Branching Bisimilarity with Explicit Divergence

Branching Bisimilarity with Explicit Div ergence Rob va n Glabbeek National ICT Austr alia, Sydney , Austr alia School of Computer Science and Engineering, University of Ne w South W ales, Sydn ey , A ustralia Bas Luttik Department of Mathematics and Computer Science, T echnische Universit eit Eindhoven, The Netherlands CWI, The Netherlands Nikola T r ˇ cka Department of Mathematics and Computer Science, T echnische Universit eit Eindhoven, The Netherlands Abstract. W e consider the relational cha racterisation o f br anching bisimilarity with explicit diver - gence. W e prove that it is an equiv a lence and that it coincides with the original definition of branch- ing bisimilarity with explicit divergence in terms of colo ured traces. W e also establish a corr espon- dence with sev eral v ariants of an action-b ased modal logic with until- and diver gence modalities. 1. Introd uction Branchin g bisimilar ity was proposed in [6]. It is a beha vioural equi v alence on processes that is com- patibl e with a notio n of abstraction from internal activi ty , while at the same preserving the branchin g structu re of processe s in a stro ng sense. W e refer the reader to [6], in p articula r to Section 10 th erein, for ample moti v ation of the rele va nce of branching bisimilarity . Branchin g bisimilarity abstracts to a lar ge extent fro m diver gence (i.e., infinite intern al activ ity). For instan ce, it identifies a process, say P , that m ay perform some internal act iv ity af ter which it returns to its initial state (i.e., P has a τ -loop) w ith a process , sa y P ′ , that admits the same behav iour as P exce pt that it cannot perform the internal acti vity leading to the in itial state (i.e., P ′ is P without the τ -loo p). This mean s that bran ching bisimilarity is no t compatible with an y temporal logic featu ring an ev entually modality : for an y des ired state th at P ′ will e ventu ally reach, the men tioned inter nal acti vity of P may be perfor med continuousl y , and thus pre ven t P from reachi ng this desired state. The notion of branc hing bisimilarity with e xplicit diver genc e (BB ∆ ), also proposed in [6], is a suit- able refinemen t of branch ing bisimilari ty that is compatib le with the well-kno wn branching -time tem- poral logic C TL ∗ without the ne xttime opera tor X (which is known to be incompatible with abstractio n from internal activit y). In fact , in [5] we hav e prov ed that it is the coarses t semantic equiv alence on labelle d transition systems with silent mov es that is a congruence for parallel composi tion (as found in process algebr as like CC S, CS P or ACP) and only equat es processes satisfy ing the same CTL ∗ − X formulas . It is also the finest equi va lence in the linear time – branc hing time spectrum of [4]. There are se veral ways to ch aracteris e a beha vioural equiv alence. The ori ginal definition of BB ∆ , in terms of colour ed traces , st ems from [6]. In [4], BB ∆ is defined in terms of a modal and a relational charac terisatio n, w hich are claimed to coin cide w ith each other and with the origi nal notion from [6]. 2 R.J. va n Glabbeek, B. Luttik, N. T r ˇ cka / Branc hing bisimilarity with explicit diver gence Of thes e three definitions of BB ∆ , the rel ational characteris ation from [4] is the most c oncise one, in the sense th at it require s the least a mount of a uxiliary concep ts. Moreov er , this definitio n is most in the style of the standard definiti ons of other kinds of bisimulation , found elsewher e in the literature. For these reason s, it is tempting to tak e it as standard definition. Although it is not hard to establish th at the m odal characteris ation from [4] is correct, in the sense that it defines an equi va lence that coincides with BB ∆ of [6], it is not at all trivi al to estab lish that the same holds for the relational characteris ation from [4]. If f act, it is non-tri vial that this relation is an equi valen ce, and that it satisfies the so-calle d stutterin g pr operty . Once these properti es hav e been establ ished, it follows that the notio n coincides w ith BB ∆ of [6]. In the re mainder of this paper , we sh all first, in Sect ion 2, briefly recapitulat e the relatio nal, colour ed- trace, and modal characteris ations of branch ing bisimilarity . Then, in S ection 3, we shall discuss the condit ion proposed in [4] that can be added to the relati onal characte risation in order to make it di ver- gence sensiti ve; w e shall then also di scuss sev eral v ariants on this cond ition. In Section 4 we establish that the re lational c haracter isation of BB ∆ all coincide, that th ey are equi vale nces and th at the y enj oy the stutter ing property . In S ection 5 we sho w that the relatio nal characterisa tions of BB ∆ coinci de with the origin al definition of BB ∆ in terms o f co loured tra ces. Finally , in Section 6, we shall e stablish a greement between the relational charac terisatio n from [4], the modal character isation from [4], and an alternati ve modal characterisa tion obtain ed by adding the di ver gence modali ty of [4] to the Hennessy -Milner logic with until propos ed in [2]. 