Decidability and Coincidence of Equivalences for Concurrency

There are two fundamental problems concerning equivalence relations in concurrency. One is: for which system classes is a given equivalence decidable? The second is: when do two equivalences coincide? Two well-known equivalences are history preserving bisimilarity (hpb) and hereditary history preserving bisimilarity (hhpb). These are both ‘independence’ equivalences: they reflect causal dependencies between events. Hhpb is obtained from hpb by adding a ‘backtracking’ requirement. This seemingly small change makes hhpb computationally far harder: hpb is well-known to be decidable for finite-state systems, whereas the decidability of hhpb has been a renowned open problem for several years; only recently it has been shown undecidable. The main aim of this thesis is to gain insights into the decidability problem for hhpb, and to analyse when it coincides with hpb; less technically, we might say, to analyse the power of the interplay between concurrency, causality, and conflict. We first examine the backtracking condition, and see that it has two dimensions: the number of transitions over which one may backtrack, and the number of backtracking moves. These dimensions translate into two hierarchies of bisimilarities; we find that both of them are strict, and that each of their levels is decidable. Our second approach is to analyse which behavioural properties of concurrent systems are crucial to the increased power of hhpb. After establishing a minimum of behavioural situations necessary to keep hpb and hhpb distinct, we study two aspects of the interplay of causality, concurrency, and conflict: three synchronization witness (SW) situations, and the notion of confusion. With the help of a composition and decomposition result we prove that in their entirety the SW situations are essential for non-coincidence (for bounded-degree systems). However, we show this is not so for confusion, which disproves the long-standing conjecture that hpb and hhpb coincide for confusion-free systems. We continue by studying two structural system classes with promising behavioural properties. First we consider basic parallel processes (BPP), with a suitable partial order semantics. These systems are infinite-state, but they restrict synchronization. Using the tableau technique, we prove the decidability and coincidence of hpb and hhpb for simple BPP (SBPP). The two bisimilarities do not coincide for the complete BPP class, but we separately achieve decidability of both (a known result for hpb, but not for hhpb). The second structural class is (safe) free choice systems, an important class in Petri net theory. These systems have a controlled interplay of concurrency and conflict, and thereby exclude confusion. Having shown that hpb and hhpb do not coincide here, we identify another interesting candidate: live strictly state machine decomposable (SSMD) free choice systems. For this class, we prove that an auxiliary bisimilarity satisfies a restricted backtracking property. As a consequence we achieve the coincidence of hpb and hhpb for a subclass of live SSMD free choice systems: the only known positive result for a class with a reasonable amount of interplay between concurrency, causality, and conflict while still admitting considerable nondeterminism.