Place Bisimilarity is Decidable, Indeed!

Abstract

Place bisimilarity ∼ p is a behavioral equivalence for finite Petri nets, proposed by Schnoebelen and co-workers in 1991. Differently from all the other behavioral relations proposed so far, a place bisimulation is not defined over the markings of a finite net, rather over its places, which are finitely many. However, place bisimilarity is not coinductive, as the union of place bisimulations may be not a place bisimulation. Place bisimilarity ∼ p was claimed decidable in [1], even if the algorithm used to this aim [2] does not characterize this equivalence, rather the unique maximal place bisimulation which is also an equivalence relation; hence, its decidability was not proved. Here we show that it is possible to decide ∼ p with a simple, yet inefficient, algorithm, which essentially scans all the place relations (which are finitely many) to check whether they are place bisimulations. Moreover, we propose a slightly coarser variant, we call d-place bisimilarity, that we conjecture to be the coarsest equivalence, fully respecting causality and branching time, to be decidable on finite Petri nets.

A Decidable Equivalence for a Turing-complete, Distributed Model of Computation

“…∼ p = {R ⊕ R is a maximal pti-place bisimulation}. However, it is not true that ∼ p = ( {R R is a maximal pti-place bisimulation}) ⊕ because the union of pti-place bisimulations may be not a pti-place bisimulation (as already observed for place bisimulation in [3,15]), so that its definition is not coinductive.…”

“…We now present pti-place bisimilarity, which conservatively extends place bisimilarity [3,15] to the case of PTI nets. First, an auxiliary definition.…”

“…We are now ready to introduce pti-place bisimulation, which is a non-interleaving behavioral relation defined over the net places. Note that for P/T nets, place bisimulation [3,15] and pti-place bisimulation coincide because I = / 0.…”

“…Despite this, we show that it is possible to define a sensible, behavioral equivalence which is actually decidable on finite PTI nets. This equivalence, we call pti-place bisimilarity, is a conservative extension of place bisimilarity on finite P/T nets, introduced in [3] as an improvement of strong bisimulation [23], (a relation proposed by Olderog in [23] on safe nets which fails to induce an equivalence relation), and recently proved decidable in [15].…”

“…We extend this idea in order to be applicable to PTI nets. Informally, a binary relation R over the set S of places is a pti-place bisimulation if for all markings m 1 and m 2 which are bijectively related via R (denoted by (m 1 , m 2 ) ∈ R ⊕ , where R ⊕ is called the additive closure of R), if m 1 can perform transition t 1 , reaching marking m ′ 1 , then m 2 can perform a transition t 2 , reaching m ′ 2 , such that • the pre-sets of t 1 and t 2 are related by R ⊕ , the label of t 1 and t 2 is the same, the post-sets of t 1 and t 2 are related by R ⊕ , and also (m ′ 1 , m ′ 2 ) ∈ R ⊕ , as required by a place bisimulation [3,15], but additionally it is required that • the inhibiting sets of t 1 and t 2 are related by R ⊕ and that, whenever (s, s ′ ) ∈ R, s belongs to the inhibiting set of t 1 if and only if s ′ belongs to the inhibiting set of t 2 ;…”