2. Branching bisimilarity W e presu ppose a set A of act ions with a special element τ ∈ A , and we pr esuppos e a labelled tra nsition system ( S, → ) with labels from A , i.e., S is a set of states and → ⊆ S × A × S is a transitio n r elation on S . Let s , s ′ ∈ S and a ∈ A . W e write s a − → s ′ for ( s , a , s ′ ) ∈ → and we ab bre viate the statement ‘ s a − → s ′ or ( a = τ and s = s ′ )’ by s ( a ) − − → s ′ . W e denote by → + the trans iti ve closure of the binary relatio n τ − → , and by ։ its reflexi ve-tr ansiti ve closure. A pa th fro m a state s is an alternati ng sequen ce s 0 , a 1 , s 1 , a 2 , s 2 , . . . , a n , s n of states and actio ns, such that s = s 0 and s k − 1 a k − − → s k for k = 1 , . . . , n . A pr ocess is giv en by a state s in a labelled transiti on system, and encompasses all the states and transi- tions reachab le from s . Relational characteri sation The definition of branc hing bisimila rity that is most widely used has a co-ind ucti ve flav our . It defines when a binary relati on on state s preserv es the beha viour of the assoc iated proces ses. It th en declares two state s to be equi va lent if ther e exi sts such a relation relat ing them. W e shall refer to this kind of charac terisatio n as a r elational char acteris ation of branchin g bisimilarity . Definition 2.1. A symmetric binary relation R on S is a branc hing bisimula tion if i t satisfies t he fo llo w- ing condit ion for all s , t ∈ S and a ∈ A : (T) if s R t and s a − → s ′ for some state s ′ , then the re e xist states t ′ and t ′′ such that t − ։ t ′′ ( a ) − − → t ′ , s R t ′′ and s ′ R t ′ . W e write s ↔ b t if there e xists a bra nching bisimulation R such that s R t . T he re lation ↔ b on state s is referre d to as (the relational characteris ation of) branching bisimilarity . R.J. va n Glabbeek, B. Luttik, N. T r ˇ cka / Branc hing bisimilarity with explicit diver gence 3 The relatio nal character isation of branching bisimila rity presente d abov e is from [4]. As shown in [1, 4, 6], it yields the same concep t of branch ing bisimilari ty as the orig inal definition in [6 ]. The technical adv antage of the above definition ov er the original definitio n is that the defined notion of branching bisimula tion is composition al : the composit ion of two branching bisimulation s is again a branching bisimula tion. Basten [1] giv es an examp le sho wing that the condition used in th e original definitio n of ↔ b of [6 ] f ails to be co mpositiona l in this sense, and t hus ar gued that establ ishing transiti vity directly for the orig inal definition is not straightforw ard. Colour ed-trace characterisat ion T o sub stantiate their claim that branching bisimilarity indeed pre- serv es the branching structur e of processes, van Glabbeek and W eijland present in [6] an altern ati ve charac terisatio n of th e notion in terms of coloured traces. Below we repeat this charact erisation . Definition 2.2. A colouri ng is an equ iv alence on S . Giv en a colourin g C and a state s ∈ S , the colour C ( s ) of s is the equi valenc e class containin g s . For π = s 0 , a 1 , s 1 , . . . , a n , s n a path from s , let C ( π ) be the altern ating sequen ce of colo urs and ac- tions obtained from C ( s 0 ) , a 1 , C ( s 1 ) , . . . , a n , C ( s n ) by contra cting all subsequenc es C, τ , C , τ , . . . , τ , C to C . The sequen ce C ( π ) is called a C -colou r ed tra ce of s . A co louring C is consisten t if two states of the same colou r always ha ve the same C -c oloured traces. W e write s ≡ c t if there exis ts a consisten t colouring C with C ( s ) = C ( t ) . In [6] it is pro ved that ≡ c coinci des with the relati onal characterisa tion ↔ b of branc hing bisimilarity . Modal characteri sation A modal charac terisatio n of a behavi oural equi va lence is a modal logic such that two processes are equi va lent if f they satisfy the same formulas of the logic. The modal logic thus cor- respon ding to a beha vioural equi valen ce then allo ws one, fo r an y two inequi valen t processes, to fo rmally exp ress a beh avi oural prope rty that disting uishes them. Whereas colourin gs or bis imulation s are goo d tools to sho w t hat two processes are equi v alent, modal formula s are better f or proving inequi v alence. The first modal charac terisatio n of a beha vioura l equi v alence is due to Hennessy and Milner [7]. They pro- vided a modal characteris ation of (stro ng) bisimilarity on image-finite labelled transition systems, using a modal lo gic that is nowa days referre d to as the Hennessy- Milner Logic . The modal cha racterisa tions of branc hing bisimilarity presented below are adapta tions of the Henness y-Milner Logic. The class of formulas Φ jb of the modal logic for branch ing bisimilarity propo sed in [4] is generated by the follo wing grammar: ϕ ::= ¬ ϕ | V Φ | ϕ a ϕ ( a ∈ A , ϕ ∈ Φ jb and Φ ⊆ Φ jb ). (1) In case the ca rdinality | S | of the set of states of our labelled transitio n syste m is less than some infinite cardin al κ , we may requ ire that | Φ | < κ in conjunct ions, thus obtainin g a set of formulas rather than a prop er class. W e shall use the follo wing standard abbre viations: ⊤ = V ∅ , ⊥ = ¬⊤ and W Φ = ¬ V {¬ ϕ | ϕ ∈ Φ } . W e define when a formula ϕ is valid in a state s (nota tion: s | = ϕ ) inducti vely as follo ws: (i) s | = ¬ ϕ iff s 6| = ϕ ; (ii) s | = V Φ iff s | = ϕ for all ϕ ∈ Φ ; 4 R.J. va n Glabbeek, B. Luttik, N. T r ˇ cka / Branc hing bisimilarity with explicit diver gence (iii) s | = ϕ a ψ iff ther e exist states s ′ and s ′′ such that s − ։ s ′′ ( a ) − − → s ′ , s ′′ | = ϕ and s ′ | = ψ . V alidity induce s an equiv alence on states: we define ≈ ⊆ S × S by s ≈ t iff ∀ ϕ ∈ Φ jb . s | = ϕ ⇔ t | = ϕ . In [4] it was shown that ≈ coincid es with ↔ b , that i s, branch ing bi similarity is characterise d by the m odal logic abov e. Clause (iii) in the definition of vali dity appea rs to be rather libera l. More strin gent alter nati ves are obtain ed by using ϕ h ˆ a i ψ or ϕ h a i ψ instead of ϕ a ψ , with the follo wing definitions: (iii ′ ) s | = ϕ h ˆ a i ψ iff eith er a = τ and s | = ψ , or there exists a seque nce of states s 0 , . . . , s n , s n +1 ( n ≥ 0 ) such that s = s 0 τ − → · · · τ − → s n a − → s n +1 , s i | = ϕ for all i = 0 , . . . , n and s n +1 | = ψ . (iii ′′ ) s | = ϕ h a i ψ iff there exists states s 0 , . . . , s n , s n +1 ( n ≥ 0 ) such that s = s 0 τ − → · · · τ − → s n ( a ) − − → s n +1 , s i | = ϕ for all i = 0 , . . . , n and s n +1 | = ψ . The modality h ˆ a i stems from D e Nicola & V aandra ger [2]. There it was shown, fo r lab elled trans ition systems with bounded nondeterminis m , that branching bisimila rity , ↔ b , is character ised by the logic with negation , binary conjuncti on and this until modality . The modality h a i is a common strengt hening of h ˆ a i and the just-b efor e modality a abo ve; it