A Study on Team Bisimulations for BPP Nets

Abstract

BPP nets, a subclass of finite P/T nets, were equipped in [13] with an efficiently decidable, truly concurrent, behavioral equivalence, called team bisi-milarity. 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 building a global model of the overall behavior of the marked BPP net. This paper has three goals. First, we provide BPP nets with various causality-based equivalences, notably a novel one, called causal-net bisimilarity, and (a version of) fully-concurrent bisimilarity [3]. Then, we define a variant equivalence, h-team bisimilarity, coarser than team bisimilarity. Then, we complete the study by comparing them with the causality-based semantics we have introduced: the main results are that team bisimilarity coincides with causal-net bisimilarity, while h-team bisimilarity with fully-concurrent bisimilarity.

Introduction

A BPP net is a simple type of finite Place/Transition Petri net [18] whose transitions have singleton pre-set. Nonetheless, as a transition can produce more tokens than the only one consumed, there can be infinitely many reachable markings of a BPP net. BPP is the acronym of Basic Parallel Processes [4], a simple CCS [11, 15] subcalculus (without the restriction operator) whose processes cannot communicate. In [12] a variant of BPP, which requires guarded sum and guarded recursion, is actually shown to represent all and only the BPP nets, up to net isomorphism, and this explains the name of this class of nets.

In a recent paper [13], we proposed a novel behavioral equivalence for BPP nets, based on a suitable generalization of the concept of bisimulation [15], originally defined over labeled transition systems (LTSs, for short). A team bisimulation R over the places of an unmarked BPP net is a relation such that if two places and are related by R, then if performs a and reaches the marking , then may perform a reaching a marking such that and are element-wise, bijectively related by R (and vice versa if moves first). Team bisimilarity is the largest team bisimulation over the places of the unmarked BPP net, and then such a relation is lifted to markings by additive closure: if place is team bisimilar to place and the marking is team bisimilar to (the base case relates the empty marking to itself), then also is team bisimilar to , where is the operator of multiset union. Note that to check if two markings are team bisimilar we need not to construct an LTS, such as the interleaving marking graph, describing the global behavior of the whole system, but only to find a bijective, team bisimilarity-preserving match among the elements of the two markings. In other words, two distributed systems, each composed of a team of sequential, non-cooperating processes (i.e., the tokens in the BPP net), are equivalent if it is possible to match each sequential component of the first system with one team-bisimilar, sequential component of the other system, as in any sports where two competing (distributed) teams have the same number of (sequential) players.

The complexity of checking whether two markings of equal size are team bisimilar is very low. First, by adapting the optimal algorithm for standard bisimulation equivalence over LTSs [19], team bisimulation equivalence over places can be computed in time, where m is the number of net transitions, p is the size of the largest post-set (i.e., p is the least natural such that for all t) and n is the number of places. Then, checking whether two markings of size k are team bisimilar can be done in time. Of course, we proved that team bisimilar markings respect the global behavior; in particular, we proved that team bisimilarity implies interleaving bisimilarity and that team bisimilarity coincides with strong place bisimilarity [1].

In this paper, we complete the comparison between team bisimilarity on markings and the causal semantics of BPP nets. In particular, we propose a novel coinductive equivalence, called causal-net bisimulation equivalence, inspired by [9], which is essentially a bisimulation semantics over the causal nets [2, 17] of the BPP net under scrutiny. We prove that team bisimilarity on markings coincides with causal-net bisimilarity, hence proving that our distributed semantics is coherent with the expected causal semantics of BPP nets. Moreover, we adapt the definition of fully-concurrent bisimulation (fc-bisimulation, for short) in [3], in order to be better suited for our aims. Fc-bisimilarity was inspired by previous notions of equivalence on other models of concurrency, in particular, by history-preserving bisimulation (hpb, for short) [8]. Moreover, we define also a slight strengthening of fc-bisimulation, called state-sensitive fc-bisimulation, which requires additionally that, for each pair of related processes, the current markings have the same size. We also prove that causal-net bisimilarity coincides with state-sensitive fc-bisimilarity. These behavioral causal semantics have been provided for BPP nets, but they can be easily adapted for general P/T nets.

The other main goal of this paper is to show that fc-bisimilarity (hence also hpb) can be characterized for BPP nets in a team-style, by means of h-team bisimulation equivalence. (The prefix h- is used to remind that h-team bisimilarity is connected to hpb.) The essential difference between a team bisimulation and an h-team bisimulation is that the former is a relation on the set of places only, while the latter is a relation on the set composed of the places and the empty marking .

