Branching Bisimilarity with Explicit Divergence
Abstract
We consider the relational characterisation of branching bisimilarity with explicit divergence. We prove that it is an equivalence and that it coincides with the original definition of branching bisimilarity with explicit divergence in terms of coloured traces. We also establish a correspondence with several variants of an actionbased modal logic with until and divergence modalities.
1 Introduction
Branching bisimilarity was proposed in [6]. It is a behavioural equivalence on processes that is compatible with a notion of abstraction from internal activity, while at the same preserving the branching structure of processes in a strong sense. We refer the reader to [6], in particular to Section 10 therein, for ample motivation of the relevance of branching bisimilarity.
Branching bisimilarity abstracts to a large extent from divergence (i.e., infinite internal activity). For instance, it identifies a process, say , that may perform some internal activity after which it returns to its initial state (i.e., has a loop) with a process, say , that admits the same behaviour as except that it cannot perform the internal activity leading to the initial state (i.e., is without the loop). This means that branching bisimilarity is not compatible with any temporal logic featuring an eventually modality: for any desired state that will eventually reach, the mentioned internal activity of may be performed continuously, and thus prevent from reaching this desired state.
The notion of branching bisimilarity with explicit divergence (BB), also proposed in [6], is a suitable refinement of branching bisimilarity that is compatible with the wellknown branchingtime temporal logic CTL without the nexttime operator (which is known to be incompatible with abstraction from internal activity). In fact, in [5] we have proved that it is the coarsest semantic equivalence on labelled transition systems with silent moves that is a congruence for parallel composition (as found in process algebras like CCS, CSP or ACP) and only equates processes satisfying the same CTL formulas. It is also the finest equivalence in the linear time – branching time spectrum of [4].
There are several ways to characterise a behavioural equivalence. The original definition of BB, in terms of coloured traces, stems from [6]. In [4], BB is defined in terms of a modal and a relational characterisation, which are claimed to coincide with each other and with the original notion from [6]. Of these three definitions of BB, the relational characterisation from [4] is the most concise one, in the sense that it requires the least amount of auxiliary concepts. Moreover, this definition is most in the style of the standard definitions of other kinds of bisimulation, found elsewhere in the literature. For these reasons, it is tempting to take it as standard definition.
Although it is not hard to establish that the modal characterisation from [4] is correct, in the sense that it defines an equivalence that coincides with BB of [6], it is not at all trivial to establish that the same holds for the relational characterisation from [4]. If fact, it is nontrivial that this relation is an equivalence, and that it satisfies the socalled stuttering property. Once these properties have been established, it follows that the notion coincides with BB of [6].
In the remainder of this paper, we shall first, in Section 2, briefly recapitulate the relational, colouredtrace, and modal characterisations of branching bisimilarity. Then, in Section 3, we shall discuss the condition proposed in [4] that can be added to the relational characterisation in order to make it divergence sensitive; we shall then also discuss several variants on this condition. In Section 4 we establish that the relational characterisation of BB all coincide, that they are equivalences and that they enjoy the stuttering property. In Section 5 we show that the relational characterisations of BB coincide with the original definition of BB in terms of coloured traces. Finally, in Section 6, we shall establish agreement between the relational characterisation from [4], the modal characterisation from [4], and an alternative modal characterisation obtained by adding the divergence modality of [4] to the HennessyMilner logic with until proposed in [2].
2 Branching bisimilarity
We presuppose a set of actions with a special element , and we presuppose a labelled transition system with labels from , i.e., is a set of states and is a transition relation on . Let and . We write for and we abbreviate the statement ‘ or ( and )’ by . We denote by the transitive closure of the binary relation , and by its reflexivetransitive closure. A path from a state is an alternating sequence of states and actions, such that and for . A process is given by a state in a labelled transition system, and encompasses all the states and transitions reachable from .
Relational characterisation
The definition of branching bisimilarity that is most widely used has a coinductive flavour. It defines when a binary relation on states preserves the behaviour of the associated processes. It then declares two states to be equivalent if there exists such a relation relating them. We shall refer to this kind of characterisation as a relational characterisation of branching bisimilarity.
Definition 2.1
A symmetric binary relation on is a branching bisimulation if it satisfies the following condition for all and :