was first conside red in [4]. T o be able to compare the exp ressi ven ess of modal logics, the follo wing definitions are propo sed by Laroussi nie, Pinchinat & Schnoeb elen [8]. Definition 2.3. T wo modal formu las ϕ and ψ that are int erpreted on state s of labelle d transition systems are eq uivalent , written ϕ ⇚ ⇛ ψ , if s | = ϕ ⇔ s | = ψ for all states s in all lab elled transiti on systems. T wo modal l ogics are equa lly expressi ve if for ev ery formula in the one th ere is an equi valen t formula in the other . As remark ed in [4], the m odalit ies h ˆ a i and h a i are equally expr essi ve, since ϕ h ˆ τ i ψ ⇚ ⇛ ψ ∨ ϕ h τ i ψ , ϕ h τ i ψ ⇚ ⇛ ϕ ∧ ϕ h ˆ τ i ψ and ϕ h a i ψ ⇚ ⇛ ϕ h ˆ a i ψ for all a 6 = τ . Note that the modalit y a can be exp ressed in terms of h a i : ϕ a ψ ⇚ ⇛ ⊤ h τ i ( ϕ h a i ψ ) . Laroussi nie, Pinchinat & S chnoe belen establ ished in [8] that the modal logic with negation , binary con- juncti on and a from [4] and the logic with ne gation, binary conjunction and h ˆ a i from [2] are equally exp ressi ve. R.J. va n Glabbeek, B. Luttik, N. T r ˇ cka / Branc hing bisimilarity with explicit diver gence 5 3. Relational characterisations of BB ∆ The notion branch ing bisimilarity d iscusse d in the previ ous sectio n abstracts from div er gence (i.e, infinite intern al activit y). In the remainde r of this paper , w e discuss a refinement of the notion of branching bisimula tion equi vale nce that takes di ver gence i nto acc ount. In this sec tion we presen t se veral conditio ns that can be added to the notion of bra nching bis imulation in order to make it div er gence sens iti ve. The induce d notions of branching bisimilarity with explicit di ver gence will all turn out to be equi valen t. Definition 3.1. [4] A symmetric binary relation R on S is a bra nchin g bisimulation w ith e xplicit diver - gen ce if it is a bran ching bisimulation (i.e., it sat isfies condition (T) of Definition 2.1) and in addition satisfies the follo wing 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 τ − → s k +1 and s k R t for all k ∈ ω , then there exist s an infinite sequence of states ( t ℓ ) ℓ ∈ ω such that t = t 0 , t ℓ τ − → t ℓ +1 for all ℓ ∈ ω , and s k R t ℓ for all k , ℓ ∈ ω . W e write s ↔ ∆ b t if there exists a branch ing bisimulation w ith ex plicit div ergen ce R such that s R t . τ τ τ τ τ τ τ s 1 t 1 τ s = s 0 t = t 0 s k t ℓ Figure 1. Condition (D). Figure 1 illu strates conditio n (D). In [4] it was cl aimed that th e notio n ↔ ∆ b defined ab ov e coincid es w ith bra nchin g bisimilar ity with e xplicit diver gen ce as defined earlie r in [ 6]. In this pap er we will su bstantia te this claim. On the way to th is end, we need to show th at ↔ ∆ b is an equ iv alence and has th e so-called stutter ing pr operty . The dif fi culty in proving that ↔ ∆ b is an equiv alence is in establishin g transiti vity . Basten’ s proof in [1] that ↔ b (i.e., branchin g bisimilarity without ex plicit di ver gence) is transiti ve, is obta ined as an immediate con sequenc e of the fact that whene ver two bin ary relations R 1 and R 2 satisfy (T), then so does their compo sition R 1 ; R 2 (see Lemma 4.3 belo w). The conditio n (D) fails to be compos itional, as we sho w in the follo wing example. Example 3.1. Consider the labell ed transit ion system depic ted on t he l eft-hand sid e of F igure 2 toget her with the branc hing bisimulations with explici t div erg ence 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 ) } and 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 rele va nt fra gment is depicte d on the right-ha nd side of Figure 2. Note that s 0 gi ves rise to a div ergen ce of which ev ery state is related by R to u 0 . Ho w e ver , since s 0 and 6 R.J. va n Glabbeek, B. Luttik, N. T r ˇ cka / Branc hing bisimilarity with explicit diver gence u 1 u 2 τ τ τ u 0 t 1 t 2 τ τ τ s 1 τ s 2 τ τ s 0 s 1 s 2 τ τ τ s 0 u 1 u 2 τ τ τ u 0 t 0 Figure 2 . The compo sition of b ranching bisimu lations with explicit divergence is not a branch ing bisimulation with explicit di vergence. u 2 are not relat ed according to R , there is no di ver gence from u 0 of which e very sta te is related to ev ery state on the di ver gence from s 0 . W e conclu de that R does not satis fy the condition (D). Our proof that ↔ ∆ b is an equi va lence proc eeds alo ng the same lin es as Basten’ s proo f in [1 ] that ↔ b is an equi v alence: we replace (D) by an alternati ve di ver gence cond ition that is compo sitiona l, pro ve that the resul ting notion of bisi milarity is an equi val ence, and then est ablish that it coin cides with ↔ ∆ b . In the remainder of this section, we shall arriv e at our compositiona l alte rnati ve for (D) through a series of adapta tions of (D). First, we observ e that (D) has a technicall y con ven ient ref ormulation : instead of requiring the exis- tence of a diver gence f rom t all the states of which enjo y ce rtain propertie s, it suf fices to re quire that there exi sts a state re achable from t by a single τ -t ransitio n with thes e properties. Formally , the reformulatio n of (D) is: (D 0 ) if s R t and there is an infinite sequen ce of states ( s k ) k ∈ ω such that s = s 0 , s k τ − → s k +1 and s k R t for all k ∈ ω , then there exists a state t ′ such that t τ − → t ′ and s k R t ′ for all k ∈ ω . τ τ τ τ s 1 t ′ τ s = s 0 s k t Figure 3. Cond ition (D 0 ). Figure 3 illustra tes condition (D 0 ). If a binary relation satisfies (D 0 ), then the div erg ence from t re- quired by (D) can be inducti vely constructed . (W e omit the inducti ve constr uction here; the proof of Proposit ion 3.1 belo w contai ns a very simila r inducti ve constru ction.) For our nex t adaptat ion we obser ve that (D 0 ) has some redundan cy . Note that it requires t ′ to be related to every state on the di ver gence fro m s . Howe ver , the univ ersal qua ntification in the conclusio n can be relax ed to an existen tial quantificati on: it suf fices to require that t has an immediate τ -succe ssor that is relate d to some state on the