The paper is organized as follows. Section 2 introduces the basic definitions about BPP nets and recalls interleaving bisimilarity. Section 3 discusses the causal semantics of BPP nets. First, the novel causal-net bisimulation is introduced, then (state-sensitive) fully-concurrent bisimilarity, as an improvement of the original one [3], which better suits our aims. Section 4 recalls the main definitions and results about team bisimilarity from [13]; in this section we also prove a novel result: causal-net bisimilarity coincides with team bisimilarity for BPP nets. Section 5 defines h-team bisimulation equivalence and studies its properties; in particular, we prove that h-team bisimilarity coincides with fc-bisimilarity. Finally, Sect. 6 discusses related literature.

Basic Definitions

Definition 1

(Multiset). Let be the set of natural numbers. Given a finite set S, a multiset over S is a function . Its support set dom(m) is . The set of all multisets over S is ranged over by m. We write if . The multiplicity of s in m is the number m(s). The size of m, denoted by |m|, is the number , i.e., the total number of its elements. A multiset m such that is called empty and is denoted by . We write if for all .

Multiset union is defined as follows: ; it is commutative, associative and has as neutral element. Multiset difference is defined as follows: . The scalar product of a number j with m is the multiset defined as . By we also denote the multiset with as its only element. Hence, a multiset m over can be represented as , where for .    

Definition 2

(BPP net). A labeled BPP net is a tuple , where

• S is the finite set of places, ranged over by s (possibly indexed),

• A is the finite set of labels, ranged over by (possibly indexed), and

• is the finite set of transitions, ranged over by t (possibly indexed).

Given a transition , we use the notation:

• to denote its pre-set s (which is a single place) of tokens to be consumed;

• l(t) for its label , and

• to denote its post-set m (which is a multiset) of tokens to be produced.

Hence, transition t can be also represented as . We also define pre-sets and post-sets for places as follows: and . Note that the pre-set (post-set) of a place is a set.    

Definition 3

(Marking, BPP net system). A multiset over S is called a marking. Given a marking m and a place s, we say that the place s contains m(s) tokens, graphically represented by m(s) bullets inside place s. A BPP net system is a tuple , where (S, A, T) is a BPP net and is a marking over S, called the initial marking. We also say that is a marked net.    

Definition 4

(Firing sequence). A transition t is enabled at m, denoted , if . The firing of t enabled at m produces the marking , written . A firing sequence starting at m is defined inductively as follows:

• is a firing sequence (where denotes an empty sequence of transitions) and

• if is a firing sequence and , then is a firing sequence.

If (for ) and is a firing sequence, then there exist such that , and is called a transition sequence starting at m and ending at . The set of reachable markings from m is

Note that the reachable markings can be countably infinite. A BPP net system is safe if each marking m reachable from the initial marking is a set, i.e., for all . The set of reachable places from s is

Note that reach(s) is always a finite set, even if is infinite.    

Example 1

Figure 1(a) shows the simplest BPP net representing a semi-counter, i.e., a counter unable to test for zero. Note that the number represented by this semi-counter is the number of tokens which are present in , i.e., in the place ready to perform dec; hence, Fig. 1(a) represents a semi-counter holding 0; note also that the number of tokens which can be accumulated in is unbounded. Indeed, the set of reachable markings for a BPP net can be countably infinite. In (b), a variant semi-counter is outlined, which holds number 2 (i.e., two tokens are ready to perform dec).    

Fig. 1.

The net representing a semi-counter in (a), and a variant in (b)

Definition 5

(Interleaving Bisimulation). Let be a BPP net. An interleaving bisimulation is a relation such that if then

• such that , such that with and ,

• such that , such that with and .

Two markings and are interleaving bisimilar, denoted by , if there exists an interleaving bisimulation R such that .    

Interleaving bisimilarity , which is defined as the union of all the interleaving bisimulations, is the largest interleaving bisimulation and also an equivalence relation.

Remark 1

(Interleaving bisimulation between two nets). The definition above covers also the case of an interleaving bisimulation between two BPP nets, say, and with , because we may consider just one single BPP net : An interleaving bisimulation is also an interleaving bisimulation on . Similar considerations hold for all the bisimulation-like definitions we propose in the following.    

