This paper axiomatises the structure of bigraphs, and proves that the resulting theory is complete. Bigraphs are graphs with double structure, representing locality and connectivity. They have been shown to represent dynamic theories for the #-calculus, mobile ambients and Petri nets, in a way that is faithful to each of those models of discrete behaviour. While the main purpose of bigraphs is to understand mobile systems, a prerequisite for this understanding is a well-behaved theory of the structure of states in such systems. The algebra of bigraph structure is surprisingly simple, as the paper demonstrates; this is because bigraphs treat locality and connectivity orthogonally

In this paper the concurrent semantics of double-pushout (DPO) graph rewriting, which is classically defined in terms of shift-equivalence classes of graph derivations, is axiomatised via the construction of a free monoidal bi-category. In contrast to a previous attempt based on 2-categories, the use of bi-categories allows to define rewriting on concrete graphs. Thus, the problem of composition of isomorphism classes of rewriting sequences is avoided. Moreover, as a first step towards the recovery of the full expressive power of the formalism via a purely algebraic description, the concept of disconnected rules is introduced, i.e., rules whose interface graphs are made of disconnected nodes and edges only. It is proved that, under reasonable assumptions, rewriting via disconnected rules enjoys similar concurrency properties like in the classical approach.

Although the algebraic semantics of place/transition Petri nets under the collective token philosophy has been fully explained in terms of (strictly) symmetric (strict) monoidal categories, the analogous construction under the individual token philosophy is not completely satisfactory because it lacks universality and also functoriality. We introduce the notion of pre-net to recover these aspects, obtaining a fully satisfactory categorical treatment centered on the notion of adjunction. This allows us to present a purely logical description of net behaviours under the individual token philosophy in terms of theories and theory morphisms in partial membership equational logic, yielding a complete match with the theory developed by the authors for the collective token view of nets.

This paper retraces, collects, and summarises the contributions of the author — both individually and in collaboration with others — on the theme of algebraic, compositional approaches to the semantics of Petri nets.

An existing procedure to compute left Kan extensions over the ground category Set also computes left Kan extensions over the ground categories Cat, 2-Cat, n-Cat for any n and indeed !-Cat. Therefore extension data structured in this manner already can make use of the left Kan extension notion of a best possible approximation. Examples include systems of labeled transition systems and certain higher dimensional rewriting systems. Keywords: Algorithms, higher dimensional structures, transition systems, labeled transition systems. A functor X: B ! S may be used to present a behavior of a base syntax, or diagram, B, over a ground category S. The ground category S is typically the category of sets and functions or else a close relative such as the category of finite sets and functions between them. A functor K: B ! E may be used to present an extension E of the base diagram B. It is then asked what is a best extension of the behavior X: B ! S to a behavior E ! S, given K: B ! E. One ans...

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