di verg ence from s . The requiremen t can be exp ressed as follows: R.J. va n Glabbeek, B. Luttik, N. T r ˇ cka / Branc hing bisimilarity with explicit diver gence 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 τ − → s k +1 and s k R t for all k ∈ ω , then there exists a state t ′ such that t τ − → t ′ and s k R t ′ for some k ∈ ω . τ τ τ τ s 1 t ′ τ s = s 0 s k t Figure 4. Cond ition (D 1 ). Conditio n (D 1 ) appears in the definition of di ver gence-sens iti ve stuttering simulation of Nejati [9]. It is illustr ated in Figure 4. W e write s ↔ ∆ 1 b t if there exists a symmetric binary rel ation R sati sfying (T) and (D 1 ) such th at s R t . Note that ev ery relation satisfyin g (D) also satis fies (D 1 ), so it follo ws that ↔ ∆ b ⊆ ↔ ∆ 1 b . The follo wing example illustrates that conditio n (D 1 ) is still not compos itional, not ev en if the com- posed rela tions satisfy (T). τ τ τ t 1 t 2 τ τ τ t 0 t 3 s 0 s 1 s 2 τ τ u 0 u 1 u 2 τ s 0 s 1 s 2 τ τ u 0 u 1 u 2 τ τ τ τ τ τ τ Figure 5. The com position of binary relations s atisfying (T) an d (D 1 ) does not necessarily satisfy (D 1 ). Example 3.2. Consider the labell ed transit ion system depic ted on t he l eft-hand sid e of F igure 5 toget her with two bi nary 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 ) } . Note that, since s 1 is not R 1 -relate d to t 0 , the div er gence s 0 τ − → s 1 τ − → s 0 τ − → s 1 τ − → · · · ne ed not be simulate d by t 0 in such a way tha t t 1 is relate d to either s 0 or s 1 . No w consider the composi tion R = R 1 ; R 2 . B oth s 0 and s 1 are R -related to u 0 , whereas the state u 1 is not R -relat ed to s 0 nor to s 1 . W e conclude that R does not satisfy (D 1 ). 8 R.J. va n Glabbeek, B. Luttik, N. T r ˇ cka / Branc hing bisimilarity with explicit diver gence The culprit in the preced ing exampl e appears to be the fact that (D 1 ) only conside rs div ergen ces from s of which ev ery state is related to t . O ur second alterna tiv e omits this restr iction. It cons iders eve ry di ver gence from s and requ ires that it is simulated by t . (D 2 ) if s R t and there is an infinite sequenc e of states ( s k ) k ∈ ω such that s = s 0 and s k τ − → s k +1 for all k ∈ ω , then there exists a state t ′ such that t τ − → t ′ and s k R t ′ for some k ∈ ω . τ τ τ τ s 1 t ′ τ s = s 0 s k t Figure 6. Cond ition (D 2 ). Figure 6 illus trates condit ion (D 2 ). In contrast to the prec eding di ver gence conditi ons, it does ha ve th e proper ty that if two relatio ns both satisfy it, then so does their relational composition . Ho weve r , to faci litate a dir ect proof of this prope rty , it is te chnically con ven ient to reformul ate condition (D 2 ) su ch that it requi res a di ver gence from t rat her than just one τ -step: (D 3 ) if s R t and there is an infinite se quence of s tates ( s k ) k ∈ ω such th at s = s 0 and s k τ − → s k +1 for all k ∈ ω , then there exi st an infinite sequenc e of states ( t ℓ ) ℓ ∈ ω and a m apping σ : ω → ω suc h that t = t 0 , t ℓ τ − → t ℓ +1 and s σ ( ℓ ) R t ℓ for all ℓ ∈ ω . τ τ τ τ τ τ τ s 1 t 1 τ s = s 0 t = t 0 s k t ℓ Figure 7. Cond ition (D 3 ). Figure 7 illust rates conditi on (D 3 ). Pro position 3.1. A binary relation R satisfies (D 2 ) if f it satisfies (D 3 ). Pro of The implicati on from right to left is trivia l. For the implication from left to right, suppose that R satisfies (D 2 ) and that s R t , and consi der an infinite sequen ce of states ( s k ) k ∈ ω such that s = s 0 and s k τ − → s k +1 for all k ∈ ω . W e construct an infinite sequence of states ( t ℓ ) ℓ ∈ ω and a mapping σ : ω → ω such that t = t 0 , t ℓ τ − → 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 be defined simultane ously by induction on l : R.J. va n Glabbeek, B. Luttik, N. T r ˇ cka / Branc hing bisimilarity with explicit diver gence 9 1. W e define t 0 = t and σ (0) = 0 ; it then clearly holds that s σ (0) R t 0 . 2. Suppose tha t the sequence ( t ℓ ) ℓ ∈ ω and t he map ping σ : ω → ω ha ve been defined u p to ℓ . Then, in particu lar , s σ ( ℓ ) R t ℓ . S ince ( s σ ( ℓ )+ k ) k ∈ ω is an infinite sequence such that s σ ( ℓ )+ k τ − → s σ ( ℓ )+ k +1 for all k ∈ ω , by (D 2 ) th ere exis ts t ′ such that t ℓ τ − → t ′ and s σ ( ℓ )+ k ′ R t ′ for so me k ′ ∈ ω . W e define t ℓ +1 = t ′ and σ ( ℓ + 1) = k ′ .  W e write s ↔ ∆ 3 b t if there exists a symmetric binary relation R satisfying (T) and (D 3 ) such that s R t . Note that (D 1 ) is a weak er requ irement than (D 2 ), and hen ce, by Propositio n 3.1 , than (D 3 ). It follo ws that ↔ ∆ 3 b ⊆ ↔ ∆ 1 b . A lso note that (D 2 ) and (D 3 ) on the one hand and (D) and (D 0 ) on the other hand are incompa rable. Using that (D 3 ) is compositiona l, it will be straightforw ard to establish th at ↔ ∆ 3 b is an equiv alence. Then, it remains to estab lish that ↔ ∆ b and ↔ ∆ 3 b coinci de. W e shall pro ve that ↔ ∆ 3 b is includ ed in ↔ ∆ b by establ ishing that ↔ ∆ 3 b is a branching bisimulati on with explic it div er gence; tha t ↔ ∆ 3 b is an equiv alence is cruc ial in the proof of this prope rty . Instea d of pro ving the con verse inclusion directly , we obtain a strong er result by establi shing that a notion of bisimilarity defined using a weake r div ergenc e condi tion and therefore including ↔ ∆ b , is include d in ↔ ∆ 3 b . T he weakest div ergenc e con dition w e encounte red so far is (D 1 ). It is, ho wev er , possible to further weaken (D 1 ): instead of requirin g that t ′ is an immedi ate τ -suc cessor , it is eno ugh require that t ′ can be reached from t by one or more τ -transitio ns. Formall y , (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 τ − → 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 . τ τ τ τ τ τ s 1 τ s = s 0 t = t 0 s k t ′ t 1 Figure 8. Cond ition (D 4 ). Figure 8 illustrates condition (D 4 ). W e write s ↔ ∆ 4 b t if there exists a symmetri c binar y relati on R satisfy ing (T) and (D 4 ) such that s R t . Clearly , ↔ ∆ 1 b ⊆ ↔ ∆ 4 b , and hence also ↔ ∆ 3 b ⊆ ↔ ∆ 4 b and ↔ ∆ b ⊆ ↔ ∆ 4 b . In the ne xt sectio n we shall also pro