Remark 2

(Comparing two marked nets). The definition above of interleaving bisimulation is defined over an unmarked BPP net, i.e., a net without the specification of an initial marking . Of course, if one desires to compare two marked nets, then it is enough to find an interleaving bisimulation (over the union of the two nets, as discussed in the previous remark), containing the pair composed of the respective initial markings. This approach is followed for all the other bisimulation-like definitions we propose.    

Example 2

Continuing Example 1 about Fig. 1, it is easy to realize that relation and and is an interleaving bisimulation.    

Causality-Based Semantics

We start with the most concrete equivalence definable over BPP nets: isomorphism equivalence.

Definition 6

(Isomorphism). Given two BPP nets and , we say that and are isomorphic via f if there exists a type-preserving bijection (i.e., a bijection such that and ), satisfying the following condition:

where f is homomorphically extended to markings (i.e., f is applied element-wise to each component of the marking: and .)

Two BPP net systems and are rooted isomorphic if the isomorphism f ensures, additionally, that .    

In order to define our approach to causality-based semantics for BPP nets, we need some auxiliary definitions, adapting those in, e.g., [3, 9, 10].

Definition 7

(Acyclic net). A BPP net is acyclic if there exists no sequence such that , for , , and for , i.e., the arcs of the net do not form any cycle.    

Definition 8

(Causal net). A BPP causal net is a marked BPP net satisfying the following conditions:

1. is acyclic;

2. (i.e., the places are not branched);

3. 

4. for all (i.e., all the arcs have weight 1).

We denote by the set , and by the set .    

Note that a BPP causal net, being a BPP net, is finite; since it is acyclic, it represents a finite computation. Note also that any reachable marking of a BPP causal net is a set, i.e., this net is safe; in fact, the initial marking is a set and, assuming by induction that a reachable marking is a set and enables t, i.e., , then also is a set, because the net is acyclic and because of the condition on the shape of the post-set of t (weights can only be 1).

Definition 9

(Partial orders of events from a causal net). From a BPP causal net , we can extract the partial order of its events , where iff there exists a sequence such that , for , , and for ; in other words, if there is a path from to .

Two partial orders and are isomorphic if there is a label-preserving, order-preserving bijection , i.e., a bijection such that and if and only if .

We also say that g is an event isomorphism between the causal nets and if it is an isomorphism between their associated partial orders of events and .    

Remark 3

As the initial marking of a causal net is fixed by its shape (according to item 3 of Definition 8), in the following, in order to make the notation lighter, we often omit the indication of the initial marking, so that the causal net is denoted by .    

Definition 10

(Moves of a causal net). Given two BPP causal nets and , we say that moves in one step to through t, denoted by , if , and ; in other words, extends by one event t.    

Definition 11

(Folding and Process). A folding from a BPP causal net into a BPP net system is a function , which is type-preserving, i.e., such that and , satisfying the following:

• and for all ;

• , i.e., ;

• , i.e., for all ;

• , i.e., for all .

A pair , where is a BPP causal net and a folding from to a BPP net system , is a process of .    

Definition 12

(Isomorphic processes). Given a BPP net system , two of its processes and are isomorphic via f if and are rooted isomorphic via bijection f (see Definition 6) and .    

Definition 13

(Moves of a process). Let be a BPP system and let , for , be two processes of . We say that moves in one step to through t, denoted by , if and .    

Causal-Net Bisimulation

We would like to define a bisimulation-based equivalence which is coarser than the branching-time semantics of isomorphism of (nondeterministic) occurrence nets (or unfoldings) [5, 7, 16] and finer than the linear-time semantics of isomorphism of causal nets [2, 17]. The proposed novel behavioral equivalence is the following causal-net bisimulation, inspired by [9].

Definition 14

(Causal-net bisimulation). Let be a BPP net. A causal-net bisimulation is a relation R, composed of triples of the form , where, for , is a process of for some , such that if then

• i)
  such that , such that

  1. ,
  2. , and ,
  3. , and ; and finally,
  4. ;
• ii)symmetrically, if moves first.

Two markings and of N are cn-bisimilar (or cn-bisimulation equivalent), denoted by , if there exists a causal-net bisimulation R containing a triple , where contains no transitions and for .    

Let us denote by . Of course, cn-bisimilarity can be seen as is a causal-net bisimulation , where is the largest causal-net bisimulation by item 4 of the following proposition.

Proposition 1

For each BPP net , the following hold:

1. the identity relation is a process of N(m)} is a causal-net bisimulation;

2. the inverse relation of a causal-net bisimulation R is a causal-net bisimulation;

3. the relational composition, up to net isomorphism, and are isomorphic processes via of two causal-net bisimulations and is a causal-net bisimulation;

4. the union of causal-net bisimulations is a causal-net bisimulation.

