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Span(Graph): a canonical feedback algebra of open transition systems

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Abstract

We show that \({\textbf{Span}}({\textbf{Graph}})_*\), an algebra for open transition systems introduced by Katis, Sabadini and Walters, satisfies a universal property. By itself, this is a justification of the canonicity of this model of concurrency. However, the universal property is itself of interest, being a formal demonstration of the relationship between feedback and state. Indeed, feedback categories, also originally proposed by Katis, Sabadini and Walters, are a weakening of traced monoidal categories, with various applications in computer science. A state bootstrapping technique, which has appeared in several different contexts, yields free such categories. We show that \({\textbf{Span}}({\textbf{Graph}})_*\) arises in this way, being the free feedback category over \({\textbf{Span}}(\textbf{Set})\). Given that the latter can be seen as an algebra of predicates, the algebra of open transition systems thus arises—roughly speaking—as the result of bootstrapping state to that algebra. Finally, we generalize feedback categories endowing state spaces with extra structure: this extends the framework from mere transition systems to automata with initial and final states.

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Notes

  1. In its original description: “the relay is designed to produce a large and permanent change in the current flowing in an electrical circuit by means of a small electrical stimulus received from the outside” ( [17], emphasis added).

  2. In other words, traces are used to talk about processes in equilibrium processes that have reached a fixed point. A theorem by Hasegawa [28] and Hyland [5] corroborates this interpretation: a trace in a Cartesian category corresponds to a fixpoint operator.

  3. This is the \(\textbf{Int}\) construction from Joyal, Street and Verity [30].

  4. Equivalently, \({\textbf{Graph}}\) is the presheaf category on the diagram \((\bullet \rightrightarrows \bullet )\), i.e. the category of functors \((\bullet \rightrightarrows \bullet ) \rightarrow \textbf{Set}\) and natural transformations between them.

  5. Here, \(\textbf{S}\) is the subcategory of isomorphisms of \(\textbf{C}\) and \(R\) is the inclusion functor.

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Authors and Affiliations

  1. Tallinn University of Technology, Ehitajate tee 5, 12616, Tallinn, Estonia

    Elena Di Lavore, Mario Román & Paweł Sobociński

  2. Free University of Bozen-Bolzano, Piazza Domenicani, 3, 39100, Bolzano, BZ, Italy

    Alessandro Gianola

  3. Università degli Studi dell’Insubria, Via Ravasi, 2, 21100, Varese, VA, Italy

    Nicoletta Sabadini

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  1. Elena Di Lavore
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Correspondence to Elena Di Lavore.

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Communicated by Gwen Salaun.

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Di Lavore, Román and Sobociński were supported by the European Union through the ESF funded Estonian IT Academy research measure (2014\(-\)2020.4.05.19-0001). This work was also supported by the Estonian Research Council grant PRG1210.

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Di Lavore, E., Gianola, A., Román, M. et al. Span(Graph): a canonical feedback algebra of open transition systems. Softw Syst Model 22, 495–520 (2023). https://doi.org/10.1007/s10270-023-01092-7

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