Document

**Published in:** LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

The logic of Dependence and Independence Bunched Implications (DIBI) is a logic to reason about conditional independence (CI); for instance, DIBI formulas can characterise CI in discrete probability distributions and in relational databases, using a probabilistic DIBI model and a similarly-constructed relational model. Despite the similarity of the two models, there lacks a uniform account. As a result, the laborious case-by-case verification of the frame conditions required for constructing new models hinders them from generalising the results to CI in other useful models such that continuous distribution. In this paper, we develop an abstract framework for systematically constructing DIBI models, using category theory as the unifying mathematical language. We show that DIBI models arise from arbitrary symmetric monoidal categories with copy-discard structure. In particular, we use string diagrams - a graphical presentation of monoidal categories - to give a uniform definition of the parallel composition and subkernel relation in DIBI models. Our approach not only generalises known models, but also yields new models of interest and reduces properties of DIBI models to structures in the underlying categories. Furthermore, our categorical framework enables a comparison between string diagrammatic approaches to CI in the literature and a logical notion of CI, defined in terms of the satisfaction of specific DIBI formulas. We show that the logical notion is an extension of string diagrammatic CI under reasonable conditions.

Tao Gu, Jialu Bao, Justin Hsu, Alexandra Silva, and Fabio Zanasi. A Categorical Approach to DIBI Models. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 17:1-17:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{gu_et_al:LIPIcs.FSCD.2024.17, author = {Gu, Tao and Bao, Jialu and Hsu, Justin and Silva, Alexandra and Zanasi, Fabio}, title = {{A Categorical Approach to DIBI Models}}, booktitle = {9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)}, pages = {17:1--17:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-323-2}, ISSN = {1868-8969}, year = {2024}, volume = {299}, editor = {Rehof, Jakob}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.17}, URN = {urn:nbn:de:0030-drops-203469}, doi = {10.4230/LIPIcs.FSCD.2024.17}, annote = {Keywords: Conditional Independence, Dependence Independence Bunched Implications, String Diagrams, Markov Categories} }

Document

**Published in:** LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Discrete probabilistic programming languages provide an expressive tool for representing and reasoning about probabilistic models. These languages typically define the semantics of a program through its posterior distribution, obtained through exact inference techniques.
While the semantics of standard programming constructs in this context is well understood, there is a gap in extending these languages with tools to reason about the asymptotic behaviour of programs. In this paper, we introduce unbounded iteration in the context of a discrete probabilistic programming language, give it a semantics, and show how to compute it exactly. This allows us to express the stationary distribution of a probabilistic function while preserving the efficiency of exact inference techniques. We discuss the advantages and limitations of our approach, showcasing their practical utility by considering examples where bounded iteration poses a challenge due to the inherent difficulty of assessing the proximity of a distribution to its stationary point.

Mateo Torres-Ruiz, Robin Piedeleu, Alexandra Silva, and Fabio Zanasi. On Iteration in Discrete Probabilistic Programming. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 20:1-20:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{torresruiz_et_al:LIPIcs.FSCD.2024.20, author = {Torres-Ruiz, Mateo and Piedeleu, Robin and Silva, Alexandra and Zanasi, Fabio}, title = {{On Iteration in Discrete Probabilistic Programming}}, booktitle = {9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)}, pages = {20:1--20:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-323-2}, ISSN = {1868-8969}, year = {2024}, volume = {299}, editor = {Rehof, Jakob}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.20}, URN = {urn:nbn:de:0030-drops-203490}, doi = {10.4230/LIPIcs.FSCD.2024.20}, annote = {Keywords: Probabilistic programming, Programming languages semantics, Unbounded iteration} }

