On the Complexity of Deciding Behavioural Equivalences and Preorders. A Survey

Abstract

This paper gives an overview of the computational complexity of all the equivalences in the linear/branching time hierarchy [vG90a] and the preorders in the corresponding hierarchy of preorders. We consider finite state or regular processes as well as infinite-state BPA [BK84b] processes.A distinction, which turns out to be important in the finite-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 infinite-state processes the results are quite different. 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.

Model-Checking Process Equivalences

“…We have presented a model checking algorithm which can be instantiated in order to yield decision procedures for the relations on finite systems. This re-establishes already known decidability results [7]. Its main contribution, though, is the -to the best of our knowledge -first framework that provides a generic and uniform algorithmic approach to process equivalence checking via defining return F(X 0 ,Y 0 ) 27: end procedure formulas.…”

“…There is a lot of potential further work into this direction. The exponential-time bound for the trace-like equivalences is not optimal since they are generally PSPACE-complete [7]. It remains to be seen whether the formulas defining them have a particular structure that would allow a PSPACE model checking algorithm for instance.…”

A Game Characterization for Contrasimilarity

“…We have not yet thought of a decisive argument for the complexity of contrasimilarity checking. Due to the results for the spectrum without internal steps [14], one may presume the coarser siblings of contrasimilarity from weak impossible futures down to weak trace equivalence to all be PSPACE-hard. But while deciding weak possible futures and nestings of possible futures equivalence is PSPACE, deciding its arbitrarily nested limit, weak bisimilarity, is PTIME.…”

Behavioural Preorders on Stochastic Systems - Logical, Topological, and Computational Aspects

“…Another notion of behavioural equivalence, which comes from automata theory, is that of trace equivalence, in which two processes are said to be equivalent if they have the same possible executions, known as traces. Generally speaking, it has proven more difficult to reason about trace equivalences than to reason about bisimulation [79].…”