if and for some state , then there exist states and such that , and .
We write if there exists a branching bisimulation such that . The relation on states is referred to as (the relational characterisation of) branching bisimilarity.
The relational characterisation of branching bisimilarity presented above is from [4]. As shown in [1, 4, 6], it yields the same concept of branching bisimilarity as the original definition in [6]. The technical advantage of the above definition over the original definition is that the defined notion of branching bisimulation is compositional: the composition of two branching bisimulations is again a branching bisimulation. Basten [1] gives an example showing that the condition used in the original definition of of [6] fails to be compositional in this sense, and thus argued that establishing transitivity directly for the original definition is not straightforward.
Colouredtrace characterisation
To substantiate their claim that branching bisimilarity indeed preserves the branching structure of processes, van Glabbeek and Weijland present in [6] an alternative characterisation of the notion in terms of coloured traces. Below we repeat this characterisation.
Definition 2.2
A colouring is an equivalence on . Given a colouring and a state , the colour of is the equivalence class containing .
For a path from , let be the alternating sequence of colours and actions obtained from by contracting all subsequences to . The sequence is called a coloured trace of . A colouring is consistent if two states of the same colour always have the same coloured traces.
We write if there exists a consistent colouring with .
In [6] it is proved that coincides with the relational characterisation of branching bisimilarity.
Modal characterisation
A modal characterisation of a behavioural equivalence is a modal logic such that two processes are equivalent iff they satisfy the same formulas of the logic. The modal logic thus corresponding to a behavioural equivalence then allows one, for any two inequivalent processes, to formally express a behavioural property that distinguishes them. Whereas colourings or bisimulations are good tools to show that two processes are equivalent, modal formulas are better for proving inequivalence. The first modal characterisation of a behavioural equivalence is due to Hennessy and Milner [7]. They provided a modal characterisation of (strong) bisimilarity on imagefinite labelled transition systems, using a modal logic that is nowadays referred to as the HennessyMilner Logic. The modal characterisations of branching bisimilarity presented below are adaptations of the HennessyMilner Logic.
The class of formulas of the modal logic for branching bisimilarity proposed in [4] is generated by the following grammar:
(1) 
In case the cardinality of the set of states of our labelled transition system is less than some infinite cardinal , we may require that in conjunctions, thus obtaining a set of formulas rather than a proper class. We shall use the following standard abbreviations: , and .
We define when a formula is valid in a state (notation: ) inductively as follows:

iff ;

iff for all ;

iff there exist states and such that , and .
Validity induces an equivalence on states: we define by
In [4] it was shown that coincides with , that is, branching bisimilarity is characterised by the modal logic above.
Clause (iii) in the definition of validity appears to be rather liberal. More stringent alternatives are obtained by using or instead of , with the following definitions:

iff either and , or there exists a sequence of states () such that , for all and .

iff there exists states () such that , for all and .
The modality stems from De Nicola & Vaandrager [2]. There it was shown, for labelled transition systems with bounded nondeterminism, that branching bisimilarity, , is characterised by the logic with negation, binary conjunction and this until modality. The modality is a common strengthening of and the justbefore modality above; it was first considered in [4].
To be able to compare the expressiveness of modal logics, the following definitions are proposed by Laroussinie, Pinchinat & Schnoebelen [8].
Definition 2.3
Two modal formulas and that are interpreted on states of labelled transition systems are equivalent, written , if for all states in all labelled transition systems. Two modal logics are equally expressive if for every formula in the one there is an equivalent formula in the other.
As remarked in [4], the modalities and are equally expressive, since
Note that the modality can be expressed in terms of :
3 Relational characterisations of BB
The notion branching bisimilarity discussed in the previous section abstracts from divergence (i.e, infinite internal activity). In the remainder of this paper, we discuss a refinement of the notion of branching bisimulation equivalence that takes divergence into account. In this section we present several conditions that can be added to the notion of branching bisimulation in order to make it divergence sensitive. The induced notions of branching bisimilarity with explicit divergence will all turn out to be equivalent.
Definition 3.1
[4] A symmetric binary relation on is a branching bisimulation with explicit divergence if it is a branching bisimulation (i.e., it satisfies condition (T) of Definition 2.1) and in addition satisfies the following condition for all and :

