Behavioural equivalence for infinite systems — Partially decidable!

Abstract

For finite-state systems non-interleaving equivalences are computationally at least as hard as interleaving equivalences. In this paper we show that when moving to infinite-state systems, this situation may change dramatically.We compare standard language equivalence for process description languages with two generalizations based on traditional approaches capturing non-interleaving behaviour, pomsets representing global causal dependency, and locality representing spatial distribution of events.We first study equivalences on Basic Parallel Processes, BPP, a process calculus equivalent to communication free Petri nets. For this simple process language our two notions of non-interleaving equivalences agree. More interestingly, we show that they are decidable, contrasting a result of Hirshfeld that standard interleaving language equivalence is undecidable. Our result is inspired by a recent result of Esparza and Kiehn, showing the same phenomenon in the setting of model checking.We follow up investigating to which extent the result extends to larger subsets of CCS and TCSP. We discover a significant difference between our non-interleaving equivalences. We show that for a certain non-trivial subclass of processes between BPP and TCSP, not only are the two equivalences different, but one (locality) is decidable whereas the other (pomsets) is not. The decidability result for locality is proved by a reduction to the reachability problem for Petri nets.

Composition and Decomposition in True-Concurrency

“…Under interleaving semantics a small fragment of a logic equivalent to CTL * is undecidable for very basic BPP; under partial order interpretation the full logic is decidable for BPP [7]. Trace equivalence on BPP is undecidable but pomset trace and location trace equivalence on BPP are shown decidable in [8]. Classical bisimilarity on BPP is PSPACE-complete [9,10]; in contrast, for BPP, distributed bisimilarity, and with it hp bisimilarity, are polynomial-time decidable [11].…”

“…BPP are infinite-state but, under truly-concurrent semantics, they have a simple tree-like structure, which has turned out to be directly exploitable: e.g. the decidability results of [8] follow by a reduction to the equivalence problem of recognizable tree languages.…”