ve that ↔ ∆ 4 b ⊆ ↔ ∆ 3 b . A cruc ial tool in our proof of this inclus ion will be the notion of stuttering closu r e of a binary relat ion R on states . The stuttering closur e of R enjo ys the so -called stuttering pr operty : if from sta te s a state s ′ can be reached through a sequence of τ -tran sitions , and both s and s ′ are R -relat ed to the same state t , then all int ermediate states between s and s ′ are R -related to t too. W e sh all pro ve a lemma to the effe ct that if a binary relatio n on st ates satisfies (T) and (D 4 ), th en its stutterin g closure s atisfies (T) and ( D 3 ), a nd u se it to establi sh the i nclusion ↔ ∆ 4 b ⊆ ↔ ∆ 3 b . An easy corollar y of the le mma is that ↔ ∆ 4 b has the stutteri ng prop erty . Here our proof also has a similarit y with Basten’ s proof in [1]; in his proof that the notions of bran ching bisimil arity 10 R.J. va n Glabbeek, B. Luttik, N. T r ˇ cka / Branc hing bisimilarity with explicit diver gence ↔ ∆ b ↔ ∆ 3 b ↔ ∆ 1 b ↔ ∆ 4 b (see Sect. 4.2) (see Sect. 4.4) Figure 9. Inclusion graph . induce d by (T) and by the origin al condition used in [6] coinc ide, es tablishin g the stutterin g pro perty is a crucia l step. Figure 9 s ho ws s ome inclusions betwee n th e differe nt v ersions of bran ching bisimilar ity with explic it di ver gence. (Note that we ne ver defined ↔ ∆ 0 b and ↔ ∆ 2 b , as these would be the same as ↔ ∆ b and ↔ ∆ 3 b , respec tiv ely .) The solid arro ws indicate inc lusions tha t hav e al ready been argue d for abov e; the dashed arro w s indicate inclusi ons that will be establish ed belo w . Remark 3.1. W e shall establish in the next section that ↔ ∆ b = ↔ ∆ 4 b . Note that, once w e hav e this, we can repl ace the second conditio n of Definition 3.1 by an y interpolant of (D) and (D 4 ), i.e., an y conditio n that is implied by (D) and implies (D 4 ), and end up with the same equi v alence. For instance, we could replac e it by condi tion (D 1 ), or by the conditi on of Gerth, Kuiper , Peled & Penczek [3]: if s R t and there is an infinite sequenc e of states ( s k ) k ∈ ω such that s = s 0 , s k τ − → s k +1 and s k R t for all k ∈ ω , then there exists a state t ′ such that t τ − → t ′ and s k R t ′ for some k > 0 . Similarly , we will pro ve tha t ↔ ∆ 3 b = ↔ ∆ 4 b , an d so we c an repl ace the second con dition of Definitio n 3.1 by an interp olant of (D 3 ) and (D 4 ). For instan ce, the cond ition if s R t and there is an infinite sequence of states ( s k ) k ∈ ω such that s = s 0 and s k τ − → 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 con venient interpolan t of (D 3 ) and (D 4 ) to use when sho wing that two states are branc hing bisimula- tion equi valen t with exp licit div erg ence. 4. BB ∆ is an equiv alence with the stuttering prop erty Our goal is now to estab lish that the relation al characteris ations of branching bisi milarity with e xplici t di ver gence introduced in the pre vious section all coin cide, that they are equi vale nces and that they en joy the stutterin g propert y . T o this e nd, we first s ho w th at ↔ ∆ 3 b is an equiv alence relat ion; conditio n (D 3 ) will enable a direct proof of this fact. Using that ↔ ∆ 3 b is an equi va lence, we obtain ↔ ∆ 3 b ⊆ ↔ ∆ b . Then, we define the notion of stutter ing clo sur e and use it to establi sh ↔ ∆ 4 b ⊆ ↔ ∆ 3 b . T ogether with the ob serv ation ↔ ∆ b ⊆ ↔ ∆ 4 b made abo ve, the cyc le of inclusion s yields that the relations ↔ ∆ b , ↔ ∆ 3 b and ↔ ∆ 4 b coinci de. It then follows that ↔ ∆ b is an equi valen ce. W e hav e not been able to find a less roundabou t way to R.J. va n Glabbeek, B. Luttik, N. T r ˇ cka / Branc hing bisimilarity with explicit diver gence 11 obtain this result . The intermediat e results needed for the equiv alence proof also yields that ↔ ∆ b has the stutter ing prope rty . 4.1. ↔ ∆ 3 b is an equi valence The proofs belo w are rather straightforw ard. Neve rtheless , the proof strategy employ ed for Lemmas 4.1 and 4.3 woul d fail for ↔ ∆ b , ↔ ∆ 1 b and ↔ ∆ 4 b . It is fo r this reason that we present all detail. Lemma 4.1. Let {R i | i ∈ I } be a family of binary relatio ns. (i) If R i satisfies (T) for all i ∈ I , then so does the union S i ∈ I R i . (ii) If R i satisfies (D 3 ) for all i ∈ I , then so does the unio n S i ∈ I R i . Pro of Let R = S i ∈ I R i . (i) Suppose that R i satisfies (T) fo r all i ∈ I . T o prove that R also satisfies (T), su ppose 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 follo ws that there are states t ′ and t ′′ such that t − ։ t ′′ ( a ) − − → t ′ , s R i t ′′ and s ′ R i t ′ , and henc e s R t ′′ and s ′ R t ′ . (ii) Suppose that R i satisfies (D 3 ) for all i ∈ I . T o prove tha t R satisfies (D 3 ), suppos e that s R t and that there is an infinite sequence of s tates ( s k ) k ∈ ω such that s = s 0 and s k τ − → s k +1 . From s R t it follo ws that s R i t for some i ∈ I . By (D 3 ) there ex ist an infinite sequence of states ( t ℓ ) ℓ ∈ ω and a mapping σ : ω → ω such that t = t 0 , t ℓ τ − → t ℓ +1 and s σ ( ℓ ) R i t ℓ for all ℓ ∈ ω , and from the latter it follo ws that s σ ( ℓ ) R t ℓ for all ℓ ∈ ω .  Lemma 4.2. Let R be a bin ary relation tha t sati sfies (T). If s R t and s − ։ s ′ , then the re is a st ate t ′ such that t − ։ t ′ and s ′ R t ′ . Pro of Let s 0 , . . . , s n be states su ch that s = s 0 τ − → · · · τ − → s n = s ′ . B y (T) an d a straightfo rward induct ion 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 be binary relation s. (i) If R 1 and R 2 both satisf y (T), then so does their composition R 1 ; R 2 . (ii) If R 1 and R 2 both satisf y (D 3 ), then so does their compos ition R 1 ; R 2 . Pro of Let R = R 1 ; R 2 . (i) T o pro ve that R sat isfies (T), suppos e s R u and s a − → s ′ . Then the re exist s 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 ′′ . W e no w distin guish two case s: (a) Suppose that a = τ and t ′′ = t ′ . Then u − ։ u ′′ ( a ) − − → u ′′ , from s R 1 t ′′ and t ′′ R 2 u ′′ it follo ws that s R u ′′ , and from s ′ R 1 t ′ and t ′ R 2 u ′′ it follo ws that s ′ R u ′′ . 