A canonical algebra of open transition systems

10/20/2020

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by   Elena Di Lavore, et al.

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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, 𝐒𝐩𝐚𝐧(𝐆𝐫𝐚𝐩𝐡)_∗, also due to Katis, Sabadini and Walters, is the free category with feedback over 𝐒𝐩𝐚𝐧(𝐒𝐞𝐭). Intuitively, this algebra of transition systems is obtained by adding state to an algebra of predicates, and therefore 𝐒𝐩𝐚𝐧(𝐆𝐫𝐚𝐩𝐡)_∗ is, in this sense, the canonical such algebra.