Branching Bisimilarity with Explicit Divergence

Transcription

1 Branching Bisimilarity with Explicit Divergence Ro van Glaeek National ICT Australia, Sydney, Australia School of Computer Science and Engineering, University of New South Wales, Sydney, Australia Bas Luttik Department of Mathematics and Computer Science, Technische Universiteit Eindhoven, The Netherlands CWI, The Netherlands Nikola Trčka Department of Mathematics and Computer Science, Technische Universiteit Eindhoven, The Netherlands Astract. We consider the relational characterisation of ranching isimilarity with explicit divergence. We prove that it is an equivalence and that it coincides with the original definition of ranching isimilarity with explicit divergence in terms of coloured traces. We also estalish a correspondence with several variants of an action-ased modal logic with until- and divergence modalities. 1. Introduction Branching isimilarity was proposed in [6]. It is a ehavioural equivalence on processes that is compatile with a notion of astraction from internal activity, while at the same preserving the ranching 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 ranching isimilarity. Branching isimilarity astracts to a large extent from divergence (i.e., infinite internal activity). For instance, it identifies a process, say P, that may perform some internal activity after which it returns to its initial state (i.e., P has a -loop) with a process, say P, that admits the same ehaviour as P except that it cannot perform the internal activity leading to the initial state (i.e., P is P without the -loop). This means that ranching isimilarity is not compatile with any temporal logic featuring an eventually modality: for any desired state that P will eventually reach, the mentioned internal activity of P may e performed continuously, and thus prevent P from reaching this desired state. The notion of ranching isimilarity with explicit divergence (BB ), also proposed in [6], is a suitale refinement of ranching isimilarity that is compatile with the well-known ranching-time temporal logic CTL without the nexttime operator X (which is known to e incompatile with astraction from internal activity). In fact, in [5] we have proved that it is the coarsest semantic equivalence on laelled transition systems with silent moves that is a congruence for parallel composition (as found in process algeras like CCS, CSP or ACP) and only equates processes satisfying the same CTL X formulas. It is also the finest equivalence in the linear time ranching time spectrum of [4]. There are several ways to characterise a ehavioural 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].

2 2 R.J. van Glaeek, B. Luttik, N. Trčka / Branching isimilarity with explicit divergence 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 isimulation, found elsewhere in the literature. For these reasons, it is tempting to take it as standard definition. Although it is not hard to estalish 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 estalish that the same holds for the relational characterisation from [4]. If fact, it is non-trivial that this relation is an equivalence, and that it satisfies the so-called stuttering property. Once these properties have een estalished, it follows that the notion coincides with BB of [6]. In the remainder of this paper, we shall first, in Section 2, riefly recapitulate the relational, colouredtrace, and modal characterisations of ranching isimilarity. Then, in Section 3, we shall discuss the condition proposed in [4] that can e 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 estalish 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 estalish agreement etween the relational characterisation from [4], the modal characterisation from [4], and an alternative modal characterisation otained y adding the divergence modality of [4] to the Hennessy-Milner logic with until proposed in [2]. 2. Branching isimilarity We presuppose a set A of actions with a special element A, and we presuppose a laelled transition system (S, ) with laels from A, i.e., S is a set of states and S A S is a transition relation on S. Let s, s S and a A. We write s a s for (s, a, s ) and we areviate the statement s a s or (a = and s = s ) y s (a) s. We denote y + the transitive closure of the inary relation, and y its reflexive-transitive closure. A path from a state s is an alternating sequence a s 0, a 1, s 1, a 2, s 2,...,a n, s n of states and actions, such that s = s 0 and s k k 1 sk for k = 1,...,n. A process is given y a state s in a laelled transition system, and encompasses all the states and transitions reachale from s. Relational characterisation The definition of ranching isimilarity that is most widely used has a co-inductive flavour. It defines when a inary relation on states preserves the ehaviour of the associated processes. It then declares two states to e equivalent if there exists such a relation relating them. We shall refer to this kind of characterisation as a relational characterisation of ranching isimilarity. Definition 2.1. A symmetric inary relation R on S is a ranching isimulation if it satisfies the following condition for all s, t S and a A: (T) if s R t and s a s for some state s, then there exist states t and t such that t t (a) t, s R t and s R t. We write s t if there exists a ranching isimulation R such that s R t. The relation on states is referred to as (the relational characterisation of) ranching isimilarity.

3 R.J. van Glaeek, B. Luttik, N. Trčka / Branching isimilarity with explicit divergence 3 The relational characterisation of ranching isimilarity presented aove is from [4]. As shown in [1, 4, 6], it yields the same concept of ranching isimilarity as the original definition in [6]. The technical advantage of the aove definition over the original definition is that the defined notion of ranching isimulation is compositional: the composition of two ranching isimulations is again a ranching isimulation. Basten [1] gives an example showing that the condition used in the original definition of of [6] fails to e compositional in this sense, and thus argued that estalishing transitivity directly for the original definition is not straightforward. Coloured-trace characterisation To sustantiate their claim that ranching isimilarity indeed preserves the ranching structure of processes, van Glaeek 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 S. Given a colouring C and a state s S, the colour C(s) of s is the equivalence class containing s. For π = s 0, a 1, s 1,...,a n, s n a path from s, let C(π) e the alternating sequence of colours and actions otained from C(s 0 ), a 1, C(s 1 ),...,a n, C(s n ) y contracting all susequences C,, C,,...,, C to C. The sequence C(π) is called a C-coloured trace of s. A colouring C is consistent if two states of the same colour always have the same C-coloured traces. We write s c t if there exists a consistent colouring C with C(s) = C(t). In [6] it is proved that c coincides with the relational characterisation of ranching isimilarity. Modal characterisation A modal characterisation of a ehavioural 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 ehavioural equivalence then allows one, for any two inequivalent processes, to formally express a ehavioural property that distinguishes them. Whereas colourings or isimulations are good tools to show that two processes are equivalent, modal formulas are etter for proving inequivalence. The first modal characterisation of a ehavioural equivalence is due to Hennessy and Milner [7]. They provided a modal characterisation of (strong) isimilarity on image-finite laelled transition systems, using a modal logic that is nowadays referred to as the Hennessy-Milner Logic. The modal characterisations of ranching isimilarity presented elow are adaptations of the Hennessy-Milner Logic. The class of formulas Φ j of the modal logic for ranching isimilarity proposed in [4] is generated y the following grammar: ϕ ::= ϕ Φ ϕ a ϕ (a A, ϕ Φ j and Φ Φ j ). (1) In case the cardinality S of the set of states of our laelled transition system is less than some infinite cardinal κ, we may require that Φ < κ in conjunctions, thus otaining a set of formulas rather than a proper class. We shall use the following standard areviations: =, = and Φ = { ϕ ϕ Φ}. We define when a formula ϕ is valid in a state s (notation: s = ϕ) inductively as follows: (i) s = ϕ iff s = ϕ; (ii) s = Φ iff s = ϕ for all ϕ Φ;

