Minimisation of event structures

Event structures are fundamental models in concurrency theory, providing a representation of events in computation and of their relations, notably concurrency, conflict and causality. In this paper we present a theory of minimisation for event structures. Working in a class of event structures that generalises many stable event structure models in the literature, (e.g., prime, asymmetric, flow and bundle event structures) we study a notion of behaviour-preserving quotient, taking hereditary history preserving bisimilarity as a reference behavioural equivalence. We show that for any event structure a uniquely determined minimal quotient always exists. We observe that each event structure can be seen as the quotient of a prime event structure, and that quotients of general event structures arise from quotients of (suitably defined) corresponding prime event structures. This gives a special relevance to quotients in the class of prime event structures, which are then studied in detail, providing a characterisation and showing that also prime event structures always admit a unique minimal quotient

Event Structure Semantics for Dynamic Graph Grammars

Dynamic graph grammars (DGGs) are a reflexive extension of Graph Grammars that have been introduced to represent mobile reflexive systems and calculi at a convenient level of abstraction. Persistent graph grammars (PGGs) are a class of graph grammars that admits a fully satisfactory concurrent semantics thanks to the fact that all so-called asymmetric conflicts (between items that are read by some productions and consumed by other) are avoided. In this paper we introduce a slight variant of DGGs, called persistent dynamic graph grammars (PDGGs), that can be encoded in PGGs preserving concurrency. Finally, PDGGs are exploited to define a concurrent semantics for the Join calculus enriched with persistent messaging (if preferred, the latter can be naively seen as dynamic nets with read arcs)

A Category Theoretical Approach to the Concurrent Semantics of Rewriting: Adhesive Categories and Related Concepts

This thesis studies formal semantics for a family of rewriting formalisms that have arisen as category theoretical abstractions of the so-called algebraic approaches to graph rewriting. The latter in turn generalize and combine features of term rewriting and Petri nets. Two salient features of (the abstract versions of) graph rewriting are a suitable class of categories which captures the structure of the objects of rewriting, and a notion of independence or concurrency of rewriting steps – as in the theory of Petri nets. Category theoretical abstractions of graph rewriting such as double pushout rewriting encapsulate the complex details of the structures that are to be rewritten by considering them as objects of a suitable abstract category, for example an adhesive one. The main difficulty of the development of appropriate categorical frameworks is the identification of the essential properties of the category of graphs which allow to develop the theory of graph rewriting in an abstract framework. The motivations for such an endeavor are twofold: to arrive at a succint description of the fundamental principles of rewriting systems in general, and to apply well-established verification and analysis techniques of the theory of Petri nets (and also term rewriting systems) to a wide range of distributed and concurrent systems in which states have a "graph-like" structure. The contributions of this thesis thus can be considered as two sides of the same coin: on the one side, concepts and results for Petri nets (and graph grammars) are generalized to an abstract category theoretical setting; on the other side, suitable classes of "graph-like" categories which capture the essential properties of the category of graphs are identified. Two central results are the following: first, (concatenable) processes are faithful partial order representations of equivalence classes of system runs which only differ w.r.t. the rescheduling of causally independent events; second, the unfolding of a system is established as the canonical partial order representation of all possible events (following the work of Winskel). Weakly ω-adhesive categories are introduced as the theoretical foundation for the corresponding formal theorems about processes and unfoldings. The main result states that an unfolding procedure for systems which are given as single pushout grammars in weakly ω-adhesive categories exists and can be characetrised as a right adjoint functor from a category of grammars to the subcategory of occurrence grammars. This result specializes to and improves upon existing results concerning the coreflective semantics of the unfolding of graph grammars and Petri nets (under an individual token interpretation). Moreover, the unfolding procedure is in principle usable as the starting point for static analysis techniques such as McMillan’s finite complete prefix method. Finally, the adequacy of weakly ω-adhesive categories as a categorical framework is argued for by providing a comparison with the notion of topos, which is a standard abstraction of the categories of sets (and graphs)

Processes and unfoldings: concurrent computations in adhesive categories

We generalise both the notion of non-sequential process and the unfolding construction (previously developed for concrete formalisms such as Petri nets and graph grammars) to the abstract setting of (single pushout) rewriting of objects in adhesive categories. The main results show that processes are in one-to-one correspondence with switch-equivalent classes of derivations, and that the unfolding construction can be characterised as a coreflection, i.e., the unfolding functor arises as the right adjoint to the embedding of the category of occurrence grammars into the category of grammars. As the unfolding represents potentially infinite computations, we need to work in adhesive categories with "well-behaved" colimits of omega-chains of monos. Compared to previous work on the unfolding of Petri nets and graph grammars, our results apply to a wider class of systems, which is due to the use of a refined notion of grammar morphism

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 22nd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2019, which took place in Prague, Czech Republic, in April 2019, held as part of the European Joint Conference on Theory and Practice of Software, ETAPS 2019. The 29 papers presented in this volume were carefully reviewed and selected from 85 submissions. They deal with foundational research with a clear significance for software science

Coreflective Concurrent Semantics for Single-Pushout Graph Grammars

The problem of extending to graph grammars the unfolding semantics originally developed by Winskel for (safe) Petri nets has been faced several times along the years, both for the single-pushout and double-pushout approaches, but only partial results were obtained. In this paper we fully extend Winskel's approach to single-pushout grammars providing them with a categorical concurrent semantics expressed as a coreflection between the category of graph grammars and the category of prime algebraic domains

Coreflective concurrent semantics for single-pushout graph grammars

Abstract. The problem of extending to graph grammars the unfolding semantics originally developed by Winskel for (safe) Petri nets has been faced several times along the years, both for the single-pushout and double-pushout approaches, but only partial results were obtained. In this paper we fully extend Winskel's approach to single-pushout grammars providing them with a categorical concurrent semantics expressed as a coreflection between the category of graph grammars and the category of prime algebraic domains. Introduction It belongs to the folklore that Graph Grammars [25] generalise Petri nets, in that they allow for a more structured representation of system states, modelled in terms of graphs rather than (multi)sets, and for a more general kind of state transformation, modelling also preservation of parts of the state, besides deletion and creation

Coreflective Concurrent Semantics for Single-Pushout Graph Grammars

The problem of extending to graph grammars the unfolding semantics originally developed by Winskel for (safe) Petri nets has been faced several times along the years, both for the single-pushout and double-pushout approaches, but only partial results were obtained. In this paper we fully extend Winskel’s approach to single-pushout grammars providing them with a categorical concurrent semantics expressed as a coreflection between the category of graph grammars and the category of prime algebraic domains