A Graph Abstract Machine Describing Event Structure Composition

Claudia Faggian, Mauro Piccolo
2007 Electronical Notes in Theoretical Computer Science  
Event structures, Game Semantics strategies and Linear Logic proof-nets arise in different domains (concurrency, semantics, proof-theory) but can all be described by means of directed acyclic graphs (dag's). They are all equipped with a specific notion of composition, interaction or normalization. We report on-going work, aiming to investigate the common dynamics which seems to underly these different structures. In this paper we focus on confusion free event structures on one side, and linear
more » ... trategies [Gir01,FM05] on the other side. We introduce an abstract machine which is based on (and generalizes) strategies interaction; it processes labelled dag's, and provides a common presentation of the composition at work in these different settings. We are in debt with Daniele Varacca for many explanations, comments, and suggestions. We are grateful to Martin Hyland, Emmanuel Beffara, and Pierre-Louis Curien for interesting discussions. We also wish to thank the referees for many usefull remarks and suggestions. 2 applied to event structures, the result is the same as the paralle composition of event structures defined by Varacca and Yoshida in [VY06] . Event structures are a causal model of concurrency (also called true concurrency models), i.e. causality, concurrency and conflict are directly represented, as opposite to interleaving models, which describe the system by means of all possible scheduling of concurrent actions. An event structure models parallel computation by means of occurrence of events, and a partial order expressing causal dependency. Non-determinism is modelled by means of a conflict relation, which expresses how the occurrence of certain events rules out the occurrence of others. Two events are concurrent if they are neither causally related, nor in conflict. Events which are in conflict live in different possible evolutions of the system. In this paper we will consider two classes of event structures: confusion free event structure (where conflict, and hence non-determinism, is well behaving), and conflict free event structures (where there is no conflict, and hence no non-determinism). Confusion free event structure, are an important class of event structures because the choices which a process can do are "local" and not influenced by independent actions. In this sense, confusion freeness generalizes confluence to systems that allow nondeterminism. A point which is central to our approach is that a confusion free event structure E can be seen as a superposition of conflict-free event structures (which we will call the slices of E): each slice represents a possible and independent evolution of the system. Because of this, if E is confusion free, the study of several properties can be reduced to the study of such properties in conflict free event structures. Games and strategies provide denotational models for programming languages and logical systems; games correspond to types (formulas), and strategies to programs (proofs). The central notion is that of interaction, which models program composition (normalization of proofs). A distinction between causal and interleaving models is appearing also in Game Semantics. In this setting, a strategy describes in an abstract way the operational behaviour of a term. In the standard approach, a strategy is described by sequences of actions (plays), which represent the traces of the computation. However, there is an active line of research in Game Semantics aiming at relaxing sequentiality, either with the purpose to have "partial order" models of programming languages or to capture concurrency [AM99, Mel04, HS02, MW05, SPP05, FM05, CF05, Lai05, GM04] . The underlying idea is to not completely specify the order in which the actions should be performed, while still being able to express constraints. Certain tasks may have to be performed before other tasks; other actions can be performed in parallel, or scheduled in any order. A strategy of this kind can be presented as a directed acyclic graph. Content. An idea which underlies the work on typed π-calulus by Honda and Yoshida is that typed processes should be seen as a sort of Hyland-Ong strategies (see for example [NYB01] ); this approach is implicit also in [VY06], on which our work builds. Varacca and Yoshida provide a typing system which guarantees that the composition of confusion free event structures is confusion free. The typing is
doi:10.1016/j.entcs.2007.04.014 fatcat:evqayqrx2fdqvcvqkwkgwbux4i