Team equivalences for finite-state machines with silent moves

Abstract

Finite-state machines, a simple class of finite Petri nets, were equipped in [16] with an efficiently decidable, truly-concurrent, bisimulation-based, behavioral equivalence, called team equivalence, which conservatively extends classic bisimulation equivalence on labeled transition systems and which is checked in a distributed manner. This paper addresses the problem of defining variants of this equivalence which are insensitive to silent moves. We define (rooted) weak team equivalence and (rooted) branching team equivalence as natural transposition to finite-state machines of Milner's weak bisimilarity [25] and van Glabbeek and Weijland's branching bisimilarity [12] on labeled transition systems. The process algebra CFM [15] is expressive enough to represent all and only the finite-state machines, up to net isomorphism. Here we first prove that the rooted versions of these equivalences are congruences for the operators of CFM, then we show some algebraic properties, and, finally, we provide finite, sound and complete, axiomatizations for them.

Introduction

By finite-state machine (FSM, for short) we mean a simple type of finite Petri net [15], [29], [33] whose transitions have singleton pre-set and singleton, or empty, post-set. The name originates from the fact that a net of this kind is isomorphic to a nondeterministic finite automaton (NFA), usually called a finite-state machine as well. However, semantically, our FSMs are richer than NFAs because, as their initial marking may be not a singleton, these nets can also exhibit concurrent behavior, while NFAs are strictly sequential. FSMs are also very similar to finite-state, labeled transition systems (LTSs, for short) [22], a class of models that are suitable for describing sequential, nondeterministic systems, and are also widely used as a semantic model for process algebras (see, e.g., [14]). On this class of models, there is widespread agreement that a very natural and convenient equivalence relation is bisimulation equivalence [28], [25]. If the LTS contains silent transitions, i.e., transitions labeled by the invisible action τ, then Milner proposed weak bisimulation equivalence [25] as a natural extension of bisimulation equivalence to this setting. However, van Glabbeek and Weijland in [12] argued that weak bisimilarity does not completely respect the branching structure of processes and so they proposed branching bisimulation equivalence as a suitable equivalence in the presence of silent moves.

In [16] we defined a new truly-concurrent equivalence relation over FSMs (without silent moves), called team equivalence, that can be computed in a distributed manner, without resorting to a global model of the overall behavior of the analyzed marked net. Since an FSM is so similar to an LTS, the basic idea we started with was to define bisimulation equivalence directly over the set of places of the unmarked net. The advantage is that bisimulation equivalence is a relation on places, rather than on markings (as it is customary for Petri nets), and so much more easily computable; more precisely, if m is the number of net transitions and n is the number of places, checking whether two places are bisimilar can be done in $O(m\text{ log }(n+1))$ time, by adapting the optimal algorithm in [30] for standard bisimulation on LTSs. After the bisimulation equivalence over the set of places has been computed, we can define, in a purely structural way, that two markings $m_1$ and $m_2$ are team equivalent if they have the same cardinality, say $|m_1|=k=|m_2|$, and there is a bisimulation-preserving, bijective mapping between the two markings, so that each of the k pairs of places $(s_1,s_2)$, with $s_1\in m_1$ and $s_2\in m_2$, is such that $s_1$ and $s_2$ are bisimilar. Team equivalence is a truly concurrent behavioral equivalence as it is sensitive to the size of the distributed state; as a matter of fact, it relates markings of the same size, only. The name team equivalence reminds us that two distributed systems, composed of a team of non-cooperating, sequential processes, are equivalent if it is possible to match each sequential component of the first system with one bisimulation-equivalent, sequential component of the other one, as in any sports where two competing (distributed) teams have the same number of (sequential) players. Once bisimilarity on places has been computed, checking whether two markings of size k are team equivalent can be computed in $O(k^2)$ time (or $O(n)$, cf. Remark 1).

Note that to check whether two markings are team equivalent we need not to construct an LTS describing the global behavior of the whole system, but only to find a suitable, bisimulation-preserving match among the local, sequential states (i.e., the elements of the markings). Nonetheless, we proved that team equivalence is coherent with the global behavior of the net. More precisely, we showed in [16] that team equivalence is finer than interleaving bisimilarity, actually it coincides with strong place bisimilarity [1] and so it respects the causal semantics of nets.