Document

**Published in:** LIPIcs, Volume 270, 10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023)

String diagrams constitute an intuitive and expressive graphical syntax that has found application in a very diverse range of fields including concurrency theory, quantum computing, control theory, machine learning, linguistics, and digital circuits. Rewriting theory for string diagrams relies on a combinatorial interpretation as double-pushout rewriting of certain hypergraphs. As previously studied, there is a "tension" in this interpretation: in order to make it sound and complete, we either need to add structure on string diagrams (in particular, Frobenius algebra structure) or pose restrictions on double-pushout rewriting (resulting in "convex" rewriting). From the string diagram viewpoint, imposing a full Frobenius structure may not always be natural or desirable in applications, which motivates our study of a weaker requirement: commutative monoid structure. In this work we characterise string diagram rewriting modulo commutative monoid equations, via a sound and complete interpretation in a suitable notion of double-pushout rewriting of hypergraphs.

Aleksandar Milosavljević, Robin Piedeleu, and Fabio Zanasi. String Diagram Rewriting Modulo Commutative (Co)Monoid Structure. In 10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 270, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{milosavljevic_et_al:LIPIcs.CALCO.2023.9, author = {Milosavljevi\'{c}, Aleksandar and Piedeleu, Robin and Zanasi, Fabio}, title = {{String Diagram Rewriting Modulo Commutative (Co)Monoid Structure}}, booktitle = {10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023)}, pages = {9:1--9:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-287-7}, ISSN = {1868-8969}, year = {2023}, volume = {270}, editor = {Baldan, Paolo and de Paiva, Valeria}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2023.9}, URN = {urn:nbn:de:0030-drops-188067}, doi = {10.4230/LIPIcs.CALCO.2023.9}, annote = {Keywords: String diagrams, Double-pushout rewriting, Commutative monoid} }

Document

**Published in:** LIPIcs, Volume 252, 31st EACSL Annual Conference on Computer Science Logic (CSL 2023)

We formulate a reverse-mode automatic differentiation (RAD) algorithm for (applied) simply typed lambda calculus in the style of Pearlmutter and Siskind [Barak A. Pearlmutter and Jeffrey Mark Siskind, 2008], using the graphical formalism of string diagrams. Thanks to string diagram rewriting, we are able to formally prove for the first time the soundness of such an algorithm. Our approach requires developing a calculus of string diagrams with hierarchical features in the spirit of functorial boxes, in order to model closed monoidal (and cartesian closed) structure. To give an efficient yet principled implementation of the RAD algorithm, we use foliations of our hierarchical string diagrams.

Mario Alvarez-Picallo, Dan Ghica, David Sprunger, and Fabio Zanasi. Functorial String Diagrams for Reverse-Mode Automatic Differentiation. In 31st EACSL Annual Conference on Computer Science Logic (CSL 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 252, pp. 6:1-6:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{alvarezpicallo_et_al:LIPIcs.CSL.2023.6, author = {Alvarez-Picallo, Mario and Ghica, Dan and Sprunger, David and Zanasi, Fabio}, title = {{Functorial String Diagrams for Reverse-Mode Automatic Differentiation}}, booktitle = {31st EACSL Annual Conference on Computer Science Logic (CSL 2023)}, pages = {6:1--6:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-264-8}, ISSN = {1868-8969}, year = {2023}, volume = {252}, editor = {Klin, Bartek and Pimentel, Elaine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2023.6}, URN = {urn:nbn:de:0030-drops-174674}, doi = {10.4230/LIPIcs.CSL.2023.6}, annote = {Keywords: string diagrams, automatic differentiation, hierarchical hypergraphs} }

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**Published in:** LIPIcs, Volume 252, 31st EACSL Annual Conference on Computer Science Logic (CSL 2023)

Whereas string diagrams for strict monoidal categories are well understood, and have found application in several fields of Computer Science, graphical formalisms for non-strict monoidal categories are far less studied. In this paper, we provide a presentation by generators and relations of string diagrams for non-strict monoidal categories, and show how this construction can handle applications in domains such as digital circuits and programming languages. We prove the correctness of our construction, which yields a novel proof of Mac Lane’s strictness theorem. This in turn leads to an elementary graphical proof of Mac Lane’s coherence theorem, and in particular allows for the inductive construction of the canonical isomorphisms in a monoidal category.

