Finite petri nets as models for recursive causal behaviour

Goltz (1988) discussed whether or not there exist finite Petri nets (with unbounded capacities) modelling the causal behaviour of certain recursive CCS terms. As a representative example, the following term is considered: \ud \ud B=(a.nilb.B)+c.nil. \ud \ud We will show that the answer depends on the chosen notion of behaviour. It was already known that the interleaving behaviour and the branching structure of terms as B can be modelled as long as causality is not taken into account. We now show that also the causal behaviour of B can be modelled as long as the branching structure is not taken into account. However, it is not possible to represent both causal dependencies and the behaviour with respect to choices between alternatives in a finite net. We prove that there exists no finite Petri net modelling B with respect to both pomset trace equivalence and failure equivalence

On Syntactic and Semantic Action Refinement

The semantic definition of action refinement on labelled event structures is compared with the notion of syntactic substitution,which can be used as another notion of action refiment in a process algebraic setting. This is done by studying a process algebra equipped with the ACP sequential composition, parallel composition with an explicit synchronization set, and an operator for action refinement. On the one hand, the language (including the renement\ud operator) is given a flow event structure semantics. On the other hand, a reduction procedure transforms a process term P into a flat term (i.e., with the refinement operator not occurring in it) red(P) by means of syntactic substitution, defined in a structural inductive way.\ud \ud The main aim of the paper is to find general conditions under which the terms P and red(P) have the same semantics. The results we present are essentially dependent on the question whether the refined action can be synchronized or not. In the latter case, P and red(P) give rise to isomorphic flow event structures under mild assumptions. The former case is considerably more difficult. We give necessary and sufficient semantic conditions under which refinement can be distributed over synchronization up to isomorphic domains of configurations. Subsequently we also give sufficient (but not necessary) syntactic conditions for reducible terms.\u

Comparing Syntactic and Semantics Action Refinement

The semantic definition of action refinement on labelled configuration structures is compared with the notion of syntactic substitution, which can be used as another notion of action refinement in a process algebraic setting. The comparison is done by studying a process algebra equipped with sequential composition, parallel composition with an explicit synchronisation set, and an operator for action refinement. On the one hand, the language (including the refinement operator) is given a configuration structure semantics. On the other hand, a reduction procedure transforms a process term P into a flat term (i.e., with the refinement operator not occurring in it) red(P) by means of syntactic substitution, defined in a structural inductive way. The main aim of the paper is to investigate general conditions under which the terms P and red(P) have the same semantics. The results we present are essentially dependent on the question whether the refined action can be synchronised or not. In the latter case, P and red(P) give rise to isomorphic configuration structures under mild assumptions. The former case is considerably more difficult, since then refinement cannot be expected to distribute over parallel composition. We give necessary and sufficient semantic conditions under which distribution still holds up to semantic equivalence. Subsequently, we also give sufficient (but not necessary) syntactic conditions for reducible terms. Finally, we generalise these results to a language with recursion.\u

Synchrony vs. Causality in Asynchronous Petri Nets

Given a synchronous system, we study the question whether the behaviour of that system can be exhibited by a (non-trivially) distributed and hence asynchronous implementation. In this paper we show, by counterexample, that synchronous systems cannot in general be implemented in an asynchronous fashion without either introducing an infinite implementation or changing the causal structure of the system behaviour.Comment: In Proceedings EXPRESS 2011, arXiv:1108.407

Symmetric and Asymmetric Asynchronous Interaction

We investigate classes of systems based on different interaction patterns with the aim of achieving distributability. As our system model we use Petri nets. In Petri nets, an inherent concept of simultaneity is built in, since when a transition has more than one preplace, it can be crucial that tokens are removed instantaneously. When modelling a system which is intended to be implemented in a distributed way by a Petri net, this built-in concept of synchronous interaction may be problematic. To investigate this we consider asynchronous implementations of nets, in which removing tokens from places can no longer be considered as instantaneous. We model this by inserting silent (unobservable) transitions between transitions and some of their preplaces. We investigate three such implementations, differing in the selection of preplaces of a transition from which the removal of a token is considered time consuming, and the possibility of collecting the tokens in a given order. We investigate the effect of these different transformations of instantaneous interaction into asynchronous interaction patterns by comparing the behaviours of nets before and after insertion of the silent transitions. We exhibit for which classes of Petri nets we obtain equivalent behaviour with respect to failures equivalence. It turns out that the resulting hierarchy of Petri net classes can be described by semi-structural properties. For two of the classes we obtain precise characterisations; for the remaining class we obtain lower and upper bounds. We briefly comment on possible applications of our results to Message Sequence Charts.Comment: 27 pages. An extended abstract of this paper was presented at the first Interaction and Concurrency Experience (ICE'08) on Synchronous and Asynchronous Interactions in Concurrent Distributed Systems, and will appear in Electronic Notes in Theoretical Computer Science, Elsevie