4 4 R.J. van Glaeek, B. Luttik, N. Trčka / Branching isimilarity with explicit divergence (iii) s = ϕ a ψ iff there exist states s and s such that s s Validity induces an equivalence on states: we define S S y s t iff ϕ Φ j. s = ϕ t = ϕ. (a) s, s = ϕ and s = ψ. In [4] it was shown that coincides with, that is, ranching isimilarity is characterised y the modal logic aove. Clause (iii) in the definition of validity appears to e rather lieral. More stringent alternatives are otained y using ϕ â ψ or ϕ a ψ instead of ϕ a ψ, with the following definitions: (iii ) s = ϕ â ψ iff either a = and s = ψ, or there exists a sequence of states s 0,...,s n, s n+1 a (n 0) such that s = s 0 s n s n+1, s i = ϕ for all i = 0,...,n and s n+1 = ψ. (iii (a) ) s = ϕ a ψ iff there exists states s 0,...,s n, s n+1 (n 0) such that s = s 0 s n s n+1, s i = ϕ for all i = 0,...,n and s n+1 = ψ. The modality â stems from De Nicola & Vaandrager [2]. There it was shown, for laelled transition systems with ounded nondeterminism, that ranching isimilarity,, is characterised y the logic with negation, inary conjunction and this until modality. The modality a is a common strengthening of â and the just-efore modality a aove; it was first considered in [4]. To e ale to compare the expressiveness of modal logics, the following definitions are proposed y Laroussinie, Pinchinat & Schnoeelen [8]. Definition 2.3. Two modal formulas ϕ and ψ that are interpreted on states of laelled transition systems are equivalent, written ϕ ψ, if s = ϕ s = ψ for all states s in all laelled 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 â and a are equally expressive, since ϕ ˆ ψ ψ ϕ ψ, ϕ ψ ϕ ϕ ˆ ψ and ϕ a ψ ϕ â ψ for all a. Note that the modality a can e expressed in terms of a : ϕ a ψ (ϕ a ψ). Laroussinie, Pinchinat & Schnoeelen estalished in [8] that the modal logic with negation, inary conjunction and a from [4] and the logic with negation, inary conjunction and â from [2] are equally expressive.

5 R.J. van Glaeek, B. Luttik, N. Trčka / Branching isimilarity with explicit divergence 5 3. Relational characterisations of BB The notion ranching isimilarity discussed in the previous section astracts from divergence (i.e, infinite internal activity). In the remainder of this paper, we discuss a refinement of the notion of ranching isimulation equivalence that takes divergence into account. In this section we present several conditions that can e added to the notion of ranching isimulation in order to make it divergence sensitive. The induced notions of ranching isimilarity with explicit divergence will all turn out to e equivalent. Definition 3.1. [4] A symmetric inary relation R on S is a ranching isimulation with explicit divergence if it is a ranching isimulation (i.e., it satisfies condition (T) of Definition 2.1) and in addition satisfies the following condition for all s, t S and a A: (D) if s R t and there is an infinite sequence of states (s k ) k ω such that s = s 0, s k s k+1 and s k R t for all k ω, then there exists an infinite sequence of states (t l ) l ω such that t = t 0, t l+1 for all l ω, and s k R t l for all k, l ω. t l We write s t if there exists a ranching isimulation with explicit divergence R such that s R t. s = s 0 s 1 s k t = t 0 t 1 t l Figure 1. Condition (D). Figure 1 illustrates condition (D). In [4] it was claimed that the notion defined aove coincides with ranching isimilarity with explicit divergence as defined earlier in [6]. In this paper we will sustantiate this claim. On the way to this end, we need to show that is an equivalence and has the so-called stuttering property. The difficulty in proving that is an equivalence is in estalishing transitivity. Basten s proof in [1] that (i.e., ranching isimilarity without explicit divergence) is transitive, is otained as an immediate consequence of the fact that whenever two inary relations R 1 and R 2 satisfy (T), then so does their composition R 1 ; R 2 (see Lemma 4.3 elow). The condition (D) fails to e compositional, as we show in the following example. Example 3.1. Consider the laelled transition system depicted on the left-hand side of Figure 2 together with the ranching isimulations with explicit divergence R 1 = {(s 0, t 0 ), (t 0, s 0 ), (s 1, t 1 ), (t 1, s 1 ), (s 2, t 2 ), (t 2, s 2 ), (s 1, t 2 ), (t 2, s 1 ), (s 2, t 1 ), (t 1, s 2 )} R 2 = {(t 0, u 0 ), (u 0, t 0 ), (t 1, u 1 ), (u 1, t 1 ), (t 2, u 2 ), (u 2, t 2 ), (t 0, u 1 ), (u 1, t 0 ), (t 1, u 0 ), (u 0, t 1 )}. The composition R = R 1 ; R 2 on the relevant fragment is depicted on the right-hand side of Figure 2. Note that s 0 gives rise to a divergence of which every state is related y R to u 0. However, since s 0 and and