12 R.J. va n Glabbeek, B. Luttik, N. T r ˇ cka / Branc hing bisimilarity with explicit diver gence (b) Suppose that t ′′ a − → t ′ . Then there e xist 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 follo ws that s R u ′′′ , and from s ′ R 1 t ′ and t ′ R 2 u ′ it follo ws that s ′ R u ′ . (ii) T o prov e 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 τ − → s k +1 for all k ∈ ω . As before , there e xists a state t such that s R 1 t and t R 2 u . From s R 1 t it follows that there exis t an infinite sequence of states ( t ℓ ) ℓ ∈ ω and a m appin g σ : ω → ω such that t = t 0 , t ℓ τ − → 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 st ates ( u m ) m ∈ ω and a m appin g ρ : ω → ω such that u = u 0 , u m τ − → u m +1 and t ρ ( m ) R 2 u m for all m ∈ ω . C learly , s σ ( ρ ( m )) R u m for all m ∈ ω .  Theor em 4.1. ↔ ∆ 3 b is an equ iv alence. Pro of The diagonal o n S (i.e., the binary relation { ( s , s ) | s ∈ S } ) is a symmetric relation that satisfies (T) and (D 2 ), so ↔ ∆ 3 b is reflex iv e. Furthermore , th at ↔ ∆ 3 b is symmetric is immediate from the require d symmetry of the witnessin g relation. T o prov e that ↔ ∆ 3 b is transi tiv e, suppos e s ↔ ∆ 3 b t and t ↔ ∆ 3 b u . T hen there exi st symmetric binary relation s R 1 and R 2 satisfy ing (T) and (D 3 ) such that s R 1 t and t R 2 u . The relati on R = ( R 1 ; R 2 ) ∪ ( R 2 ; R 1 ) is clearly symmetric and, by Lemmas 4.1 an d 4.3, satisfies (T) and (D 3 ). Henc e, since s R u , it follo ws that s ↔ ∆ 3 b u .  4.2. ↔ ∆ 3 b is included in ↔ ∆ b T o pro ve the inclusio n ↔ ∆ 3 b ⊆ ↔ ∆ b we establ ish that ↔ ∆ 3 b is a branching bisimulatio n with ex plicit di ver gence. Lemma 4.4. The relation ↔ ∆ 3 b satisfies (T) and (D 3 ). Pro of Directly from the definition it follows that ↔ ∆ 3 b is the union of all symmetric relations satisfying (T) and (D 3 ), so, using Lemma 4.1, ↔ ∆ 3 b itself satisfies (T) and (D 3 ).  In fa ct, it is now clea r that ↔ ∆ 3 b is the lar gest symmetric binary relation satisfyin g (T) and (D 3 ). Lemma 4.5. The relation ↔ ∆ 3 b satisfies (D). Pro of Suppo se that s ↔ ∆ 3 b t and that there is an infinite sequence of states ( s k ) k ∈ ω such that s = s 0 , s k τ − → s k +1 and s k ↔ ∆ 3 b t for all k ∈ ω . A ccord ing to Lemm a 4.4 , the relation ↔ ∆ 3 b satisfies (D 3 ), so there exist an infinite sequenc e of states ( t ℓ ) ℓ ∈ ω and a mapping σ : ω → ω such that t = t 0 , t ℓ τ − → t ℓ +1 and s σ ( ℓ ) ↔ ∆ 3 b t ℓ for all ℓ ∈ ω . By Theorem 4.1, ↔ ∆ 3 b is an equi valenc e, so it follo w s from s k ↔ ∆ 3 b t , s σ ( ℓ ) ↔ ∆ 3 b t and s σ ( ℓ ) ↔ ∆ 3 b t ℓ that s k ↔ ∆ 3 b t ℓ for all k , ℓ ∈ ω . Hence ↔ ∆ 3 b satisfies (D).  R.J. va n Glabbeek, B. Luttik, N. T r ˇ cka / Branc hing bisimilarity with explicit diver gence 13 Theor em 4.2. ↔ ∆ 3 b ⊆ ↔ ∆ b . Pro of By Theorem 4.1, the relation ↔ ∆ 3 b is symmetric. By L emma 4.4, it satisfies (T), and by Lemma 4.5 it satisfies (D). So ↔ ∆ 3 b is a branch ing bisimul ation with ex plicit div er gence, and hence s ↔ ∆ 3 b t implies s ↔ ∆ b t .  4.3. Stutte ring closure Definition 4.1. A bin ary relatio n R has the stuttering pr operty if, whenev er t 0 τ − → · · · τ − → t n , s R t 0 and s R t n , then s R t i for all i = 0 , . . . , n . The follo wing operation con vert s any binary relation R on S into a larger relation ˆ R that has the stuttering proper ty . Definition 4.2. Let R be a bin ary relation on S . The stutterin g closur e ˆ R of R is defined by ˆ R = { ( s , t ) | ∃ s ♭ , s ♯ , t ♭ , t ♯ ∈ S . s ♭ − ։ s − ։ s ♯ & t ♭ − ։ t − ։ t ♯ & s ♭ R t ♯ & s ♯ R t ♭ } . t ♯ s ♯ s s ♭ t ♭ t Figure 10. Stuttering closure. Figure 10 illustrates the notion o f st uttering clos ure. Clearly R ⊆ ˆ R . W e e stablish a few basic prop erties of the stutte ring closure. Lemma 4.6. The stutte ring closure of a binary relation has the stuttering pr operty . Pro of Let R be a binary relation and let ˆ R be its stutterin g closure . T o sho w that ˆ R has the stuttering proper ty , suppose that t 0 τ − → · · · τ − → t n , s ˆ R t 0 and s ˆ R 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 exis t states s ♭ and t ♯ n such that s ♭ − ։ s , t n − ։ t ♯ n and s ♭ R t ♯ n . No w , since s ♭ − ։ s − ։ s ♯ and t ♭ 0 − ։ t i − ։ t ♯ n for all i = 0 , . . . , n , it follo ws that s ˆ R t i .  Remark 4.1. The stuttering closure ˆ R of a binary relatio n R is (contrary to what our terminology may sugge st) not necessar ily the smalle st relation containin g R with the stutterin g propert y . For a count erex- ample, cons ider a transition system with states s ♭ , s ♯ , t ♭ and t ♯ and tran sitions s ♭ τ − → s ♯ and t ♭ τ − → t ♯ ; the binary relatio n R = { ( s ♭ , t ♯ ) , ( t ♯ , s ♭ ) , ( s ♯ , t ♭ ) , ( t ♭ , s ♯ ) , ( s ♯ , t ♯ ) , ( t ♯ , s ♯ ) } has the stut tering proper ty , but ˆ R has additi onally the pairs ( s ♭ , t ♭ ) and ( t ♭ , s ♭ ) . 14 R.J. va n Glabbeek, B. Luttik, N. T r ˇ cka / Branc hing bisimilarity with explicit diver gence Lemma 4.7. The stutterin g closure ˆ R of a symmetri c binary relation R is symmetric. Pro of Suppo se s ˆ R 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 follo ws that t ♭ R s ♯ and t ♯ R s ♭ . Hence t ˆ R s .  Lemma 4.8. Let ˆ R be the stuttering closure of R . If R satisfies (T) and s ˆ R t , then there exists t ′ such that t − ։ t ′ and s R t ′ . Pro of Suppo se s ˆ R 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 fo llo ws by L emma 4.2 th at th ere ex ists t ′ such that ( t − ։ ) t ♯ − ։ t ′ and s R t ′ .  Lemma 4.9. If R satisfies (T), then so does its stutterin g closure ˆ R . Pro of Suppo se that s ˆ R t and that s a − → s ′ for some s ′ . Then by Lemma 4.8 there exis ts t † such that t − ։ t † and s R t † . Hence , since s a − → s ′ , it follo w s by (T) that there exist state s t ′′ and t ′ such that ( t − ։ ) t † − ։ t ′′ ( a ) − − → t ′ , s R t ′′ and s ′ R t ′ . No w , s R t ′′ and s ′ R t ′ respec tiv ely imply s ˆ R t ′′ and s ′ ˆ R t ′ .  