Team bisimilarity, and its associated modal logic, for BPP nets

Abstract

BPP nets, a subclass of finite Place/Transition Petri nets, are equipped with an efficiently decidable, truly concurrent, bisimulation-based, behavioral equivalence, called team bisimilarity. This equivalence is a very intuitive extension of classic bisimulation equivalence (over labeled transition systems) to BPP nets and it is checked in a distributed manner, without necessarily building a global model of the overall behavior of the marked BPP net. An associated distributed modal logic, called team modal logic (TML, for short), is presented and shown to be coherent with team bisimilarity: two markings are team bisimilar if and only if they satisfy the same TML formulae. As the process algebra BPP (with guarded summation and guarded body of constants) is expressive enough to represent all and only the BPP nets, we provide algebraic laws for team bisimilarity as well as a finite, sound and complete, axiomatization.

True Concurrency Can Be Easy

“…In this way, we would be able to have a compositional semantics, fully respecting causality and the branching structure of systems, for the class of all the finite Petri nets. It is possible to prove that net bisimilarity ∼ n coincides with team bisimilarity over BPP nets [17], which in turn coincides with place bisimilarity [1] on this class of nets. Interestingly enough, the definition of net bisimulation has some similarities with that of place bisimulation, as also this relation requires that the size of the related markings is the same and that the pre-sets of the matching transitions be related.…”

“…However, we expect that our simpler characterization of ∼ cn in term of net bisimilarity ∼ n may offer some hints towards a more efficient algorithm, yet exponential as well. Net bisimilarity is decidable on BPP nets, because on this class of nets it coincides with team bisimilarity [17,20], which is decidable in polynomial time.…”

Place Bisimilarity is Decidable, Indeed!

“…When R is an equivalence relation, it is rather easy to check whether two markings are related by R ⊕ . An algorithm, described in [10], establishes whether an R-preserving bijection exists between the two markings m 1 and m 2 of equal size k in O(k 2 ) time. Another algorithm, described in [15], checks whether (m 1 , m 2 ) ∈ R ⊕ in O(n) time, where n is the size of S. However, these performant algorithms heavily rely on the fact that R is an equivalence relation, hence also subtractive (case 4 of Proposition 2).…”

“…k). Now we list some useful, and less obvious, properties of additively closed place relations (proof in [10]). Proposition 3.…”

“…On the subclass of BPP nets (i.e., nets whose transitions have singleton pre-set), place bisimilarity coincides with team bisimilarity [10], which is actually coinductive and, for this reason, very efficiently decidable (precisely in O(m • n • log(n + 1)) time, where n is the number of places and m of transitions), characterized by a simple modal logic, and even axiomatizable over the process algebra BPP with guarded summation and guarded recursion [9]. Moreover, d-place bisimilarity specializes to h-team bisimilarity [11], which coincides with fully-concurrent bisimilarity on this subclass of P/T nets.…”