6 6 R.J. van Glaeek, B. Luttik, N. Trčka / Branching isimilarity with explicit divergence s 0 s 1 s 2 s 0 s 1 s 2 t 0 t 1 t 2 u 0 u 1 u 2 u 0 u 1 u 2 Figure 2. The composition of ranching isimulations with explicit divergence is not a ranching isimulation with explicit divergence. u 2 are not related according to R, there is no divergence from u 0 of which every state is related to every state on the divergence from s 0. We conclude that R 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) y an alternative divergence condition that is compositional, prove that the resulting notion of isimilarity is an equivalence, and then estalish 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 oserve that (D) has a technically convenient reformulation: instead of requiring the existence of a divergence from t all the states of which enjoy certain properties, it suffices to require that there exists a state reachale from t y a single -transition with these properties. Formally, the reformulation of (D) is: (D 0 ) if s R t and there is an infinite sequence of states (s k ) k ω such that s = s 0, s k s k+1 and s k R t for all k ω, then there exists a state t such that t t and s k R t for all k ω. s = s 0 s 1 s k t t Figure 3. Condition (D 0 ). Figure 3 illustrates condition (D 0 ). If a inary relation satisfies (D 0 ), then the divergence from t required y (D) can e inductively constructed. (We omit the inductive construction here; the proof of Proposition 3.1 elow contains a very similar inductive construction.) For our next adaptation we oserve that (D 0 ) has some redundancy. Note that it requires t to e related to every state on the divergence from s. However, the universal quantification in the conclusion can e relaxed to an existential quantification: it suffices to require that t has an immediate -successor that is related to some state on the divergence from s. The requirement can e expressed as follows:

7 R.J. van Glaeek, B. Luttik, N. Trčka / Branching isimilarity with explicit divergence 7 (D 1 ) if s R t and there is an infinite sequence of states (s k ) k ω such that s = s 0, s k s k+1 and s k R t for all k ω, then there exists a state t such that t t and s k R t for some k ω. s = s 0 s 1 s k t t Figure 4. Condition (D 1 ). Condition (D 1 ) appears in the definition of divergence-sensitive stuttering simulation of Nejati [9]. It is illustrated in Figure 4. We write s 1 t if there exists a symmetric inary relation R satisfying (T) and (D 1 ) such that s R t. Note that every relation satisfying (D) also satisfies (D 1 ), so it follows that 1. The following example illustrates that condition (D 1 ) is still not compositional, not even if the composed relations satisfy (T). s 1 s 0 s 2 s 1 s 0 s 2 t 0 t 1 t 2 t 3 u 0 u 1 u 2 u 0 u 1 u 2 Figure 5. The composition of inary relations satisfying (T) and (D 1 ) does not necessarily satisfy (D 1 ). Example 3.2. Consider the laelled transition system depicted on the left-hand side of Figure 5 together with two inary relations satisfying (T) and (D 1 ): R 1 = {(s 0, t 0 ), (t 0, s 0 ), (s 0, t 2 ), (t 2, s 0 ), (s 1, t 3 ), (t 3, s 1 )} {(s 2, t i ), (t i, s 2 ) 0 i 3} and R 2 = {(t i, u i ), (u i, t i ) 0 i 2} {(t 3, u 0 ), (u 0, t 3 )}. s 0 s 1 need not e Note that, since s 1 is not R 1 -related to t 0, the divergence s 0 s 1 simulated y t 0 in such a way that t 1 is related to either s 0 or s 1. Now consider the composition R = R 1 ; R 2. Both s 0 and s 1 are R-related to u 0, whereas the state u 1 is not R-related to s 0 nor to s 1. We conclude that R does not satisfy (D 1 ).

8 8 R.J. van Glaeek, B. Luttik, N. Trčka / Branching isimilarity with explicit divergence The culprit in the preceding example appears to e the fact that (D 1 ) only considers divergences from s of which every state is related to t. Our second alternative omits this restriction. It considers every divergence from s and requires that it is simulated y t. (D 2 ) if s R t and there is an infinite sequence of states (s k ) k ω such that s = s 0 and s k s k+1 for all k ω, then there exists a state t such that t t and s k R t for some k ω. s = s 0 s 1 s k t t Figure 6. Condition (D 2 ). Figure 6 illustrates condition (D 2 ). In contrast to the preceding divergence conditions, it does have the property that if two relations oth 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 2 ) such that it requires a divergence from t rather than just one -step: (D 3 ) if s R t and there is an infinite sequence of states (s k ) k ω such that s = s 0 and s k s k+1 for all k ω, then there exist an infinite sequence of states (t l ) l ω and a mapping σ : ω ω such that t = t 0, t l t l+1 and s σ(l) R t l for all l ω. s = s 0 s 1 s k t = t 0 t 1 t l Figure 7. Condition (D 3 ). Figure 7 illustrates condition (D 3 ). Proposition 3.1. A inary relation R satisfies (D 2 ) iff it satisfies (D 3 ). Proof The implication from right to left is trivial. For the implication from left to right, suppose that R satisfies (D 2 ) and that s R t, and consider an infinite sequence of states (s k ) k ω such that s = s 0 and s k s k+1 for all k ω. We construct an infinite sequence of states (t l ) l ω and a mapping σ : ω ω such that t = t 0, t l t l+1 and s σ(l) R t l for all l ω. The infinite sequence (t l ) l ω and the mapping σ : ω ω can e defined simultaneously y induction on l:

