On the Complexity of Deciding Behavioural Equivalences

We give an overview of the computational complexity of all the equivalences in van Glabbeek's linear/branching time hierarchy and the preorders in the corresponding hierarchy of preorders. We consider nite state or regular processes as well as innnite-state BPA and BPP processes. A distinction, which turns out to be important in the nite-state processes, is that of simulation-like equivalences/preorders vs. trace-like equivalences and preorders. Here we survey various known complexity results for these relations. For regular processes, all simulation-like equivalences and preorders are decidable in polynomial time whereas all trace-like equivalences and preorders are PSPACE-Complete. We also consider interesting special classes of regular processes such as deterministic, determinate, unary, locally unary, and tree-like processes and survey the known complexity results in these special cases. For innnite-state processes the results are quite diierent. For the class of context-free processes or BPA processes any preorder or equivalence beyond bisimulation is undecidable but bisimulation equivalence is polynomial time decidable for normed BPA processes and is known to be elementarily decidable in the general case. For the class of BPP processes, all preorders and equivalences apart from bisimilarity are undecidable. However, bisimilarity is decidable in this case and is known to be decidable in polynomial time for normed BPP processes. 1. Introduction Within concurrency theory, a number of preorders and equivalence relations between processes have been considered in various approaches to the semantics of concurrency and automatic veriication. In this paper, we shall consider the equivalences that have come out of the study of interleaving semantics in the context of process calculi in the tradition of CCS Mil80] and CSP Hoa84]. Most of these preorders and equivalences rst arose in the literature of comparative concurrency semantics vG90a; BIM90; GV92]. In this particular area of concurrency semantics, the main emphasis is on full abstraction and various notions of equivalences have been found to be fully abstract for diierent