Proof

Trivial for 1, 2 and 4. For case 3, assume that and that . Since is a causal-net bisimulation and , we have that such that

1. ,

2. , and ,

3. , and ; and finally,

4. ;

Since and are isomorphic via f, it follows that , so that the move is derivable, too. As and is a causal-net bisimulation, for , such that

1. ,

2. , and ,

3. , and ; and finally,

4. .

Note that and are isomorphic via , where extends f in the obvious way (notably, by mapping transition t to ). As , it follows that and are isomorphic via f. Therefore, , so that the move is derivable, too. Since and are isomorphic via f, transition , can be matched by , where , so that and are isomorphic via . Hence, if and , then such that

1. ,

2. , and ,

3. , and ; and finally,

4. .

The symmetric case when moves first is analogous, hence omitted. Therefore, is a causal-net bisimulation, indeed.    

Proposition 2

For each BPP net , relation is an equivalence relation.

Proof

Standard, by exploiting Proposition 1.    

Example 3

Consider the nets in Fig. 1. Clearly the net in a) with initial marking and the net in b) with initial marking are not isomorphic; however, we can prove that they have isomorphic unfoldings [5, 7, 16]; moreover, , even if the required causal-net bisimulation contains infinitely many triples.    

Example 4

Consider the nets in Fig. 2. Of course, the initial markings and do not generate isomorphic unfoldings; however, .    

Fig. 2.

Two cn-bisimilar BPP nets

Example 5

Look at Fig. 3. Of course, , even if they generate the same causal nets. In fact, might be matched by either with or with , so that it is necessary that or ; but this is impossible, because only can perform both b and c. Moreover, as they generate different causal nets.    

Fig. 3.

Some non-cn-bisimilar BPP nets

(State-Sensitive) Fully-Concurrent Bisimulation

Behavioral equivalences for distributed systems, usually, observe only the events. Hence, causal-net bisimulation, which also observes the structure of the distributed state, may be considered too concrete an equivalence. We disagree with this view, as the structure of the distributed state is not less observable than the events this distributed system can perform. Among the equivalences not observing the state, the most prominent is fully-concurrent bisimulation (fc-bisimulation, for short) [3]. As we think that the definition in [3] is not very practical (as it assumes implicitly a universal quantification over the infinite set of all the possible extensions of the current process), we prefer to offer here an equivalent definition, by considering a universal quantification over the finite set of the net transitions only. We define also a novel, slightly stronger version, called state-sensitive fc-bisimulation equivalence, that we prove to coincide with cn-bisimilarity.

Definition 15

(Fully-concurrent bisimulation). Let be a BPP net. An fc-bisimulation is a relation R, composed of triples of the form , where, for , is a process of for some and g is an event isomorphism between and , such that if then

• i)
  such that , such that

  1. ;
  2. , and ,
  3. , and ;
  4. , and finally,
  5. ;
• ii)symmetrically, if moves first.

Two markings and of N are fc-bisimilar, denoted by , if there exists an fc-bisimulation R containing a triple , where contains no transitions, is empty and for .    

Let us denote by . Of course, , where relation

is the largest fully-concurrent bisimulation. Similarly to what done in Proposition 1, we can prove that (i) the identity relation is a process of N(m) and id is the identity event isomorphism on is an fc-bisimulation; that (ii) the inverse relation of an fc-bisimulation R is an fc-bisimulation; that (iii) the composition and are isomorphic processes via and are isomorphic processes via of two fc-bisimulations and is an fc-bisimulation; and finally, that (iv) the union of a family of fc-bisimulations is an fc-bisimulation.

Proposition 3

For each BPP net , relation is an equivalence relation.    

Example 6

In Example 5 about Fig. 3 we argued that ; however, , because, even if they do not generate the same causal net, still they generate isomorphic partial orders of events. On the contrary, because, even if they generate the same causal nets, the two markings have a different branching structure. Note that the deadlock place and the empty marking are fc-bisimilar.    

Definition 16

(State-sensitive fully-concurrent bisimulation). An fc-bisimulation R is state-sensitive if for each triple , the maximal markings have equal size, i.e., . Two markings and of N are sfc-bisimilar, denoted by , if there exists a state-sensitive fc-bisimulation R containing a triple , where contains no transitions, is empty and for .    