Paul Wilson, Dan Ghica, and Fabio Zanasi. String Diagrams for Non-Strict Monoidal Categories. In 31st EACSL Annual Conference on Computer Science Logic (CSL 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 252, pp. 37:1-37:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{wilson_et_al:LIPIcs.CSL.2023.37, author = {Wilson, Paul and Ghica, Dan and Zanasi, Fabio}, title = {{String Diagrams for Non-Strict Monoidal Categories}}, booktitle = {31st EACSL Annual Conference on Computer Science Logic (CSL 2023)}, pages = {37:1--37:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-264-8}, ISSN = {1868-8969}, year = {2023}, volume = {252}, editor = {Klin, Bartek and Pimentel, Elaine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2023.37}, URN = {urn:nbn:de:0030-drops-174981}, doi = {10.4230/LIPIcs.CSL.2023.37}, annote = {Keywords: String Diagrams, Strictness, Coherence} }

Document

**Published in:** LIPIcs, Volume 228, 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)

This paper develops a formal string diagram language for monoidal closed categories. Previous work has shown that string diagrams for freely generated symmetric monoidal categories can be viewed as hypergraphs with interfaces, and the axioms of these categories can be realized by rewriting systems. This work proposes hierarchical hypergraphs as a suitable formalization of string diagrams for monoidal closed categories. We then show double pushout rewriting captures the axioms of these closed categories.

Mario Alvarez-Picallo, Dan Ghica, David Sprunger, and Fabio Zanasi. Rewriting for Monoidal Closed Categories. In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 29:1-29:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{alvarezpicallo_et_al:LIPIcs.FSCD.2022.29, author = {Alvarez-Picallo, Mario and Ghica, Dan and Sprunger, David and Zanasi, Fabio}, title = {{Rewriting for Monoidal Closed Categories}}, booktitle = {7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)}, pages = {29:1--29:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-233-4}, ISSN = {1868-8969}, year = {2022}, volume = {228}, editor = {Felty, Amy P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2022.29}, URN = {urn:nbn:de:0030-drops-163108}, doi = {10.4230/LIPIcs.FSCD.2022.29}, annote = {Keywords: string diagrams, rewriting, hierarchical hypergraph, monoidal closed category} }

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(Co)algebraic pearls

**Published in:** LIPIcs, Volume 211, 9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021)

Farkas' lemma is a celebrated result on the solutions of systems of linear inequalities, which finds application pervasively in mathematics and computer science. In this work we show how to formulate and prove Farkas' lemma in diagrammatic polyhedral algebra, a sound and complete graphical calculus for polyhedra. Furthermore, we show how linear programs can be modeled within the calculus and how some famous duality results can be proved.

Filippo Bonchi, Alessandro Di Giorgio, and Fabio Zanasi. From Farkas' Lemma to Linear Programming: an Exercise in Diagrammatic Algebra ((Co)algebraic pearls). In 9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 211, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bonchi_et_al:LIPIcs.CALCO.2021.9, author = {Bonchi, Filippo and Di Giorgio, Alessandro and Zanasi, Fabio}, title = {{From Farkas' Lemma to Linear Programming: an Exercise in Diagrammatic Algebra}}, booktitle = {9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021)}, pages = {9:1--9:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-212-9}, ISSN = {1868-8969}, year = {2021}, volume = {211}, editor = {Gadducci, Fabio and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2021.9}, URN = {urn:nbn:de:0030-drops-153643}, doi = {10.4230/LIPIcs.CALCO.2021.9}, annote = {Keywords: String diagrams, Farkas Lemma, Duality, Linear Programming} }