9 R.J. van Glaeek, B. Luttik, N. Trčka / Branching isimilarity with explicit divergence 9 1. We define t 0 = t and σ(0) = 0; it then clearly holds that s σ(0) R t Suppose that the sequence (t l ) l ω and the mapping σ : ω ω have een defined up to l. Then, in particular, s σ(l) R t l. Since (s σ(l)+k ) k ω is an infinite sequence such that s σ(l)+k s σ(l)+k+1 for all k ω, y (D 2 ) there exists t such that t l t and s σ(l)+k R t for some k ω. We define t l+1 = t and σ(l + 1) = k. We write s 3 t if there exists a symmetric inary relation R satisfying (T) and (D 3 ) such that s R t. Note that (D 1 ) is a weaker requirement than (D 2 ), and hence, y Proposition 3.1, than (D 3 ). It follows that 3 1. Also note that (D 2 ) and (D 3 ) on the one hand and (D) and (D 0 ) on the other hand are incomparale. Using that (D 3 ) is compositional, it will e straightforward to estalish that 3 is an equivalence. Then, it remains to estalish that and 3 coincide. We shall prove that 3 is included in y estalishing that 3 is a ranching isimulation with explicit divergence; that 3 is an equivalence is crucial in the proof of this property. Instead of proving the converse inclusion directly, we otain a stronger result y estalishing that a notion of isimilarity defined using a weaker divergence condition and therefore including, is included in 3. The weakest divergence condition we encountered so far is (D 1 ). It is, however, possile to further weaken (D 1 ): instead of requiring that t is an immediate -successor, it is enough require that t can e reached from t y one or more -transitions. Formally, (D 4 ) if s R t and there is an infinite sequence of states (s k ) k ω such that s = s 0, s k s k+1 and s k R t for all k ω, then there exists a state t such that t + t and s k R t for some k ω. s = s 0 s 1 s k t = t 0 t 1 t Figure 8. Condition (D 4 ). Figure 8 illustrates condition (D 4 ). We write s 4 t if there exists a symmetric inary relation R satisfying (T) and (D 4 ) such that s R t. Clearly, 1 4, and hence also 3 4 and 4. In the next section we shall also prove that 4 3. A crucial tool in our proof of this inclusion will e the notion of stuttering closure of a inary relation R on states. The stuttering closure of R enjoys the so-called stuttering property: if from state s a state s can e reached through a sequence of -transitions, and oth s and s are R-related to the same state t, then all intermediate states etween s and s are R-related to t too. We shall prove a lemma to the effect that if a inary relation on states satisfies (T) and (D 4 ), then its stuttering closure satisfies (T) and (D 3 ), and use it to estalish the inclusion 4 3. An easy corollary of the lemma is that 4 has the stuttering property. Here our proof also has a similarity with Basten s proof in [1]; in his proof that the notions of ranching isimilarity

10 10 R.J. van Glaeek, B. Luttik, N. Trčka / Branching isimilarity with explicit divergence 4 (see Sect. 4.4) 1 3 (see Sect. 4.2) Figure 9. Inclusion graph. induced y (T) and y the original condition used in [6] coincide, estalishing the stuttering property is a crucial step. Figure 9 shows some inclusions etween the different versions of ranching isimilarity with explicit divergence. (Note that we never defined 0 and 2, as these would e the same as and 3, respectively.) The solid arrows indicate inclusions that have already een argued for aove; the dashed arrows indicate inclusions that will e estalished elow. Remark 3.1. We shall estalish in the next section that = 4. Note that, once we have this, we can replace the second condition of Definition 3.1 y any interpolant of (D) and (D 4 ), i.e., any condition that is implied y (D) and implies (D 4 ), and end up with the same equivalence. For instance, we could replace it y condition (D 1 ), or y the condition of Gerth, Kuiper, Peled & Penczek [3]: if s R t and there is an infinite sequence of states (s k ) k ω such that s = s 0, s k s k+1 and s k R t for all k ω, then there exists a state t such that t t and s k R t for some k > 0. Similarly, we will prove that 3 = 4, and so we can replace the second condition of Definition 3.1 y an interpolant of (D 3 ) and (D 4 ). For instance, the condition if s R t and there is an infinite sequence of states (s k ) k ω such that s = s 0 and s k s k+1 for all k ω, then there exists a state t such that t + t and s k R t for some k 0 is a convenient interpolant of (D 3 ) and (D 4 ) to use when showing that two states are ranching isimulation equivalent with explicit divergence. 4. BB is an equivalence with the stuttering property Our goal is now to estalish that the relational characterisations of ranching isimilarity 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 3 is an equivalence relation; condition (D 3 ) will enale a direct proof of this fact. Using that 3 is an equivalence, we otain 3. Then, we define the notion of stuttering closure and use it to estalish 4 3. Together with the oservation 4 made aove, the cycle of inclusions yields that the relations, 3 and 4 coincide. It then follows that is an equivalence. We have not een ale to find a less roundaout way to

11 R.J. van Glaeek, B. Luttik, N. Trčka / Branching isimilarity with explicit divergence 11 otain this result. The intermediate results needed for the equivalence proof also yields that stuttering property. has the is an equivalence The proofs elow are rather straightforward. Nevertheless, the proof strategy employed for Lemmas 4.1 and 4.3 would fail for, 1 and 4. It is for this reason that we present all detail. Lemma 4.1. Let {R i i I} e a family of inary relations. (i) If R i satisfies (T) for all i I, then so does the union i I R i. (ii) If R i satisfies (D 3 ) for all i I, then so does the union i I R i. Proof Let R = i I R i. (i) Suppose that R i satisfies (T) for all i I. To prove that R also satisfies (T), suppose that s R t and s a s for some state s. Then s R i t for some i I. Since R i satisfies (T), it follows that there are states t and t such that t t (a) t, s R i t and s R i t, and hence s R t and s R t. (ii) Suppose that R i satisfies (D 3 ) for all i I. To prove that R satisfies (D 3 ), suppose that s R t and that there is an infinite sequence of states (s k ) k ω such that s = s 0 and s k s k+1. From s R t it follows that s R i t for some i I. By (D 3 ) there exist an infinite sequence of states (t l ) l ω and a mapping σ : ω ω such that t = t 0, t l t l+1 and s σ(l) R i t l for all l ω, and from the latter it follows that s σ(l) R t l for all l ω. Lemma 4.2. Let R e a inary relation that satisfies (T). If s R t and s s, then there is a state t such that t t and s R t. Proof Let s 0,...,s n e states such that s = s 0 s n = s. By (T) and a straightforward induction on n there exist states t 0,...,t n such that t = t 0 t n = t and s i R t i for all i n. Lemma 4.3. Let R 1 and R 2 e inary relations. (i) If R 1 and R 2 oth satisfy (T), then so does their composition R 1 ; R 2. (ii) If R 1 and R 2 oth satisfy (D 3 ), then so does their composition R 1 ; R 2. Proof Let R = R 1 ; R 2. (i) To prove that R satisfies (T), suppose s R u and s a s. Then there exists a state t such that s R 1 t and t R 2 u. Since R 1 satisfies (T), there exist states t and t such that t t (a) t, s R 1 t and s R 1 t. By Lemma 4.2 there is a state u such that u u and t R 2 u. We now distinguish two cases: (a) Suppose that a = and t = t. Then u u (a) u, from s R 1 t and t R 2 u it follows that s R u, and from s R 1 t and t R 2 u it follows that s R u.

