Georg Struth

Catoids and modal convolution algebras

Abstract We show how modal quantales arise as convolution algebras Q^X of functions from catoids X, that is, multisemigroups with a source map \ell and a target map r, into modal quantales Q, which can be seen as weight or value algebras. In the tradition of boolean algebras with operators we study modal correspondences between algebraic laws in X, Q and Q^X. The class of catoids we introduce generalises Schweizer and Sklar’s function systems and object-free categories to a setting isomorphic to algebras of ternary relations, as they are used for boolean algebras with operators and substructural logics.

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A Kleene theorem for higher-dimensional automata

By Uli Fahrenberg, Christian Johansen, Georg Struth, Krzysztof Ziemiański


In 33rd international conference on concurrency theory (CONCUR 2022)

Abstract We prove a Kleene theorem for higher-dimensional automata (HDAs). It states that the languages they recognise are precisely the rational subsumption-closed sets of interval pomsets. The rational operations include a gluing composition, for which we equip pomsets with interfaces. For our proof, we introduce HDAs with interfaces as presheaves over labelled precube categories and use tools inspired by algebraic topology, such as cylinders and (co)fibrations. HDAs are a general model of non-interleaving concurrency, which subsumes many other models in this field.

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Posets with interfaces as a model for concurrency

Abstract We introduce posets with interfaces (iposets) and generalise their standard serial composition to a new gluing composition. In the partial order semantics of concurrency, interfaces and gluing allow modelling events that extend in time and across components. Alternatively, taking a decompositional view, interfaces allow cutting through events, while serial composition may only cut through edges of a poset. We show that iposets under gluing composition form a category, which generalises the monoid of posets under serial composition up to isomorphism.

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