Document

**Published in:** LIPIcs, Volume 211, 9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021)

Logic programming and its variations are widely used for formal reasoning in various areas of Computer Science, most notably Artificial Intelligence. In this paper we develop a systematic and unifying perspective for (ground) classical, probabilistic, weighted logic programs, based on categorical algebra. Our departure point is a formal distinction between the syntax and the semantics of programs, now regarded as separate categories. Then, we are able to characterise the various variants of logic program as different models for the same syntax category, i.e. structure-preserving functors in the spirit of Lawvere’s functorial semantics.
As a first consequence of our approach, we showcase a series of semantic constructs for logic programming pictorially as certain string diagrams in the syntax category. Secondly, we describe the correspondence between probabilistic logic programs and Bayesian networks in terms of the associated models. Our analysis reveals that the correspondence can be phrased in purely syntactical terms, without resorting to the probabilistic domain of interpretation.

Tao Gu and Fabio Zanasi. Functorial Semantics as a Unifying Perspective on Logic Programming. In 9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 211, pp. 17:1-17:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{gu_et_al:LIPIcs.CALCO.2021.17, author = {Gu, Tao and Zanasi, Fabio}, title = {{Functorial Semantics as a Unifying Perspective on Logic Programming}}, booktitle = {9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021)}, pages = {17:1--17:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-212-9}, ISSN = {1868-8969}, year = {2021}, volume = {211}, editor = {Gadducci, Fabio and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2021.17}, URN = {urn:nbn:de:0030-drops-153723}, doi = {10.4230/LIPIcs.CALCO.2021.17}, annote = {Keywords: string diagrams, functorial semantics, logic programming} }

Document

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

Probabilistic logic programming is increasingly important in artificial intelligence and related fields as a formalism to reason about uncertainty. It generalises logic programming with the possibility of annotating clauses with probabilities. This paper proposes a coalgebraic perspective on probabilistic logic programming. Programs are modelled as coalgebras for a certain functor F, and two semantics are given in terms of cofree coalgebras. First, the cofree F-coalgebra yields a semantics in terms of derivation trees. Second, by embedding F into another type G, as cofree G-coalgebra we obtain a "possible worlds" interpretation of programs, from which one may recover the usual distribution semantics of probabilistic logic programming.

Tao Gu and Fabio Zanasi. A Coalgebraic Perspective on Probabilistic Logic Programming. In 8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 139, pp. 10:1-10:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{gu_et_al:LIPIcs.CALCO.2019.10, author = {Gu, Tao and Zanasi, Fabio}, title = {{A Coalgebraic Perspective on Probabilistic Logic Programming}}, booktitle = {8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019)}, pages = {10:1--10:21}, 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.2019.10}, URN = {urn:nbn:de:0030-drops-114387}, doi = {10.4230/LIPIcs.CALCO.2019.10}, annote = {Keywords: probabilistic logic programming, coalgebraic semantics, distribution semantics} }

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Tool Paper

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

We introduce cartographer, a tool for editing and rewriting string diagrams of symmetric monoidal categories. Our approach is principled: the layout exploits the isomorphism between string diagrams and certain cospans of hypergraphs; the implementation of rewriting is based on the soundness and completeness of convex double-pushout rewriting for string diagram rewriting.

Paweł Sobociński, Paul W. Wilson, and Fabio Zanasi. CARTOGRAPHER: A Tool for String Diagrammatic Reasoning (Tool Paper). In 8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 139, pp. 20:1-20:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{sobocinski_et_al:LIPIcs.CALCO.2019.20, author = {Soboci\'{n}ski, Pawe{\l} and Wilson, Paul W. and Zanasi, Fabio}, title = {{CARTOGRAPHER: A Tool for String Diagrammatic Reasoning}}, booktitle = {8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019)}, pages = {20:1--20:7}, 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.2019.20}, URN = {urn:nbn:de:0030-drops-114482}, doi = {10.4230/LIPIcs.CALCO.2019.20}, annote = {Keywords: tool, string diagram, symmetric monoidal category, graphical reasoning} }