if and there is an infinite sequence of states such that , and for all , then there exists an infinite sequence of states such that , for all , and for all .
We write if there exists a branching bisimulation with explicit divergence such that .
Figure 1 illustrates condition (D). In [4] it was claimed that the notion defined above coincides with branching bisimilarity with explicit divergence as defined earlier in [6]. In this paper we will substantiate this claim. On the way to this end, we need to show that is an equivalence and has the socalled stuttering property.
The difficulty in proving that is an equivalence is in establishing transitivity. Basten’s proof in [1] that (i.e., branching bisimilarity without explicit divergence) is transitive, is obtained as an immediate consequence of the fact that whenever two binary relations and satisfy (T), then so does their composition (see Lemma 4.3 below). The condition (D) fails to be compositional, as we show in the following example.
Example 3.1
Consider the labelled transition system depicted on the lefthand side of Figure 2 together with the branching bisimulations with explicit divergence
The composition on the relevant fragment is depicted on the righthand side of Figure 2. Note that gives rise to a divergence of which every state is related by to . However, since and are not related according to , there is no divergence from of which every state is related to every state on the divergence from . We conclude that does not satisfy the condition (D).
Our proof that is an equivalence proceeds along the same lines as Basten’s proof in [1] that is an equivalence: we replace (D) by an alternative divergence condition that is compositional, prove that the resulting notion of bisimilarity is an equivalence, and then establish that it coincides with . In the remainder of this section, we shall arrive at our compositional alternative for (D) through a series of adaptations of (D).
First, we observe that (D) has a technically convenient reformulation: instead of requiring the existence of a divergence from all the states of which enjoy certain properties, it suffices to require that there exists a state reachable from by a single transition with these properties. Formally, the reformulation of (D) is:

if and there is an infinite sequence of states such that , and for all , then there exists a state such that and for all .
Figure 3 illustrates condition (D). If a binary relation satisfies (D), then the divergence from required by (D) can be inductively constructed. (We omit the inductive construction here; the proof of Proposition 3.1 below contains a very similar inductive construction.)
For our next adaptation we observe that (D) has some redundancy. Note that it requires to be related to every state on the divergence from . However, the universal quantification in the conclusion can be relaxed to an existential quantification: it suffices to require that has an immediate successor that is related to some state on the divergence from . The requirement can be expressed as follows:

if and there is an infinite sequence of states such that , and for all , then there exists a state such that and for some .
Condition (D) appears in the definition of divergencesensitive stuttering simulation of Nejati [9]. It is illustrated in Figure 4. We write if there exists a symmetric binary relation satisfying (T) and (D) such that . Note that every relation satisfying (D) also satisfies (D), so it follows that .
The following example illustrates that condition (D) is still not compositional, not even if the composed relations satisfy (T).
Example 3.2
Consider the labelled transition system depicted on the lefthand side of Figure 5 together with two binary relations satisfying (T) and (D):
Note that, since is not related to , the divergence need not be simulated by in such a way that is related to either or .
Now consider the composition . Both and are related to , whereas the state is not related to nor to . We conclude that does not satisfy (D).
The culprit in the preceding example appears to be the fact that (D) only considers divergences from of which every state is related to . Our second alternative omits this restriction. It considers every divergence from and requires that it is simulated by .

if and there is an infinite sequence of states such that and for all , then there exists a state such that and for some .
Figure 6 illustrates condition (D). In contrast to the preceding divergence conditions, it does have the property that if two relations both satisfy it, then so does their relational composition. However, to facilitate a direct proof of this property, it is technically convenient to reformulate condition (D) such that it requires a divergence from rather than just one step:

if and there is an infinite sequence of states such that and for all , then there exist an infinite sequence of states and a mapping such that , and for all .
Proof.
The implication from right to left is trivial. For the implication from left to right, suppose that satisfies (D) and that , and consider an infinite sequence of states such that and for all . We construct an infinite sequence of states and a mapping such that , and for all .
The infinite sequence and the mapping can be defined simultaneously by induction on :

We define and ; it then clearly holds that .

