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A Canonical Algebra of Open Transition Systems

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Formal Aspects of Component Software (FACS 2021)

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Abstract

Feedback and state are closely interrelated concepts. Categories with feedback, originally proposed by Katis, Sabadini and Walters, are a weakening of the notion of traced monoidal categories, with several pertinent applications in computer science. The construction of the free such categories has appeared in several different contexts, and can be considered as state bootstrapping. We show that a categorical algebra for open transition systems, \(\mathbf {Span}(\mathbf {Graph})_*\), also due to Katis, Sabadini and Walters, is the free category with feedback over \(\mathbf {Span}(\mathbf {Set})\). This algebra of transition systems is obtained by adding state to an algebra of predicates, and therefore \(\mathbf {Span}(\mathbf {Graph})_*\) is the canonical such algebra.

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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Notes

  1. 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” ([12], emphasis added).

  2. 2.

    In other words, traces are used to talk about processes in equilibrium, processes that have reached a fixed point. A theorem by Hasegawa [22] and Hyland [5] corroborates this interpretation: a trace in a cartesian category corresponds to a fixpoint operator.

  3. 3.

    This is the \(\mathbf {Int}\) construction from [24].

  4. 4.

    As in Sect. 2, \(\partial _{A} = \mathsf {fbk}(\sigma _{A,A})\).

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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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  4. Nicoletta Sabadini
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  1. Grenoble Alpes University, Saint-Martin-d'Hères, France

    Gwen Salaün

  2. Eindhoven University of Technology, Eindhoven, The Netherlands

    Anton Wijs

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Di Lavore, E., Gianola, A., Román, M., Sabadini, N., Sobociński, P. (2021). A Canonical Algebra of Open Transition Systems. In: Salaün, G., Wijs, A. (eds) Formal Aspects of Component Software. FACS 2021. Lecture Notes in Computer Science(), vol 13077. Springer, Cham. https://doi.org/10.1007/978-3-030-90636-8_4

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