12 12 R.J. van Glaeek, B. Luttik, N. Trčka / Branching isimilarity with explicit divergence a () Suppose that t t. Then there exist states u and u such that u u (a) u, t R 2 u and t R 2 u. So, u u (a) u, from s R 1 t and t R 2 u it follows that s R u, and from s R 1 t and t R 2 u it follows that s R u. (ii) To prove that R satisfies (D 3 ), suppose that s R u and that there is an infinite sequence of states (s k ) k ω such that s = s 0, s k s k+1 for all k ω. As efore, there exists a state t such that s R 1 t and t R 2 u. From s R 1 t it follows that there exist an infinite sequence of states (t l ) l ω and a mapping σ : ω ω such that t = t 0, t l t l+1 and s σ(l) R t l for all l ω. Hence, since t R 2 u, it follows that there exist an infinite sequence of states (u m ) m ω and a mapping ρ : ω ω such that u = u 0, u m u m+1 and t ρ(m) R 2 u m for all m ω. Clearly, s σ(ρ(m)) R u m for all m ω. Theorem is an equivalence. Proof The diagonal on S (i.e., the inary relation {(s, s) s S}) is a symmetric relation that satisfies (T) and (D 2 ), so 3 is reflexive. Furthermore, that 3 is symmetric is immediate from the required symmetry of the witnessing relation. To prove that 3 is transitive, suppose s 3 t and t 3 u. Then there exist symmetric inary relations R 1 and R 2 satisfying (T) and (D 3 ) such that s R 1 t and t R 2 u. The relation R = (R 1 ; R 2 ) (R 2 ; R 1 ) is clearly symmetric and, y Lemmas 4.1 and 4.3, satisfies (T) and (D 3 ). Hence, since s R u, it follows that s 3 u is included in To prove the inclusion 3 we estalish that 3 divergence. is a ranching isimulation with explicit Lemma 4.4. The relation 3 satisfies (T) and (D 3 ). Proof Directly from the definition it follows that 3 is the union of all symmetric relations satisfying (T) and (D 3 ), so, using Lemma 4.1, 3 itself satisfies (T) and (D 3 ). In fact, it is now clear that 3 is the largest symmetric inary relation satisfying (T) and (D 3 ). Lemma 4.5. The relation 3 satisfies (D). Proof Suppose that s 3 t and that there is an infinite sequence of states (s k ) k ω such that s = s 0, s k s k+1 and s k 3 t for all k ω. According to Lemma 4.4, the relation 3 satisfies (D 3 ), so there exist an infinite sequence of states (t l ) l ω and a mapping σ : ω ω such that t = t 0, t l t l+1 and s σ(l) 3 t l for all l ω. By Theorem 4.1, 3 is an equivalence, so it follows from s k 3 t, s σ(l) 3 t and s σ(l) 3 t l that s k 3 t l for all k, l ω. Hence 3 satisfies (D).

13 R.J. van Glaeek, B. Luttik, N. Trčka / Branching isimilarity with explicit divergence 13 Theorem Proof By Theorem 4.1, the relation 3 is symmetric. By Lemma 4.4, it satisfies (T), and y Lemma 4.5 it satisfies (D). So 3 is a ranching isimulation with explicit divergence, and hence s 3 t implies s t Stuttering closure Definition 4.1. A inary relation R has the stuttering property if, whenever t 0 and s R t n, then s R t i for all i = 0,...,n. t n, s R t 0 The following operation converts any inary relation R on S into a larger relation ˆR that has the stuttering property. Definition 4.2. Let R e a inary relation on S. The stuttering closure ˆR of R is defined y ˆR = {(s, t) s, s, t, t S. s s s & t t t & s R t & s R t }. s s s t t t Figure 10. Stuttering closure. Figure 10 illustrates the notion of stuttering closure. Clearly R ˆR. We estalish a few asic properties of the stuttering closure. Lemma 4.6. The stuttering closure of a inary relation has the stuttering property. Proof Let R e a inary relation and let ˆR e its stuttering closure. To show that ˆR has the stuttering property, suppose that t 0 t n, s ˆR t 0 and s ˆR t n. Then, on the one hand, there exist states s and t 0 such that s s, t 0 t 0 and s R t 0, and on the other hand there exist states s and t n such that s s, t n t n and s R t n. Now, since s s s and t 0 t i t n for all i = 0,...,n, it follows that s ˆR t i. Remark 4.1. The stuttering closure ˆR of a inary relation R is (contrary to what our terminology may suggest) not necessarily the smallest relation containing R with the stuttering property. For a counterexample, consider a transition system with states s, s, t and t and transitions s s and t t ; the inary relation R = {(s, t ), (t, s ), (s, t ), (t, s ), (s, t ), (t, s )} has the stuttering property, ut ˆR has additionally the pairs (s, t ) and (t, s ).