Of course, also the above definition is defined coinductively; as we can prove an analogous of Proposition 1, it follows that is an equivalence relation, too.

Theorem 1

(cn-bisimilarity and sfc-bisimilarity coincide). For each BPP net , if and only if .

Proof

). If , then there exists a causal-net bisimulation R such that it contains a triple , where contains no transitions and for . Relation , where id is the identity event isomorphism on , is a state-sensitive fc-bisimulation. Since contains the triple , it follows that .

) (Sketch). If , then there exists a state-sensitive fc-bisimulation containing a triple , where contains no transitions, is empty and for , with . Hence, and are isomorphic, where the isomorphism function is a suitably chosen bijection from to .1

We build the candidate causal-net bisimulation R inductively, by first including the triple ; hence, if R is a causal-net bisimulation, then .

Since and is a state-sensitive fully-concurrent bisimulation, if , then such that

1. ;

2. , and ,

3. , and ;

4. , and finally,

5. , with .

It is necessary that the isomorphism has been chosen in such a way that . As and , it is necessary that and have the same post-set size; hence, and are isomorphic and the bijection can be extended to bijection f with the pair and also with a suitably chosen bijection between the post-sets of these two transitions. Hence, we include into R also the triple . Symmetrically, if moves first.

By iterating this procedure, we add (possibly unboundedly many) triples to R. It is an easy observation to realize that R is a causal-net bisimulation.    

Remark 4

For general P/T nets, is finer than . E.g., consider the nets and . Of course, , but .    

Deadlock-Free BPP Nets and Fully-Concurrent Bisimilarity

We first define a cleaning-up operation on a BPP net N, yielding a net d(N) where all the deadlock places of N are removed. Then, we show that two markings and of N are fc-bisimilar if and only if the markings and , obtained by removing all the deadlock places in and respectively, are state-senstive fc-bisimilar in d(N).

Definition 17

(Deadlock-free BPP net). For each BPP net , we define its associated deadlock-free net d(N) as the tuple (d(S), A, d(T)) where

• , where if and only if ;

• , where and is the marking obtained from by removing all its deadlock places.

A BPP net is deadlock-free if all of its places are not a deadlock, i.e., and so .    

Formally, given a marking , we define d(m) as the marking

For instance, let us consider the FSM in Fig. 3(c). Then, , or . Of course, d(m) is a multiset on d(S).

Example 7

Let us consider the BPP net , where , and , where and . Then, its associated deadlock-free net is . Note that and .    

Proposition 4

(Fc-bisimilarity and sfc-bisimilarity coincide on deadlock-free nets). For each deadlock-free BPP net , if and only if .

Proof

. Of course, a state-sensitive fc- bisimulation is also an fc-bisimulation.

. If there are no deadlock places, an fc-bisimulation must be state sensitive. In fact, if two related markings have a different size, then, since no place is a deadlock and the BPP net transitions have singleton pre-set, they would originate different partial orders of events.    

Proposition 5

Given a BPP net and its associated deadlock-free net , two markings and of N are fc-bisimilar if and only if and in d(N) are sfc-bisimilar.

Proof

). If , then there exists an fc-bisimulation on N containing a triple , where contains no transitions, is empty and for .

Relation such that is the restriction of on the places of , for , and is such that iff is an fc-bisimulation on d(N). By Proposition 4, R is actually a state-sensitive fully-concurrent bisimulation on d(N). Note that R contains the triple such that for , and so .

). If , then there exists an sfc-bisimulation R on d(N) containing a triple , where contains no transitions, is empty and for .

Relation is a process of for some , for such that is the restriction of on the places of , for , and g is such that iff is an fc-bisimulation on N. Note that contains the triple such that, for , , , and so .    

Team Bisimulation Equivalence

In this section, we recall the main definitions and results about team bisimulation equivalence, outlined in [13]. We also include one novel, main result: causal-net bisimilarity coincides with team bisimilarity.

Additive Closure and Its Properties

Definition 18

(Additive closure). Given a BPP net and a place relation , we define a marking relation , called the additive closure of R, as the least relation induced by the following axiom and rule.

   

Note that, by definition, two markings are related by only if they have the same size; in fact, the axiom states that the empty marking is related to itself, while the rule, assuming by induction that and have the same size, ensures that and have the same size. An alternative way to define that two markings and are related by is to state that can be represented as , can be represented as and for .

It is possible to prove that if R is an equivalence relation, then its additive closure is also an equivalence relation. Moreover, if , then , i.e., the additive closure is monotonic.