Suppose that the sequence and the mapping have been defined up to . Then, in particular, . Since is an infinite sequence such that for all , by (D) there exists such that and for some . We define and .
We write if there exists a symmetric binary relation satisfying (T) and (D) such that . Note that (D) is a weaker requirement than (D), and hence, by Proposition 3.1, than (D). It follows that . Also note that (D) and (D) on the one hand and (D) and (D) on the other hand are incomparable.
Using that (D) is compositional, it will be straightforward to establish that is an equivalence. Then, it remains to establish that and coincide. We shall prove that is included in by establishing that is a branching bisimulation with explicit divergence; that is an equivalence is crucial in the proof of this property. Instead of proving the converse inclusion directly, we obtain a stronger result by establishing that a notion of bisimilarity defined using a weaker divergence condition and therefore including , is included in . The weakest divergence condition we encountered so far is (D). It is, however, possible to further weaken (D): instead of requiring that is an immediate successor, it is enough require that can be reached from by one or more transitions. Formally,

if and there is an infinite sequence of states such that , and for all , then there exists a state such that and for some .
Figure 8 illustrates condition (D). We write if there exists a symmetric binary relation satisfying (T) and (D) such that . Clearly, , and hence also and .
In the next section we shall also prove that . A crucial tool in our proof of this inclusion will be the notion of stuttering closure of a binary relation on states. The stuttering closure of enjoys the socalled stuttering property: if from state a state can be reached through a sequence of transitions, and both and are related to the same state , then all intermediate states between and are related to too. We shall prove a lemma to the effect that if a binary relation on states satisfies (T) and (D), then its stuttering closure satisfies (T) and (D), and use it to establish the inclusion . An easy corollary of the lemma is that has the stuttering property. Here our proof also has a similarity with Basten’s proof in [1]; in his proof that the notions of branching bisimilarity induced by (T) and by the original condition used in [6] coincide, establishing the stuttering property is a crucial step.
Figure 9 shows some inclusions between the different versions of branching bisimilarity with explicit divergence. (Note that we never defined and , as these would be the same as and , respectively.) The solid arrows indicate inclusions that have already been argued for above; the dashed arrows indicate inclusions that will be established below.
Remark 3.1
We shall establish in the next section that . Note that, once we have this, we can replace the second condition of Definition 3.1 by any interpolant of (D) and (D), i.e., any condition that is implied by (D) and implies (D), and end up with the same equivalence. For instance, we could replace it by condition (D), or by the condition of Gerth, Kuiper, Peled & Penczek [3]:

if and there is an infinite sequence of states such that , and for all , then there exists a state such that and for some .
Similarly, we will prove that , and so we can replace the second condition of Definition 3.1 by an interpolant of (D) and (D). For instance, the condition

if and there is an infinite sequence of states such that and for all , then there exists a state such that and for some
is a convenient interpolant of (D) and (D) to use when showing that two states are branching bisimulation equivalent with explicit divergence.
4 Bb is an equivalence with the stuttering property
Our goal is now to establish that the relational characterisations of branching bisimilarity with explicit divergence introduced in the previous section all coincide, that they are equivalences and that they enjoy the stuttering property. To this end, we first show that is an equivalence relation; condition (D) will enable a direct proof of this fact. Using that is an equivalence, we obtain . Then, we define the notion of stuttering closure and use it to establish . Together with the observation made above, the cycle of inclusions yields that the relations , and coincide. It then follows that is an equivalence. We have not been able to find a less roundabout way to obtain this result. The intermediate results needed for the equivalence proof also yields that has the stuttering property.
4.1 is an equivalence
The proofs below are rather straightforward. Nevertheless, the proof strategy employed for Lemmas 4.1 and 4.3 would fail for , and . It is for this reason that we present all detail.
Lemma 4.1
Proof.
Let .

Suppose that satisfies (D) for all . To prove that satisfies (D), suppose that and that there is an infinite sequence of states such that and . From it follows that for some . By (D) there exist an infinite sequence of states and a mapping such that , and for all , and from the latter it follows that for all .
Lemma 4.2
Let be a binary relation that satisfies (T). If and , then there is a state such that and .
Proof.
Let be states such that . By (T) and a straightforward induction on there exist states such that and for all .
Lemma 4.3
Proof.
Let .

To prove that satisfies (T), suppose and . Then there exists a state such that and . Since satisfies (T), there exist states and such that , and . By Lemma 4.2 there is a state such that and . We now distinguish two cases:

Suppose that and . Then , from and it follows that , and from and it follows that .

Suppose that . Then there exist states and such that , and . So, , from and