4.4. Closing the cycle of inclusions Using the notio n of stuttering closure we can no w prov e ↔ ∆ 4 b ⊆ ↔ ∆ 3 b , there by closing the cycle of inclus ions. T o prov e the inclusio n w e establish that if R is a witne ssing relat ion for ↔ ∆ 4 b , the n ˆ R is a witnessin g relation for ↔ ∆ 3 b . Lemma 4.10. If R satisfies (T) and (D 4 ), then ˆ R satisfies (D 3 ). Pro of Suppose that R satisfies (T) and (D 4 ). By Propositi on 3.1 it suf fices to establi sh that ˆ R satisfies (D 2 ). S uppos e that s ˆ R t and there exists an infinite seq uence of state s ( s k ) k ∈ ω such that s = s 0 and s k τ − → s k +1 for all k ∈ ω . W e hav e to sho w that there exists a state t ′ such that t τ − → t ′ and s k ˆ R t ′ for some k ∈ ω . As s ˆ R t , by Lemma 4.8 there exist t 0 , . . . , t n such that t = t 0 τ − → · · · τ − → t n and s R t n . By Lemma 4.6, s ˆ R t i for all i = 0 , . . . , n , so if n > 0 , then we can take t ′ = t 1 . W e procee d w ith th e assumpti on that n = 0 ; so s R t . First suppose that s k R t for all k ∈ ω . Then by conditio n (D 4 ) there exist t 0 , . . . , t m such that t = t 0 τ − → · · · τ − → t m with m > 0 and s k R t m for some k ∈ ω . As s k ˆ R t 0 and s k ˆ R t m , it follo w s by Lemma 4.6 that s k ˆ R t i for all i = 0 , . . . , n . Hence, in particu lar , s k ˆ R t 1 , so we can tak e 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 k 0 +1 and t are not related by R . Since s k 0 R t and s k 0 τ − → s k 0 +1 , by condit ion (T) of Definition 3.1 there exist states t 0 , . . . , t m , t m +1 such that t = t 0 τ − → · · · τ − → t m ( τ ) − − → t m +1 , s k 0 R t m and s k 0 +1 R t m +1 . Since s k 0 +1 and t are not related by R , it follo w s that t 0 6 = t m +1 , so either m > 0 or t m τ − → t m +1 . In case m > 0 , since s k 0 ˆ R t 0 and s k 0 ˆ R t m , by L emma 4.6 it fol lows that s k 0 ˆ R t 1 , so we can take t ′ = t 1 . In case m = 0 and t = t m τ − → t m +1 , we can tak e t ′ = t m +1 .  R.J. va n Glabbeek, B. Luttik, N. T r ˇ cka / Branc hing bisimilarity with explicit diver gence 15 Theor em 4.3. ↔ ∆ 4 b ⊆ ↔ ∆ 3 b . Pro of S uppose that s ↔ ∆ 4 b t . T hen there exists a binary relation R satisfying (T ) and (D 4 ), such that s R t . By Lemma 4.7 the stuttering closure ˆ R of R is symmetric, by Lemm a 4.9 it satisfies (T), and by Lemma 4.10 it satis fies (D 3 ). More ov er , s ˆ R t . Hence, s ↔ ∆ 3 b t .  The inc lusions already establish ed in Section 3 togeth er w ith the in clusion s establish ed in Theorems 4.2 and 4.3 yiel d the followin g corollary (see also Figure 9). Cor ollary 4.1. ↔ ∆ b = ↔ ∆ 4 b = ↔ ∆ 3 b .  Cor ollary 4.2. The relation ↔ ∆ b is an equi valen ce.  Recall that the proof strategy employed in L emma 4.1(ii) to sho w that any union of binary relations satisfy ing (D 3 ) al so satis fies (D 3 ), f ails with (D) or (D 4 ) in stead of (D 3 ). In fact, it is eas y to show that these results do not e ven hold. Therefore, we could not directly in fer fr om the definiti on of ↔ ∆ b that it is itself a branching bisimulatio n with explicit div er gence. But now it follo ws, for ↔ ∆ b = ↔ ∆ 3 b satisfies (T) and (D 3 ) by Lemma 4.4, and hence also the weake r condi tion (D 4 ). It sa tisfies (D) by Lemma 4.5. Cor ollary 4.3. ↔ ∆ b is the lar gest symmetric relation satisfying (T) and (D 4 ). It e ven satisfies (D ), (D 3 ) and (D 2 ). It th erefore is the larg est branch ing bisimula tion with explicit di ver gence.  The follo wing coroll ary is anoth er consequen ce, which we need in the next secti on. Cor ollary 4.4. The relation ↔ ∆ b has the stutter ing pr operty . Pro of Since ↔ ∆ b satisfies (T) and (D 4 ), it s st uttering clos ure c ↔ ∆ b satisfies (T) and (D 3 ) b y Lemmas 4.9 and 4.10. Moreov er , c ↔ ∆ b is symmetric by L emma 4.7. Therefore c ↔ ∆ b is included in ↔ ∆ 3 b (cf. the proof of Lemma 4.4). As ↔ ∆ b ⊆ c ↔ ∆ b ⊆ ↔ ∆ 3 b we find ↔ ∆ b = c ↔ ∆ b . T hus, by Lemma 4.6, ↔ ∆ b has the stutter ing prope rty .  5. Colour ed-trace characterisation of BB ∆ W e no w recal l from [6] the origina l characte risation in terms of coloure d traces of branch ing bisimilarity with e xplicit di ver gence, and establish that it co incides with the rela tional character isations of Sectio n 3. Definition 5.1. Let C be a colouring . A st ate s is C -diver gent if there exi sts an in finite seq uence of s tates ( s k ) k ∈ ω such that s = s 0 , s k τ − → s k +1 and C ( s k ) = C ( s ) for all k ∈ ω . A consistent colouring is said to pr eserve diver gence if no C -div er gent state has the same colour as a nondi ver gent state. W e write s ≡ ∆ c t if there exists a consist ent, di ver gence preser ving colouring C with C ( s ) = C ( t ) . W e pro ve that ≡ ∆ c = ↔ ∆ b . Lemma 5.1. Let C be a colou ring such that two states with the same colo ur ha ve th e same C -coloured traces of length three (i.e. colo ur - action - colour). Then C is con sistent. 16 R.J. va n Glabbeek, B. Luttik, N. T r ˇ cka / Branc hing bisimilarity with explicit diver gence Pro of Suppose C ( s 0 ) = C ( t 0 ) and C 0 , a 1 , C 1 , . . . , a n , C n is a colou red trace of s 0 . T hen, 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 . W ith induction on i , for i = 1 , . . . , n we find s tates t i with C ( s i ) = C ( t i ) and paths ρ i from t i − 1 to t i such th at C ( ρ i ) = C i − 1 , a i , C i . Namely , th e assump tion about C allo ws us to find ρ i gi ven t i − 1 , an d then t i is defined as the last state of ρ i . Concatena ting all the p aths ρ i yields a path ρ from t 0 with C ( ρ ) = C 0 , a 1 , C 1 , . . . , a n , C n .  