Document

**Published in:** LIPIcs, Volume 140, 30th International Conference on Concurrency Theory (CONCUR 2019)

Turi and Plotkin’s bialgebraic semantics is an abstract approach to specifying the operational semantics of a system, by means of a distributive law between its syntax (encoded as a monad) and its dynamics (an endofunctor). This setup is instrumental in showing that a semantic specification (a coalgebra) satisfies desirable properties: in particular, that it is compositional.
In this work, we use the bialgebraic approach to derive well-behaved structural operational semantics of string diagrams, a graphical syntax that is increasingly used in the study of interacting systems across different disciplines. Our analysis relies on representing the two-dimensional operations underlying string diagrams in various categories as a monad, and their bialgebraic semantics in terms of a distributive law for that monad.
As a proof of concept, we provide bialgebraic compositional semantics for a versatile string diagrammatic language which has been used to model both signal flow graphs (control theory) and Petri nets (concurrency theory). Moreover, our approach reveals a correspondence between two different interpretations of the Frobenius equations on string diagrams and two synchronisation mechanisms for processes, à la Hoare and à la Milner.

Filippo Bonchi, Robin Piedeleu, Pawel Sobocinski, and Fabio Zanasi. Bialgebraic Semantics for String Diagrams. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 37:1-37:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bonchi_et_al:LIPIcs.CONCUR.2019.37, author = {Bonchi, Filippo and Piedeleu, Robin and Sobocinski, Pawel and Zanasi, Fabio}, title = {{Bialgebraic Semantics for String Diagrams}}, booktitle = {30th International Conference on Concurrency Theory (CONCUR 2019)}, pages = {37:1--37:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-121-4}, ISSN = {1868-8969}, year = {2019}, volume = {140}, editor = {Fokkink, Wan and van Glabbeek, Rob}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2019.37}, URN = {urn:nbn:de:0030-drops-109398}, doi = {10.4230/LIPIcs.CONCUR.2019.37}, annote = {Keywords: String Diagram, Structural Operational Semantics, Bialgebraic semantics} }

Document

**Published in:** LIPIcs, Volume 140, 30th International Conference on Concurrency Theory (CONCUR 2019)

Kleene algebra with tests (KAT) is an algebraic framework for reasoning about the control flow of sequential programs. Generalising KAT to reason about concurrent programs is not straightforward, because axioms native to KAT in conjunction with expected axioms for concurrency lead to an anomalous equation. In this paper, we propose Kleene algebra with observations (KAO), a variant of KAT, as an alternative foundation for extending KAT to a concurrent setting. We characterise the free model of KAO, and establish a decision procedure w.r.t. its equational theory.

Tobias Kappé, Paul Brunet, Jurriaan Rot, Alexandra Silva, Jana Wagemaker, and Fabio Zanasi. Kleene Algebra with Observations. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 41:1-41:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kappe_et_al:LIPIcs.CONCUR.2019.41, author = {Kapp\'{e}, Tobias and Brunet, Paul and Rot, Jurriaan and Silva, Alexandra and Wagemaker, Jana and Zanasi, Fabio}, title = {{Kleene Algebra with Observations}}, booktitle = {30th International Conference on Concurrency Theory (CONCUR 2019)}, pages = {41:1--41:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-121-4}, ISSN = {1868-8969}, year = {2019}, volume = {140}, editor = {Fokkink, Wan and van Glabbeek, Rob}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2019.41}, URN = {urn:nbn:de:0030-drops-109431}, doi = {10.4230/LIPIcs.CONCUR.2019.41}, annote = {Keywords: Concurrent Kleene algebra, Kleene algebra with tests, free model, axiomatisation, decision procedure} }

Document

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

This paper proposes a formal definition of influence in Bayesian reasoning, based on the notions of state (as probability distribution), predicate, validity and conditioning. Our approach highlights how conditioning a joint entwined/entangled state with a predicate on one of its components has 'crossover' influence on the other components. We use the total variation metric on probability
distributions to quantitatively measure such influence. These insights are applied to give a rigorous explanation of the fundamental concept of d-separation in Bayesian networks.