14 14 R.J. van Glaeek, B. Luttik, N. Trčka / Branching isimilarity with explicit divergence Lemma 4.7. The stuttering closure ˆR of a symmetric inary relation R is symmetric. Proof Suppose s ˆR t; then there exist states s, s, t and t such that s s s, t t t, s R t and s R t. Since R is symmetric, it follows that t R s and t R s. Hence t ˆR s. Lemma 4.8. Let ˆR e the stuttering closure of R. If R satisfies (T) and s ˆR t, then there exists t such that t t and s R t. Proof Suppose s ˆR t; then there exist states s, s, t and t such that s s s, t t t, s R t and s R t. From s R t and s s it follows y Lemma 4.2 that there exists t such that (t )t t and s R t. Lemma 4.9. If R satisfies (T), then so does its stuttering closure ˆR. Proof Suppose that s ˆR t and that s a s for some s. Then y Lemma 4.8 there exists t such that t t and s R t. Hence, since s a s, it follows y (T) that there exist states t and t such that (t )t t (a) t, s R t and s R t. Now, s R t and s R t respectively imply s ˆR t and s ˆR t Closing the cycle of inclusions Using the notion of stuttering closure we can now prove 4 3, therey closing the cycle of inclusions. To prove the inclusion we estalish that if R is a witnessing relation for 4, then ˆR is a witnessing relation for 3. Lemma If R satisfies (T) and (D 4 ), then ˆR satisfies (D 3 ). Proof Suppose that R satisfies (T) and (D 4 ). By Proposition 3.1 it suffices to estalish that ˆR satisfies (D 2 ). Suppose that s ˆR t and there exists an infinite sequence of states (s k ) k ω such that s = s 0 and s k s k+1 for all k ω. We have to show that there exists a state t such that t t and s k ˆR t for some k ω. As s ˆR t, y Lemma 4.8 there exist t 0,...,t n such that t = t 0 t n and s R t n. By Lemma 4.6, s ˆR t i for all i = 0,...,n, so if n > 0, then we can take t = t 1. We proceed with the assumption that n = 0; so s R t. First suppose that s k R t for all k ω. Then y condition (D 4 ) there exist t 0,...,t m such that t = t 0 t m with m > 0 and s k R t m for some k ω. As s k ˆR t0 and s k ˆR tm, it follows y Lemma 4.6 that s k ˆR ti for all i = 0,...,n. Hence, in particular, s k ˆR t1, so we can take t = t 1. In the remaining case there is a k 0 ω such that s k R t for all k k 0 while s k0 +1 and t are not related y R. Since s k0 R t and s k0 s k0 +1, y condition (T) of Definition 3.1 there exist states ( ) t 0,...,t m, t m+1 such that t = t 0 t m t m+1, s k0 R t m and s k0 +1 R t m+1. Since s k0 +1 and t are not related y R, it follows that t 0 t m+1, so either m > 0 or t m t m+1. In case m > 0, since s k0 ˆR t0 and s k0 ˆR tm, y Lemma 4.6 it follows that s k0 ˆR t1, so we can take t = t 1. In case m = 0 and t = t m t m+1, we can take t = t m+1.

15 R.J. van Glaeek, B. Luttik, N. Trčka / Branching isimilarity with explicit divergence 15 Theorem Proof Suppose that s 4 t. Then there exists a inary relation R satisfying (T) and (D 4 ), such that s R t. By Lemma 4.7 the stuttering closure ˆR of R is symmetric, y Lemma 4.9 it satisfies (T), and y Lemma 4.10 it satisfies (D 3 ). Moreover, s ˆR t. Hence, s 3 t. The inclusions already estalished in Section 3 together with the inclusions estalished in Theorems 4.2 and 4.3 yield the following corollary (see also Figure 9). Corollary 4.1. = 4 = 3. Corollary 4.2. The relation is an equivalence. Recall that the proof strategy employed in Lemma 4.1(ii) to show that any union of inary relations satisfying (D 3 ) also satisfies (D 3 ), fails with (D) or (D 4 ) instead of (D 3 ). In fact, it is easy to show that these results do not even hold. Therefore, we could not directly infer from the definition of that it is itself a ranching isimulation with explicit divergence. But now it follows, for = 3 satisfies (T) and (D 3 ) y Lemma 4.4, and hence also the weaker condition (D 4 ). It satisfies (D) y Lemma 4.5. Corollary 4.3. is the largest symmetric relation satisfying (T) and (D 4). It even satisfies (D), (D 3 ) and (D 2 ). It therefore is the largest ranching isimulation with explicit divergence. The following corollary is another consequence, which we need in the next section. Corollary 4.4. The relation has the stuttering property. Proof Since satisfies (T) and (D 4), its stuttering closure satisfies (T) and (D 3 ) y Lemmas 4.9 and Moreover, is symmetric y Lemma 4.7. Therefore is included in 3 (cf. the proof of Lemma 4.4). As 3 we find =. Thus, y Lemma 4.6, has the stuttering property. 5. Coloured-trace characterisation of BB We now recall from [6] the original characterisation in terms of coloured traces of ranching isimilarity with explicit divergence, and estalish that it coincides with the relational characterisations of Section 3. Definition 5.1. Let C e a colouring. A state s is C-divergent if there exists an infinite sequence of states (s k ) k ω such that s = s 0, s k s k+1 and C(s k ) = C(s) for all k ω. A consistent colouring is said to preserve divergence if no C-divergent state has the same colour as a nondivergent state. We write s c t if there exists a consistent, divergence preserving colouring C with C(s) = C(t). We prove that c =. Lemma 5.1. Let C e a colouring such that two states with the same colour have the same C-coloured traces of length three (i.e. colour - action - colour). Then C is consistent.

16 16 R.J. van Glaeek, B. Luttik, N. Trčka / Branching isimilarity with explicit divergence Proof Suppose C(s 0 ) = C(t 0 ) and C 0, a 1, C 1,...,a n, C n is a coloured trace of s 0. Then, for i = 1,...,n, there are states s i and paths π i from s i 1 to s i, such that C(π i ) = C i 1, a i, C i. With induction on i, for i = 1,...,n we find states t i with C(s i ) = C(t i ) and paths ρ i from t i 1 to t i such that C(ρ i ) = C i 1, a i, C i. Namely, the assumption aout C allows us to find ρ i given t i 1, and then t i is defined as the last state of ρ i. Concatenating all the paths ρ i yields a path ρ from t 0 with C(ρ) = C 0, a 1, C 1,...,a n, C n. Theorem 5.1. c =. Proof : Let C e a consistent colouring that preserves divergence. It suffices to show that C is a ranching isimulation with explicit divergence. Suppose s C t, i.e. C(s) = C(t), and s a s for some state s. In case a = and C(s ) = C(s) we have s C t and condition (T) is satisfied. So suppose a or C(s ) C(s). Then s, and therefore also t, has a coloured trace C(s), a, C(s ). This implies that there are states t 0,...,t n for some n 0 and t (a) with t = t 0 t 1 t n t such that C(t i ) = C(s) for i = 0,..., n and C(t ) = C(s ). Hence (T) is satisfied. Now suppose s C t and there is an infinite sequence of states (s k ) k ω such that s = s 0, s k s k+1 and s k C t for all k ω. Then C(s k ) = C(s) for all k ω. Hence s, and therefore also t, is C-divergent. Thus there exists an infinite sequence of states (t l ) l ω such that t = t 0, t l t l+1 and C(t l ) = C(t) for all l ω. It follows that s k C t l for all k, l ω. Hence also (D) is satisfied. : It suffices to show that is a consistent, divergence preserving colouring. By Corollary 4.2 it is an equivalence. We also use that it satisfies (T) and (D) (Corollary 4.3) and has the stuttering property (Corollary 4.4). We invoke Lemma 5.1 for proving consistency. Suppose that s and t have the same colour, i.e., s t, and let C, a, D e a -coloured trace of s. Then a or C D, and there are states s and s with s s a s, such that s, s C and s D. As satisfies (T), y Lemma 4.2 there is a state t with t t and s t. Therefore there exist states t and t such that (t )t t (a) t, s t and s t. As has the stuttering property and t s s t, all states on the -path from t to t have the same colour as s. Hence C, a, D is a -coloured trace of t. Now suppose s and t have the same colour and s is -divergent. Then there is an infinite sequence of states (s k ) k ω such that s = s 0, s k s k+1 and s k s t for all k ω. As satisfies (D), this implies that there exists an infinite sequence of states (t l ) l ω such that t = t 0, t l t l+1 and s k t l for all k, l ω. It follows that t l t for all l ω, and hence t is -divergent. 6. Modal characterisations of BB We shall now estalish agreement etween the relational and modal characterisations of BB proposed in [4]. The class of formulas Φ j of the modal logic for BB proposed in [4] is generated y the grammar otained y adding the following clause to the grammar in (1) of Section 2: ϕ ::= ϕ (ϕ Φ j ). (2) We extend the inductive definition of validity in Section 2 with:

17 R.J. van Glaeek, B. Luttik, N. Trčka / Branching isimilarity with explicit divergence 17 (iv) s = ϕ iff there exists an infinite sequence (s k ) k ω of states such that s s 0, s k s k+1 and s k = ϕ for all k ω. Again, validity induces an equivalence on states: we define S S y s t iff ϕ Φ j. s = ϕ t = ϕ. We shall now estalish that coincides with. Theorem 6.1. For all states s and t: s t iff s t. Proof To estalish the implication from left to right, we prove y structural induction on ϕ that if s t and s = ϕ, then t = ϕ. There are four cases to consider. 1. Suppose ϕ = ψ and s = ϕ. Then s = ψ. As t s, it follows y the induction hypothesis that t = ψ, and hence t = ϕ. 2. Suppose s = Ψ. Then, for all ψ Ψ, s = ψ, and y induction t = ψ. This yields t = φ. 3. Suppose ϕ = ψaχ and s = ϕ. Then there exist states s and s such that s s (a) s, s = ψ and s = χ. By Lemma 4.2, there exists a state t such that t t and s t. From this it follows that there exist states t and t such that t t (a) t, s t and s t, for if a = and s = s we can take t = t = t and otherwise, since s t, the states t and t exist y (T). It follows y the induction hypothesis that t = ψ and t = χ, and hence t = ϕ. 4. Suppose ϕ = ψ and s = ϕ. Then there exists an infinite sequence (s k ) k ω such that s s 0, s k s k+1 and s k = ψ for all k ω. By Lemma 4.2, there exists a state t 0 such that t t 0 and s 0 t 0. From Corollary 4.3 it follows that satisfies (D 3), so there exist an infinite sequence of states (t l ) l ω and a mapping σ : ω ω such that t l t l+1 and s σ(l) t l for all l ω. By the induction hypothesis t l = ψ for all l ω, and hence t = ϕ. For the implication from right to left, it suffices y Corollary 4.1 to prove that is symmetric and satisfies the conditions (T) and (D 4 ). That is symmetric is clear from its definition. To estalish condition (T) of Definition 3.1, suppose that s t and s s a, and define sets T and T as follows: T = {t S t t & s t }; and T = {t S t S. t t (a) t & s t }. For each t T let ϕ t e a formula such that s = ϕ t and t = ϕ t, and let ϕ = {ϕ t t T }. Similarly, for each t T let ψ t e a formula with s = ψ t and t = ψ t, and let ψ = {ψ t t T }. Note that s = ϕ a ψ, and hence, since s t, also t = ϕ a ψ. So, there exist states t and t such that t t (a) t, t = ϕ and t = ψ. From t = ϕ it follows that t T, so s t ; from t = ψ it follows that t T, so s t. Therey, condition (T) is estalished. To estalish condition (D 4 ), suppose that s t and that there exists an infinite sequence (s k ) k ω such that s = s 0, s k s k+1 and s k t for all k ω. Define the set T y T = {t S t t & s t }.

