Walter Vogler

A survey of behaviour and equivalence preserving refinements of Petri nets

Results on refinements of places and transitions in Petri nets are surveyed. Such refinements may either transform a net to a refined net with the same behaviour, where behaviour often means safeness or liveness. Or they may transform semantically equivalent nets to refined nets which are semantically equivalent again. Here the semantics of a net is a description of the possible runs incorporating information on choices to a varying degree.

Bisimulation and action refinement

For event structures with silent moves we consider several types of bisimulation that incorporate ‘true’ concurrency to a varying degree, and show how each can be lifted in a uniform way to a congruence w.r.t. action refinement. We prove that we have constructed the coarsest congruences that respect interleaving, pomset and history preserving bisimulation.

Behaviour preserving refinements of Petri nets

In a hierarchic design of a Petri net a host net, which is a ”rough” model, is refined by replacing a transition by a daughter net, that simulates the transition. For the independent design of host and daughter net those daughter nets are of interest that guarantee that an arbitrary host has the same behaviour as the respective refined net. We characterize these daughter nets, called modules, prove that it is decidable whether a net is a module and show how firing sequences and reachable markings of module, host and refined net are interrelated. Our results shed some light on the problem what a homomorphism of Petri nets should be and allow the generation of live Petri nets.

Executions: a new partial-order semantics of Petri nets

Abstract Executions, a new partial-order semantics of P/T nets, are defined as a generalization of the processes of safe nets. Various relations between executions and processes are established; especially, it is shown that for each net N there is a safe net SN ( N ) such that the processes of SN ( N ) are isomorphic to the executions of N . Furthermore, executions are related to other partial-order semantics of nets in much the same way as processes of safe nets are. It is shown that nets are conflict-free (in some sense) if and only if they have just one maximal execution.

Decidable Boundedness Problems for Hyperedge-Replacement Graph Grammar

Consider a class C of hyperedge-replacement graph grammars and a numeric function on graphs like the number of edges, the degree (i.e., the maximum of the degrees of all nodes of a graph), the number of simple paths, the size of a maximum set of independent nodes, etc. Each such function induces a Boundedness Problem for the class C: Given a grammar HRG in C, are the function values of all graphs in the language L(HRG), generated by HRG, bounded by an integer or not? We show that the Boundedness Problem is decidable if the corresponding function is compatible with the derivation process of the grammars in C and if it is composed of maxima, sums, and products in a certain way. This decidability result applies particularly to the examples listed above.

The Step Failure Semantics

The (linear) failure semantics is a well known model for the theoretical version of Hoares CSP. We generalize this semantics by taking steps (i.e. multisets of simultaneously occurring actions) instead of single actions as the basic execution unit. Hence opposed to the linear semantics, where parallelism is modelled as arbitrary interleaving in order to avoid technical complication, the step failure semantics models true parallelism and is equally easy to manage. Opposed to the linear model here divergence is treated uniformly.

Partial words versus processes: a short comparison

In this note I want to draw attention to partial words, a less known partial order semantics of Petri nets. Partial words have recently proven to be of importance, e.g. in the study of action refinement. An important result of A. Kiehn relates partial words and Petri net processes, and a short proof of this result is given here.

Local Checking of Trace Synchroniziability

Trace theory has been developed to describe the behaviour of concurrent systems. For a modular approach synchronization of traces is of special interest. We characterize those trace monoids for which synchronization can be described locally.

Asynchronous Communication of Petri Nets and the Refinement of Transitions

In this paper we study nets whose boundaries in a context net consist of places only, i.e. nets that communicate asynchronously with their environment. Such nets are called equivalent, if exchanging them in any context preserves deadlock-freeness. We characterize this equivalence internally, i.e. without referring to all possible contexts. Since this equivalence is undecidable in general, we then define a subclass of nets that are to some degree deterministic. For nets from this subclass we can show that the equivalence is decidable and that the exchange of equivalent nets does not only preserve deadlock-freeness, but gives nets that are even bisimilar; these results especially apply to the behaviour preserving refinement of transitions.

On the synchronization of traces

Trace theory has been developed to describe the behavior of concurrent systems. Synchronization of traces is of special interest for a modular approach. We characterize those trace monoids for which synchronization can be described locally.