The main goal of this paper is to extend our previous definition of (strong) bisimulation on places (and of team equivalence) to FSMs with silent moves, taking inspiration from Milner's weak bisimulation [25], [14] and van Glabbeek and Weijland's branching bisimulation [12], [14]. Therefore, we first define weak bisimulation on places and its associated weak team equivalence, together with the variants requiring the so-called rootedness condition (i.e., an initial silent move can be matched only by a nonempty sequence of silent moves, as in rooted weak bisimilarity [25], [14]). Then, we define branching bisimulation on places and its associated branching team equivalence, together with the variants requiring the rootedness condition (i.e., the first move is to be matched strongly, as in rooted branching bisimilarity [12], [14]). The originality of our proposal is not in the technical definition of (rooted) weak/branching bisimulations on places (which are, indeed, almost identical to the original ones on LTSs), rather on the fact that these relations are defined over the places of an unmarked net, rather than on the reachable markings of a marked net. Moreover, the main originality of our proposal is in the definition of (rooted) weak/branching team equivalences; these equivalences are all computed in a structural way, without building a model of the global behavior. Nonetheless, we will prove that these are coherent with the global behavior; in particular, they are finer than the corresponding (rooted) weak/branching interleaving bisimulation equivalences over FSMs (see Section 2 for details), which are equivalences relations defined over the net markings. The key feature common to all the new team equivalences we present in this paper, is that, contrary to the weak/branching interleaving bisimulation equivalences, to a silent move of a single sequential component of the marking $m_1$, the marking $m_2$ may reply only with a (possibly empty) sequence of silent transitions which are local to one of its sequential components.

In [15] we proved that the class of FSMs can be “alphabetized” by means of the process algebra CFM: not only the net semantics of each CFM term is an FSM, but also, given a FSM N, we can single out a CFM term $p_N$ such that its net semantics is an FSM isomorphic to N. This means that we can define team equivalences also over CFM process terms. CFM is a simple process algebra: it is actually a slight extension to finite-state CCS [25] and a subcalculus of both regular CCS and BPP [14].

Based on our previous work [17], where we provided a sound and complete axiomatization for (strong) team equivalence over CFM, the goals of the second part of this paper are three: we want (i) to prove that rooted weak/branching team equivalence is a congruence for the CFM operators, (ii) to study the algebraic properties of these equivalences and, finally, (iii) to provide them with a sound and complete, finite axiomatization for CFM. These axiomatizations are not too surprising: it is enough to add to (a slightly revised version of) the finite axiomatization of rooted weak/branching bisimulation equivalence for finite-state CCS [26], [11], three axioms for parallel composition stating that it is associative, commutative and with 0 as neutral element. However, the technical treatment is different (and somehow simpler) than [26], [11], as we base our axiomatization on guarded process constants (e.g., $C\doteq a.C$) rather than on the recursion construct (with possible unguarded variables; e.g., $\mu Xa.X+X$). To the best of our knowledge, these are the first axiomatizations of a truly concurrent behavioral equivalence in the presence of silent moves.

The paper is organized as follows. Section 2 introduces the basic definitions about finite-state machines and some well-known behavioral equivalences on them: (rooted) weak interleaving bisimilarity and (rooted) branching interleaving bisimilarity. Section 3 copes with the distributed equivalence checking problem for (rooted) weak equivalence; first, (rooted) weak bisimulation over places of an unmarked net is defined; then, (rooted) weak team equivalence is introduced and some examples are presented, together with a proof that it is finer than (rooted) weak interleaving bisimilarity; moreover, the minimization of an FSM w.r.t. weak bisimilarity on places is defined. Section 4 copes with the similar distributed equivalence checking problem for (rooted) branching bisimilarity: we define first (rooted) branching bisimilarity on places, then (rooted) branching team equivalence and its minimized net. Section 5 introduces the process algebra CFM, its syntax, its net semantics and recalls the so-called representability theorem from [15]. Section 6 shows that rooted weak/branching team equivalences are congruences for the CFM operators and studies their algebraic properties. Section 7 presents the finite axiomatizations of rooted weak/branching team equivalences for CFM, proving that they are sound and complete. Finally, Section 8 discusses some related literature and future research.