Team Bisimulation on Places

Definition 19

(Team bisimulation). Let be a BPP net. A team bisimulation is a place relation such that if then for all

• such that , such that and ,

• such that , such that and .

Two places s and are team bisimilar (or team bisimulation equivalent), denoted , if there exists a team bisimulation R such that .    

Example 8

Continuing Example 1 about Fig. 1, it is easy to see that relation is a team bisimulation. In fact, is a team bisimulation pair because, to transition , can respond with , and ; symmetrically, if moves first. Also is a team bisimulation pair because, to transition , can respond with , and ; symmetrically, if moves first. Also is a team bisimulation pair: to transition , responds with , and . Similarly for the pair . Hence, relation R is a team bisimulation, indeed.

The team bisimulation R above is a very simple, finite relation, proving that and are team bisimulation equivalent. In Example 2, in order to show that and are interleaving bisimilar, we had to introduce a complex relation, with infinitely many pairs. In Example 3 we argued that , even if we did not provide any causal-net bisimulation (which would be composed of infinitely many triples).    

Example 9

Consider the nets in Fig. 2. Of course, because the finite relation is a team bisimulation. Actually, all the places are pairwise team bisimilar. In Example 4 we argued that , but the justifying causal-net bisimulation would contain infinitely many triples.    

Example 10

Consider the nets in Fig. 4. It is easy to see that is a team bisimulation. This example shows that team bisimulation is compatible with duplication of behavior and fusion of places.    

Fig. 4.

Two team bisimilar BPP nets

It is not difficult to prove [13] that (i) the identity relation is a team bisimulation; that (ii) the inverse relation of a team bisimulation R is a team bisimulation; that (iii) the relational composition of two team bisimulations and is a team bisimulation; and, finally, that (iv) the union of team bisimulations is a team bisimulation. Remember that if there exists a team bisimulation containing the pair . This means that is the union of all team bisimulations, i.e.,

Hence is also a team bisimulation, the largest such relation. Moreover, by the property listed above, relation is an equivalence relation.

Remark 5

(Complexity 1). It is well-known that the optimal algorithm for computing bisimilarity over a finite-state LTS with n states and m transitions has time complexity [19]; this very same partition refinement algorithm can be easily adapted also for team bisimilarity over BPP nets; it is enough to start from an initial partition composed of two blocks: S and , and to consider the little additional cost due to the fact that the reached markings are to be related by the additive closure of the current partition; this extra cost is related to the size of the post-set of the net transitions; if p is the size of the largest post-set of the net transitions, then the time complexity is , where m is the number of the net transitions and n is the number of the net places.    

Team Bisimilarity over Markings

Starting from team bisimulation equivalence , which has been computed over the places of an unmarked BPP net N, we can lift it over the markings of N in a distributed way: is team bisimulation equivalent to if these two markings are related by the additive closure of , i.e., if , usually denoted by .

Of course, if , then . Moreover, for any BPP net , relation is an equivalence relation.

Remark 6

(Complexity 2). Once has been computed once and for all for the given net (in time), the algorithm in [13] checks whether two markings and are team bisimulation equivalent in time, where k is the size of the markings. In fact, if is implemented as an adjacency matrix, then the complexity of checking if two markings and (represented as an array of places with multiplicities) are related by is , because the problem is essentially that of finding for each element of a matching, -related element of . Moreover, if we want to check whether other two markings of the same net are team bisimilar, we can reuse the already computed relation, so that the time complexity is again quadratic.    

Example 11

Continuing Example 8 about the semi-counters, the marking is team bisimilar to the following markings of the net in (b): , or , or , or , or , or .    

Of course, two markings and are not team bisimilar if there is no bijective, team-bisimilar-preserving mapping between them; this is the case when and have different size, or if the algorithm in [13] ends with b holding false, i.e., by singling out a place in (the residual of) which has no matching team bisimilar place in (the residual of) .

The following theorem provides a characterization of team bisimulation equivalence as a suitable bisimulation-like relation over markings. It is interesting to observe that this characterization gives a dynamic interpretation of team bisimulation equivalence, while Definition 18 gives a structural definition of team bisimulation equivalence as the additive closure of . The proof is outlined in [13].

Theorem 2

Let be a BPP net. Two markings and are team bisimulation equivalent, , if and only if and

• such that , such that , , , and , and symmetrically,

• such that , such that , , , and .

   

By the theorem above, it is clear that is an interleaving bisimulation.

