Time Process Equivalences for Time Petri Nets

In the core of every theory of systems lies a notion of equivalence between systems: it indicates which particular aspects of systems behaviors are considered to be observable. In concurrency theory, a variety of observational equivalences has been promoted, and the relationships between them have been quite wellunderstood. In order to investigate the performance of systems (e.g. the maximal time used for the execution of certain activities and average waiting time for certain requests), many time extensions have been de ned for a non-interleaving model of Petri nets. On the other hand, there are few mentions of a fusion of timing and partial order semantics, in the Petri net literature. In [9], processes of timed Petri nets (under the asap hypothesis) have been de ned by an algebra of the so-called weighted pomsets. The paper [8] has provided and compared timed step sequence and timed process semantics for timed Petri nets. A method to compute all valid timings for a causal net process of a time Petri net has been put forward in [3]. Branching processes (unfoldings) of time Petri nets have been constructed in [7]. To the best of our knowledge, the incorporation of timing into equivalence notions on Petri nets is even less advanced. In this regard, the paper [4] is a welcome exception, where the testing approach has been extended to Petri nets with associating clocks to tokens and time intervals to arcs from places to transitions. A comparison of di erent subclasses of time Petri nets has been made in [5], on the base of timed interleaving language and bisimulation equivalences. The papers [1,2] contributed to the classi cation of the wealth of observational equivalences of linear time branching time spectrum, based on interleaving, causal tree and partial order semantics, for dense time extensions of event structures with/without internal actions. The intention of the note is towards developing, studying and comparing trace and bisimulation equivalences based on interleaving, step, partial order, and net-process semantics in the setting of time Petri nets (elementary net systems enriched with the time static intervals on transitions, and with some niteness