Concurrency and Probability: Removing Confusion, Compositionally

Assigning a satisfactory truly concurrent semantics to Petri nets with confusion and distributed decisions is a long standing problem, especially if one wants to fully replace nondeterminism with probability distributions and no stochastic structure is desired/allowed. Here we propose a general solution based on a recursive, static decomposition of (finite, occurrence) nets in loci of decision, called structural branching cells (s-cells). Each scell exposes a set of alternatives, called transactions, that can be equipped with a general probabilistic distribution, i.e., as desired, nondeterminism can be entirely replaced by probability. The solution is formalised as a transformation from a given Petri net to another net whose transitions are the transactions of the s-cells and whose places are the places of the original net, with some auxiliary structure for bookkeeping. The resulting net is confusion-free, namely if a transition is enabled, then all its conflicting alternatives are also enabled. Thus sets of conflicting alternatives can be equipped with probability distributions, while nonintersecting alternatives are purely concurrent and do not introduce any nondeterminism: they are Church-Rosser and their probability distributions are independent. The validity of the construction is witnessed by a tight correspondence result with the recent approach by Abbes and Benveniste (AB) based on recursively stopped configurations in event structures. Some advantages of our approach over AB’s are that: i) s-cells are defined statically and locally in a compositional way, whereas AB’s branching cells are defined dynamically (by executing the event structure) and globally (by manipulating the entire event structure); ii) their recursively stopped configurations correspond to possible executions, but the existing concurrency is not made explicit. Instead, our resulting nets are equipped with an original concurrency structure exhibiting a so-called complete concurrency property.