Corollary 1

(Team bisimilarity is finer than interleaving bisimilarity). Let be a BPP net. If , then .    

Team Bisimilarity and Causal-Net Bisimilarity Coincide

Theorem 3

(Team bisimilarity implies cn-bisimilarity). Let be a BPP net. If , then .

Proof

Let is a process of and is a process of such that , for all . We want to prove that R is a causal-net bisimulation. First, observe that, any triple of the form , where is a BPP causal net with no transitions, and , for all , belongs to R and its existence is justified by the hypothesis . Note also that if the relation R is a causal-net bisimulation, then this triple ensures that . Now assume . In order to be a causal-net bisimulation triple, it is necessary that

i)

such that , such that

1.

,

2.

, and ,

3.

, and ; and finally,

4.

;

ii)

symmetrically, if moves first.

Let be any transition such that and let . Since by hypothesis we have that , for all , if , then there exists such that . Hence, there exists such that , , , so that, by Theorem 2, and . Therefore, it is really possible to extend the causal net to the causal net through a suitable transition t such that , as required above, and to extend and to and , respectively, in such a way that , for all because .

Summing up, for the move , we have that because , for all , as required. Symmetrically, if moves first.    

Theorem 4

(Cn-bisimilarity implies team bisimilarity). Let be a BPP net. If then .

Proof

If , then there exists a causal-net bisimulation R containing a triple , where is a BPP causal net which has no transitions and for .

Let us consider . If we prove that is a team bisimulation, then, since for each , it follows that . As , we also have that .

Let us consider a pair . Hence, there exist a triple and a place such that and . If moves, e.g., , then , where . Since R is a causal-net bisimulation, such that

1. ,

2. , and ,

3. , and ; and finally,

4. ;

Note that t is such that , and so . This means that , where ; in other words, . Note also that extends by mapping t to and, similarly, extends by mapping t to ; in this way, and . Since , it follows that the set is a subset of , so that .

Summing up, for , if , then such that ; symmetrically, if moves first. Therefore, is a team bisimulation.    

Corollary 2

(Team bisimilarity and cn-bisimilarity coincide). Let be a BPP net. Then, if and only if .

Proof

By Theorems 3 and 4, we get the thesis.    

Corollary 3

(Team bisimilarity and sfc-bisimilarity coincide). Let be a BPP net. Then, if and only if .

Proof

By Corollary 2 and Theorem 1, we get the thesis.    

Therefore, our characterization of cn-bisimilarity and sfc-bisimilarity, which are, in our opinion, the intuitively correct (strong) causal semantics for BPP nets, is quite appealing because it is based on the very simple technical definition of team bisimulation on the places of the unmarked net, and, moreover, offers a very efficient algorithm to check if two markings are cn-bisimilar (see Remarks 5 and 6).

H-Team Bisimulation

We provide the definition of h-bisimulation on places for unmarked BPP nets, adapting the definition of team bisimulation on places (cf. Definition 19). In this definition, the empty marking is considered as an additional place, so that the relation is defined not on S, rather on ; therefore, the symbols and that occur in the definition below can only denote either the empty marking or a single place.

Definition 20

(H-team bisimulation). Let be a BPP net. An h-team bisimulation is a place relation such that if then for all

• such that , such that and ,

• such that , such that and .

and are h-team bisimilar (or h-team bisimulation equivalent), denoted , if there exists an h-team bisimulation R such that .    

Since a team bisimulation is also an h-team bisimulation, we have that team bisimilarity implies h-team bisimilarity . This implication is strict as illustrated in the following example.

Example 12

Consider the nets in Fig. 3. It is not difficult to realize that and are h-team bisimilar because is an h-team bisimulation. In fact, can reach by performing a, and can reply by reaching the empty marking , and . In Example 6 we argued that and in fact we will prove that h-team bisimilarity coincide with fc-bisimilarity.    

Remark 7

(Additive closure properties). Note that the additive closure of an h-team bisimulation R does not ensure that if two markings are related by , then they must have the same size. For instance, considering the above relation , we have that, e.g., , because is the identity for multiset union. However, the other properties of the additive closure described in Sect. 4.1 hold also for these more general place relations.    

It is not difficult to prove that, for any BPP net , the following hold:

1. The identity relation is an h-team bisimulation;

2. the inverse relation of an h-team bisimulation R is an h-team bisimulation;

3. the relational composition of two h-team bisimulations and is an h-team bisimulation;

4. the union of h-team bisimulations is an h-team bisimulation.

