth at most two are the ones excluding the clique with four vertices as a minor. Equivalently, they are the graphs whose biconnected components are series-parallel.
We turn those graphs into a free algebra, answering positively a question by Courcelle and Engelfriet, in the case of treewidth two.
First we propose a syntax for denoting them: in addition to series and parallel compositions, it suffices to consider the neutral elements of those operations and a unary transpose operation. Then we give a finite equational presentation and we prove it complete: two terms from the syntax are congruent if and only if they denote the same graph.

Enric Cosme Llópez and Damien Pous. K4-free Graphs as a Free Algebra. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 76:1-76:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{cosmellopez_et_al:LIPIcs.MFCS.2017.76, author = {Cosme Ll\'{o}pez, Enric and Pous, Damien}, title = {{K4-free Graphs as a Free Algebra}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {76:1--76:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.76}, URN = {urn:nbn:de:0030-drops-80883}, doi = {10.4230/LIPIcs.MFCS.2017.76}, annote = {Keywords: Universal Algebra, Graph theory, Axiomatisation, Tree decompositions, Graph minors} }

Document

**Published in:** LIPIcs, Volume 72, 7th Conference on Algebra and Coalgebra in Computer Science (CALCO 2017)

Distributive laws between functors are a fundamental tool in the theory of coalgebras. In the context of coinduction in complete lattices, they correspond to the so-called compatible functions, which enable enhancements of the coinductive proof technique. Amongst these, the greatest compatible function, called the companion, has recently been shown to satisfy many good properties.
Categorically, the companion of a functor corresponds to the final object in a category of distributive laws. We show that every accessible functor on a locally presentable category has a companion. Central to this and other constructions in the paper is the presentation of distributive laws as coalgebras for a certain functor. This functor itself has again, what we call, a second-order companion. We show how this companion interacts with the various monoidal structures on functor categories. In particular, both the first- and second-order companion give rise to monads. We use these results to obtain an abstract GSOS-like extension result for specifications involving the second-order companion.

Henning Basold, Damien Pous, and Jurriaan Rot. Monoidal Company for Accessible Functors. In 7th Conference on Algebra and Coalgebra in Computer Science (CALCO 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 72, pp. 5:1-5:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{basold_et_al:LIPIcs.CALCO.2017.5, author = {Basold, Henning and Pous, Damien and Rot, Jurriaan}, title = {{Monoidal Company for Accessible Functors}}, booktitle = {7th Conference on Algebra and Coalgebra in Computer Science (CALCO 2017)}, pages = {5:1--5:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-033-0}, ISSN = {1868-8969}, year = {2017}, volume = {72}, editor = {Bonchi, Filippo and K\"{o}nig, Barbara}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2017.5}, URN = {urn:nbn:de:0030-drops-80379}, doi = {10.4230/LIPIcs.CALCO.2017.5}, annote = {Keywords: coalgebras, distributive laws, accessible functors, monoidal categories} }

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**Published in:** LIPIcs, Volume 85, 28th International Conference on Concurrency Theory (CONCUR 2017)

Concurrent Kleene algebras support equational reasoning about
computing systems with concurrent behaviours. Their natural
semantics is given by series(-parallel) rational pomset languages, a
standard true concurrency semantics, which is often associated with
processes of Petri nets. We use constructions on Petri nets to
provide two decision procedures for such pomset languages motivated
by the equational and the refinement theory of concurrent Kleene
algebra. The contribution to the first problem lies in a much
simpler algorithm and an EXPSPACE complexity bound. Decidability
of the second, more interesting problem is new and, in fact,
EXPSPACE-complete.

Paul Brunet, Damien Pous, and Georg Struth. On Decidability of Concurrent Kleene Algebra. In 28th International Conference on Concurrency Theory (CONCUR 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 85, pp. 28:1-28:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{brunet_et_al:LIPIcs.CONCUR.2017.28, author = {Brunet, Paul and Pous, Damien and Struth, Georg}, title = {{On Decidability of Concurrent Kleene Algebra}}, booktitle = {28th International Conference on Concurrency Theory (CONCUR 2017)}, pages = {28:1--28:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-048-4}, ISSN = {1868-8969}, year = {2017}, volume = {85}, editor = {Meyer, Roland and Nestmann, Uwe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2017.28}, URN = {urn:nbn:de:0030-drops-77881}, doi = {10.4230/LIPIcs.CONCUR.2017.28}, annote = {Keywords: Concurrent Kleene algebra, series-parallel pomsets, Petri nets} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

An axiomatisation of Nominal Kleene Algebra has been proposed by Gabbay and Ciancia, and then shown to be complete and decidable by Kozen et al. However, one can think of at least four different formulations for a Kleene Algebra with names: using freshness conditions or a presheaf structure (types), and with explicit permutations or not. We formally show that these variations are all equivalent.
Then we introduce an extension of Nominal Kleene Algebra, motivated by relational models of programming languages. The idea is to let letters (i.e., atomic programs) carry a set of names, rather than being reduced to a single name. We formally show that this extension is at least as expressive as the original one, and that it may be presented with or without a presheaf structure, and with or without syntactic permutations. Whether this extension is strictly more
expressive remains open.
All our results were formally checked using the Coq proof assistant.

Paul Brunet and Damien Pous. A Formal Exploration of Nominal Kleene Algebra. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 22:1-22:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{brunet_et_al:LIPIcs.MFCS.2016.22, author = {Brunet, Paul and Pous, Damien}, title = {{A Formal Exploration of Nominal Kleene Algebra}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {22:1--22:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.22}, URN = {urn:nbn:de:0030-drops-64379}, doi = {10.4230/LIPIcs.MFCS.2016.22}, annote = {Keywords: Nominal sets, Kleene algebra, equational theory, Coq} }

Document

**Published in:** LIPIcs, Volume 42, 26th International Conference on Concurrency Theory (CONCUR 2015)

Up-to techniques are useful tools for optimising proofs of behavioural equivalence of processes. Bisimulations up-to context can be safely used in any language specified by GSOS rules. We showed this result in a previous paper by exploiting the well-known observation by Turi and Plotkin that such languages form bialgebras. In this paper, we prove the soundness of up-to contextual closure for weak bisimulations of systems specified by cool rule formats, as defined by Bloom to ensure congruence of weak bisimilarity. However, the weak transition systems obtained from such cool rules give rise to lax bialgebras, rather than to bialgebras. Hence, to reach our goal, we extend our previously developed categorical framework to an ordered setting.

Filippo Bonchi, Daniela Petrisan, Damien Pous, and Jurriaan Rot. Lax Bialgebras and Up-To Techniques for Weak Bisimulations. In 26th International Conference on Concurrency Theory (CONCUR 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 42, pp. 240-253, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bonchi_et_al:LIPIcs.CONCUR.2015.240, author = {Bonchi, Filippo and Petrisan, Daniela and Pous, Damien and Rot, Jurriaan}, title = {{Lax Bialgebras and Up-To Techniques for Weak Bisimulations}}, booktitle = {26th International Conference on Concurrency Theory (CONCUR 2015)}, pages = {240--253}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-91-0}, ISSN = {1868-8969}, year = {2015}, volume = {42}, editor = {Aceto, Luca and de Frutos Escrig, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2015.240}, URN = {urn:nbn:de:0030-drops-53709}, doi = {10.4230/LIPIcs.CONCUR.2015.240}, annote = {Keywords: Up-to techniques, weak bisimulation, (lax) bialgebras} }

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