Bart Jacobs and Fabio Zanasi. A Formal Semantics of Influence in Bayesian Reasoning. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 21:1-21:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{jacobs_et_al:LIPIcs.MFCS.2017.21, author = {Jacobs, Bart and Zanasi, Fabio}, title = {{A Formal Semantics of Influence in Bayesian Reasoning}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {21:1--21:14}, 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.2017.21}, URN = {urn:nbn:de:0030-drops-80896}, doi = {10.4230/LIPIcs.MFCS.2017.21}, annote = {Keywords: probability distribution, Bayesian network, influence} }

Document

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

Calculi of string diagrams are increasingly used to present the syntax and algebraic structure of various families of circuits, including signal flow graphs, electrical circuits and quantum processes. In many such approaches, the semantic interpretation for diagrams is given in terms of relations or corelations (generalised equivalence relations) of some kind. In this paper we show how semantic categories of both relations and corelations can be characterised as colimits of simpler categories. This modular perspective is important as it simplifies the task of giving a complete axiomatisation for semantic equivalence of string diagrams. Moreover, our general result unifies various theorems that are independently found in literature and are relevant for program semantics, quantum computation and control theory.

Brendan Fong and Fabio Zanasi. A Universal Construction for (Co)Relations. In 7th Conference on Algebra and Coalgebra in Computer Science (CALCO 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 72, pp. 12:1-12:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{fong_et_al:LIPIcs.CALCO.2017.12, author = {Fong, Brendan and Zanasi, Fabio}, title = {{A Universal Construction for (Co)Relations}}, booktitle = {7th Conference on Algebra and Coalgebra in Computer Science (CALCO 2017)}, pages = {12:1--12:16}, 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.2017.12}, URN = {urn:nbn:de:0030-drops-80334}, doi = {10.4230/LIPIcs.CALCO.2017.12}, annote = {Keywords: corelation, prop, string diagram} }

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

Concurrent Kleene Algebra (CKA) is a mathematical formalism to study programs that exhibit concurrent behaviour. As with previous extensions of Kleene Algebra, characterizing the free model is crucial in order to develop the foundations of the theory and potential applications. For CKA, this has been an open question for a few years and this paper makes an important step towards an answer. We present a new automaton model and a Kleene-like theorem that relates a relaxed version of CKA to series-parallel pomset languages, which are a natural candidate for the free model. There are two substantial differences with previous work: from expressions to automata, we use Brzozowski derivatives, which enable a direct construction of the automaton; from automata to expressions, we provide a syntactic characterization of the automata that denote valid CKA behaviours.

Tobias Kappé, Paul Brunet, Bas Luttik, Alexandra Silva, and Fabio Zanasi. Brzozowski Goes Concurrent - A Kleene Theorem for Pomset Languages. In 28th International Conference on Concurrency Theory (CONCUR 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 85, pp. 25:1-25:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{kappe_et_al:LIPIcs.CONCUR.2017.25, author = {Kapp\'{e}, Tobias and Brunet, Paul and Luttik, Bas and Silva, Alexandra and Zanasi, Fabio}, title = {{Brzozowski Goes Concurrent - A Kleene Theorem for Pomset Languages}}, booktitle = {28th International Conference on Concurrency Theory (CONCUR 2017)}, pages = {25:1--25:16}, 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.2017.25}, URN = {urn:nbn:de:0030-drops-77913}, doi = {10.4230/LIPIcs.CONCUR.2017.25}, annote = {Keywords: Kleene theorem, Series-rational expressions, Automata, Brzozowski derivatives, Concurrency, Pomsets} }

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