18 18 R.J. van Glaeek, B. Luttik, N. Trčka / Branching isimilarity with explicit divergence For each t T let ϕ t e a formula such that s = ϕ t and t = ϕ t, and let ϕ = {ϕ t t T }. Since s 0 = s = ϕ and s k t s, it follows that s k = ϕ for all k ω, and therefore s = ϕ. Hence, t = ϕ, so there exists an infinite sequence (t l ) l ω such that t t 0, t l t l+1 and t l = ϕ for all l ω. It follows that t l T, so s t l, for all l ω, and hence s k s t l for all k, l ω. It follows, in particular, that t + t 1 and s k t 1 for some k ω. Therey, also condition (D 4 ) is estalished. We already mentioned in Section 2 the result of Laroussinie, Pinchinat & Schnoeelen [8] that the modal logic with negation, inary conjunction and â and the logic with negation, inary conjunction and a are equally expressive. Below, we adapt their method to show that replacing a y â or a in the modal logic for BB proposed in [4] also yields an equally expressive logic. Henceforth we denote y Φ u the set of formulas generated y the grammar that is otained when replacing ϕ a ϕ y ϕ a ϕ in the grammar for Φ j (see (1) in Section 2 and (2) at the eginning of this section). The central idea, from [8], is that any formula in Φ j can e written as a Boolean comination of formulas that propagate either upwards or downwards along a path of -transitions. A formula ϕ that propagates upwards, i.e., with the property that if s t and s = ϕ, then also t = ϕ, we shall call an upward formula. A formula ϕ that propagates downwards, i.e., with the property that if s t and t = ϕ, then also s = ϕ, we shall call a downward formula. Lemma 6.1. Every ϕ Φ j is equivalent with a formula of the form Φ, where each formula in Φ is a conjunction of an upward and a downward formula. Proof Note that ψ a χ and ψ are downward formulas and that the negation of a downward formula is an upward formula. Furthermore, a conjunction of upward formulas is again an upward formula and a conjunction of downward formulas is again a downward formula. It follows, y the standard laws of Boolean algera, that the formula ϕ is equivalent to a formula of the desired shape. The proof that for every formula ϕ Φ u there exists an equivalent formula ϕ Φ j proceeds y induction on the structure of ϕ, and the only nontrivial case is when ϕ = ψ a χ. According to the induction hypothesis, for ψ and χ there exist equivalent formulas in Φ j, so, y Lemma 6.1, ψ is equivalent to a disjunction of conjunctions of upward and downward formulas. The proof in [8] then relies on these disjunctions eing finite. To generalise it to infinite disjunctions, we shall use the following lemma. Lemma 6.2. Let Φ e a set of formulas and let ϕ e a formula. Then Proof ( Φ) a ϕ {( Φ ) a ϕ Φ a finite suset of Φ}. ( ) Suppose s = ( Φ) a ϕ. Then there exist states s 0,...,s n, s n+1 such that s = s 0 (a) s n s n+1, s i = Φ for all i = 0,...,n and s n+1 = ϕ. Since s i = Φ, we can associate with every s i (i = 0,...,n) a formula ϕ i Φ such that s i = ϕ i. Let Φ = {ϕ i i = 0,...,n}; then Φ is a finite suset of Φ such that s i = Φ for every i = 0,...,n. It follows that s = ( Φ ) a ϕ, and hence s = {( Φ ) a ϕ Φ a finite suset of Φ}.

19 R.J. van Glaeek, B. Luttik, N. Trčka / Branching isimilarity with explicit divergence 19 ( ) If s = {( Φ ) a ϕ Φ a finite suset of Φ}, then s = ( Φ ) a ϕ for some finite suset Φ (a) of Φ. So there exist states s 0,...,s n, s n+1 such that s = s 0 s n s n+1, s i = Φ for all i = 0,...,n and s n+1 = ϕ. Since s i = Φ implies s i = Φ for all i = 0,...,n, it follows that s = ( Φ) a ϕ. We now adapt the method in [8] and show that replacing a y â or a in the modal logic for BB proposed in [4] yields an equally expressive logic. Theorem 6.2. For every formula ϕ Φ u there exists an equivalent formula ϕ Φ j. Proof The proof is y structural induction on ϕ; the only nontrivial case is when ϕ = ψ a χ. By the induction hypothesis there exist formulas ψ, χ Φ j such that ψ ψ and χ χ. By Lemma 6.1, ψ Ψ, where each formula in Ψ is a conjunction of an upward and a downward formula. Hence, y the evident congruence property of and Lemma 6.2, ϕ {( Ψ ) a χ Ψ a finite suset of Ψ}. Clearly, it now suffices to estalish that ( Ψ ) a χ is equivalent to a formula in Φ j, for all finite susets Ψ of Ψ. Recall that Ψ consists of conjunctions of an upward and a downward formula, so we can assume that Ψ = {ψi u ψd i i = 1,...,n}; we proceed y induction on the cardinality of Ψ. If Ψ = 0, then ( ) Ψ a χ, and Φ j. Suppose Ψ > 0. By the induction hypothesis there exists, for every i = 1,...,n, a formula ϕ i Φ j such that ( Ψ {ψ u i ψ d i }) a χ ϕ i. Then, it is easy to verify that ( Ψ ) a χ n i=1 ( ( )) ψi u ψi d a χ ψi d ϕ i, and the right-hand side formula is in Φ j. Some intuition for this last step is offered in [8]. In the same vain, there is also an ovious strengthening of the divergence modality. Let e the unary divergence modality with the following definition: (iv ) s = ϕ iff there exists an infinite sequence (s k ) k ω of states such that s = s 0, s k s k+1 and s k = ϕ for all k ω. We denote y Φ j the set of formulas generated y the grammar in (1) with ϕ replaced y ϕ. Note that the modality can e expressed in terms of : ϕ ϕ.

20 20 R.J. van Glaeek, B. Luttik, N. Trčka / Branching isimilarity with explicit divergence t 0 t 1 t 2 t 3 a 1 a 2 a 3 a 4 s = s s 1 s 2 s u 0 u 1 u 2 u 3 Figure 11. A divergence. A crucial step in our adaptation of the method of Laroussinie, Pinchinat & Schnoeelen aove consisted of showing that infinite disjunctions in the left argument of a can e avoided. If infinite disjunctions could also e avoided as an argument of, then a further adaptation of the method would e possile, showing that replacing y in the modal logic for BB would yield a logic with equal expressivity. However, the following example suggests that infinite disjunctions under cannot always e avoided. Example 6.1. Let a 1, a 2, a 3,... and 0, 1, 2,... e infinite sequences of distinct actions and consider the formula ( ) ϕ = ( ( a i ) ( i )). i=0 The formula ϕ holds in a state iff there exists an infinite -path such that in every state there is an i 0 such that the action i is still possile, whereas the action a i is not. Note that ϕ holds in the state s of the transition system in Figure 11; each of the disjuncts ( a i ) ( i ) holds in precisely one state. We conjecture that the formula of Example 6.1 is not equivalent to a formula in Φ j, and that, hence, replacing y in the modal logic for BB yields a strictly more expressive logic. We conclude the paper with a proof that the equivalence S S induced on states y validity of formulas in Φ j, defined y s t iff ϕ Φ j. s = ϕ t = ϕ, nevertheless also coincides with. Theorem 6.3. For all states s and t: s t iff s t. Proof For the implication from left to right, we prove y structural induction on ϕ that if s t and s = ϕ, then t = ϕ. We only treat the case ϕ = ψ, for the cases ϕ = ψ, ϕ = Ψ and ϕ = ψ a χ are already treated in the proof of Theorem 6.1. So, suppose ϕ = ψ and s = ϕ. Then there exists an infinite sequence (s k ) k ω of states such that s = s 0, s k s k+1 and s k = ψ for all k ω. From