Relation is the union of all h-team bisimulations, i.e.,

Hence, is also an h-team bisimulation, the largest such relation. Moreover, by the observations above, relation is an equivalence relation.

Starting from h-team bisimulation equivalence , which has been computed over the places (and the empty marking) of an unmarked BPP net N, we can lift it over the markings of N in a distributed way: is h-team bisimulation equivalent to if these two markings are related by the additive closure of , i.e., if , usually denoted by . Since is an equivalence relation, then also relation is an equivalence relation.

Remark 8

(Complexity 3). Computing is not more difficult than computing . The partition refinement algorithm in [19] can be adapted also in this case. It is enough to consider the empty marking as an additional, special place which is h-team bisimilar to each deadlock place. Hence, the initial partition considers two sets: one composed of all the deadlock places and , the other one with all the non-deadlock places. Therefore, the time complexity is also in this case , where m is the number of the net transitions, n is the number of the net places and p the size of the largest post-set of the net transitions.

Once has been computed once and for all for the given net, the complexity of checking whether two markings and are h-team bisimulation equivalent, according to the algorithm in [13], is , where k is the size of the largest marking, since the problem is essentially that of finding for each element (not -related to ) of a matching, -related element of (and then checking that all the remaining elements of and are -related to ).    

H-Team Bisimilarity and Fully-Concurrent Bisimilarity Coincide

In this section, we first show that h-team bisimilarity over a BPP net N coincides with team-bisimilarity over its associated deadlock-free net d(N). A consequence of this result is that h-team bisimilarity coincides with fc-bisimilarity on BPP nets.

Proposition 6

Given a BPP net and its associated deadlock-free net , two markings and of N are h-team bisimilar if and only if and in d(N) are team bisimilar.

Proof

). If , then there exists an h-team bisimulation on N such that . If we take relation , then it is easy to see that is a team bisimulation on d(N), so that , hence .

). If , then there exists a team bisimulation on d(N) such that . Now, take relation , where the set is . It is easy to observe that is an h-team bisimulation on N, so that , hence .    

Theorem 5

(Fully concurrent bisimilarity and h-team bisimilarity coincide). Given a BPP net , if and only if .

Proof

By Proposition 5, in N if and only if in the associated deadlock-free net d(N). By Corollary 3, iff in d(N). By Proposition 6, in d(N) if and only if in N.    

Conclusion

Team bisimilarity is the most natural, intuitive and simple extension of LTS bisimilarity to BPP nets; it also has a very low complexity, actually lower than any other equivalence for BPP nets. Moreover, it coincides with causal-net bisimilarity and state-sensitive fully-concurrent bisimilarity, hence it corresponds to the intuitively correct bisimulation-based causal semantics for BPP nets. Moreover, it coincides also with structure-preserving bisimilarity, because our causal-net bisimilarity is rather similar to its process-oriented characterization in [9]. From a technical point of view, team bisimulation seems a sort of egg of Columbus: a simple (actually, a bit surprising in its simplicity) solution for a presumedly hard problem. This paper is not only an addition to [13], where team bisimilarity was originally introduced, but also an extension to a team-style characterization of fully-concurrent bisimilarity, namely h-team bisimilarity.

We think that state-sensitive fc-bisimilarity (hence, also team bisimilarity) is more accurate than fc-bisimilarity (hence, h-team bisimilarity) because it is resource-aware, i.e., it is sensitive to the number of resources that are present in the net. This more concrete equivalence is justified in, e.g., the area of information flow security [14].

Our complexity results for fc-bisimilarity in terms of the equivalent h-team bisimilarity (cf. Remark 8), seem comparable with those in [6], where, by using an event structure [20] semantics, Fröschle et al. show that history-preserving bisimilarity (hpb, for short) is decidable for the BPP process algebra with guarded summation in time, where n is the size of the involved BPP terms. However, this value n is strictly larger than the size of the corresponding BPP net. In fact, in [6] the size of a BPP term p is defined as “the total number of occurrences of symbols (including parentheses)”, where p is defined by means of a concrete syntax. E.g., has size 11, while the net semantics for p generates one place and one transition (and 2 tokens). For a comparison of team bisimilarity with other equivalences for BPP, we refer you to [13].

In [13] we presented a modal logic characterization of and also a finite axiomatization for the process algebra BPP (with guarded sum and guarded recursion). As a future work, we plan to extend these results to , hence equipping fc-bisimilarity (and hpb) with a logic characterization and an axiomatic one for the process algebra BPP.