Partial Order Semantics and Read Arcs

We study a new partial order semantics of Petri nets with read arcs, where read arcs model reading without consuming, which is often more adequate than the destructive-read-and-rewrite modelled in ordinary nets. As basic observations we take ST-traces, which are sequences of transition starts and ends. We deene processes of our nets and derive two partial orders modelling causality and start precedence. These partial orders are related to observations and system states just as in the ordinary approach the single partial order of a process is related to ring sequences and reachable markings. Our approach also supports a new view of concurrency as captured by steps. 1 Introduction Describing the runs of a concurrent system by sequences of actions ignores the possible concurrency of these actions, which can be important e.g. for judging the temporal ee-ciency of the system. Alternatively to this so-called interleaving approach, one can take step sequences, where a step consists of simultaneous actions, or partial orders to describe runs { resulting in a so-called`true concurrency' semantics. We will use safe Petri nets to model concurrent systems; for these models, the most prominent partial order semantics are so-called processes. A process of a net N is essentially a very simple net consisting of events (transition rings in N) and conditions (tokens in N produced during the run); the process gives a partial order on these events and conditions. The beauty of the approach is that operationally deened entities of N can now be derived order-theoretically: Each linearization of the events is a ring sequence of N, and vice versa, each ring sequence of N is a linearization of a unique process. We can view the process as a run and its linearizations as observations of the run; essentially by Szpilrajn's Theorem, we can reconstruct the partial order of the events simply as intersection of the total orders given by all these observations. Furthermore, unordered conditions are Work on this paper was partially supported by the DFG (Project`Halbordnungstesten'). An extended abstract is to appear in the proceedings of MFCS 97.