Theor em 5.1. ≡ ∆ c = ↔ ∆ b . Pro of “ ⊆ ”: Let C be a con sistent col ouring tha t preser ves div erg ence. It suf fi ces to sho w that C is a branch ing bisimulation with ex plicit div ergenc e. Suppose s C t , i.e. C ( s ) = C ( t ) , and s a − → s ′ for some state s ′ . In case a = τ and C ( s ′ ) = C ( s ) we ha ve s ′ C t and conditio n (T) is satisfied. So suppo se a 6 = τ or C ( s ′ ) 6 = C ( s ) . T hen s , and therefore also t , has a coloured trace C ( s ) , a, C ( s ′ ) . T his implies that there are states t 0 , . . . , t n for some n ≥ 0 and t ′ with t = t 0 τ − → t 1 τ − → · · · τ − → t n ( a ) − − → t ′ such that C ( t i ) = C ( s ) for i = 0 , ..., n and C ( t ′ ) = C ( s ′ ) . Henc e (T) is satisfied . No w suppose s C t and there is an infinite sequenc e of states ( s k ) k ∈ ω such that s = s 0 , s k τ − → s k +1 and s k C t for all k ∈ ω . Then C ( s k ) = C ( s ) for all k ∈ ω . H ence s , and theref ore also t , is C -di ver gent. Thus there exi sts an infinite sequen ce of states ( t ℓ ) ℓ ∈ ω such that t = t 0 , t ℓ τ − → t ℓ +1 and C ( t ℓ ) = C ( t ) for all ℓ ∈ ω . It follo ws that s k C t ℓ for all k , ℓ ∈ ω . Hence also (D) is satis fied. “ ⊇ ”: It suffices to show that ↔ ∆ b is a c onsisten t, div ergen ce preserv ing colou ring. By Corollary 4.2 it is an equ iv alence. W e also use t hat it satisfies (T) and (D) (Corol lary 4.3) and has the stutteri ng proper ty (Corollar y 4.4 ). W e in vok e Lemma 5.1 for pro ving consisten cy . Suppose that s and t ha ve the same colour , i.e., s ↔ ∆ b t , and let C , a , D be a ↔ ∆ b -colou red trace of s . T hen a 6 = τ or C 6 = D , and there are states s ′′ and s ′ with s − ։ s ′′ a − → s ′ , such that s ′′ , s ∈ C and s ′ ∈ D . As ↔ ∆ b satisfies (T), by Lemma 4.2 there is a state t † with t − ։ t † and s ′′ ↔ ∆ b t † . Therefore there exist states t ′′ and t ′ such that ( t − ։ ) t † − ։ t ′′ ( a ) − − → t ′ , s ′′ ↔ ∆ b t ′′ and s ′ ↔ ∆ b t ′ . As ↔ ∆ b has the stutter ing property and t ′′ ↔ ∆ b s ′′ ↔ ∆ b s ↔ ∆ b t , all states o n th e τ -path from t to t ′′ ha ve the same colou r as s . Hence C, a , D is a ↔ ∆ b -colou red trace of t . No w su ppose s and t ha ve the sa me col our and s is ↔ ∆ b -di ver gent. Then there is an infinite s equence of stat es ( s k ) k ∈ ω such that s = s 0 , s k τ − → s k +1 and s k ↔ ∆ b s ↔ ∆ b t for all k ∈ ω . As ↔ ∆ b satisfies (D), this implies that there exist s an infinite seque nce of states ( t ℓ ) ℓ ∈ ω such that t = t 0 , t ℓ τ − → t ℓ +1 and s k ↔ ∆ b t ℓ for all k , ℓ ∈ ω . It follo ws that t ℓ ↔ ∆ b t for all ℓ ∈ ω , and hence t is ↔ ∆ b -di ver gent.  6. Modal characterisations of BB ∆ W e shall now establ ish agreement between the relationa l and modal characterisa tions of BB ∆ propo sed in [4]. The class of formulas Φ ∆ jb of the mod al logic f or BB ∆ propo sed in [4] is g enerated b y the g rammar obtain ed by addin g the follo wing clause to the grammar in (1) of Section 2: ϕ ::= ∆ ϕ ( ϕ ∈ Φ ∆ jb ). (2) W e exten d the inducti ve definition of va lidity in Section 2 w ith: R.J. va n Glabbeek, B. Luttik, N. T r ˇ cka / Branc hing bisimilarity with explicit diver gence 17 (i v) s | = ∆ ϕ iff there ex ists an infinite sequence ( s k ) k ∈ ω of states such that s − ։ s 0 , s k τ − → s k +1 and s k | = ϕ for all k ∈ ω . Again, v alidit y induces an equi val ence on states: we define ≈ ∆ ⊆ S × S by s ≈ ∆ t iff ∀ ϕ ∈ Φ ∆ jb . s | = ϕ ⇔ t | = ϕ . W e shall no w establish that ≈ ∆ coinci des with ↔ ∆ b . Theor em 6.1. For all state s s and t : s ↔ ∆ b t iff s ≈ ∆ t . Pro of T o establis h the implicatio n from left to right, w e prov e by structura l inductio n on ϕ that if s ↔ ∆ b t and s | = ϕ , then t | = ϕ . T here are four cases to consid er . 1. Suppose ϕ = ¬ ψ and s | = ϕ . Then s 6| = ψ . As t ↔ ∆ b s , it fol lo ws by the induction hyp othesis that t 6| = ψ , and hence t | = ϕ . 2. Suppose s | = V Ψ . T hen, for all ψ ∈ Ψ , s | = ψ , and by induction t | = ψ . This yields t | = φ . 3. Suppose ϕ = ψ a χ and s | = ϕ . T hen there e xist states s ′ and s ′′ such that s − ։ s ′′ ( a ) − − → s ′ , s ′′ | = ψ and s ′ | = χ . By Lemma 4.2, there e xists a state t † such that t − ։ t † and s ′′ ↔ ∆ b t † . From this it follo ws that th ere exis t stat es t ′ and t ′′ such that t − ։ t ′′ ( a ) − − → t ′ , s ′′ ↔ ∆ b t ′′ and s ′ ↔ ∆ b t ′ , for if a = τ and s ′ = s ′′ we can take t ′ = t ′′ = t † and otherwise , since s ′′ ↔ ∆ b t † , the states t ′ and t ′′ exi st by (T). It follo w s by the induction hypothes is that t ′′ | = ψ and t ′ | = χ , and hence t | = ϕ . 4. Suppose ϕ = ∆ ψ and s | = ϕ . Then there exist s an infinite sequence ( s k ) k ∈ ω such that s − ։ s 0 , s k τ − → s k +1 and s k | = ψ for all k ∈ ω . By Lemma 4.2, there e xists a s tate t 0 such th at t − ։ t 0 and s 0 ↔ ∆ b t 0 . F rom C orollar y 4.3 it follo ws that ↔ ∆ b satisfies (D 3 ), s o there e xist an in finite seque nce of st ates ( t ℓ ) ℓ ∈ ω and a mapping σ : ω → ω such that t ℓ τ − → t ℓ +1 and s σ ( ℓ ) ↔ ∆ b t ℓ for a ll ℓ ∈ ω . By the induct ion hypot hesis t ℓ | = ψ for all ℓ ∈ ω , and hence t | = ϕ . For the implication from right to left, it suf fi ces by Corollary 4.1 to prove that ≈ ∆ is symmetric and satisfies the cond itions (T) and (D 4 ). That ≈ ∆ is symmetric is clear from its definit ion. T o establ ish conditio n (T) of Definit ion 3.1, suppo se that s ≈ ∆ t and s a − → s ′ , an d define sets T ′′ and T ′ as follo w s: T ′′ = { t ′′ ∈ S | t − ։ t ′′ & s 6≈ ∆ t ′′ } ; and T ′ = { t ′ ∈ S | ∃ t ′′ ∈ S . t − ։ t ′′ ( a ) − − → t ′ & s ′ 6≈ ∆ t ′ } . For eac h t ′′ ∈ T ′′ let ϕ t ′′ be a formula such that s | = ϕ t ′′ and t ′′ 6| = ϕ t ′′ , and let ϕ = V { ϕ t ′′ | t ′′ ∈ T ′′ } . Similarly , for each t ′ ∈ T ′ let ψ t ′ be a formula with s ′ | = ψ t ′ and t ′ 6| = ψ t ′ , an d let ψ = V { 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 follo ws that t ′′ 6∈ T ′′ , so s ≈ ∆ t ′′ ; from t ′ | = ψ it follo ws that t ′ 6∈ T ′ , so s ′ ≈ ∆ t ′ . Thereb y , condition (T) is establish ed. T o establish condition (D 4 ), suppose that s ≈ ∆ t and that ther e e xists an infinite seq uence ( s k ) k ∈ ω such that s = s 0 , s k τ − → s k +1 and s k ≈ ∆ t for all k ∈ ω . Define the set T ∞ by T ∞ = { t ′ ∈ S | t − ։ t ′ & s 6≈ ∆ t ′ } . 18 R.J. va n Glabbeek, B. Luttik, N. T r ˇ cka / Branc hing bisimilarity with explicit diver gence For each t ′ ∈ T ∞ let ϕ t ′ be a formula such that s | = ϕ t ′ and t ′ 6| = ϕ t ′ , and let ϕ = V { ϕ t ′ | t ′ ∈ T ∞ } . Since s 0 = s | = ϕ and s k ≈ ∆ t ≈ ∆ s , it follo ws that s k | = ϕ for all k ∈ ω , and therefor e s | = ∆ ϕ . Hence, t | = ∆ ϕ , so there e xists an infinite seq uence ( t ℓ ) ℓ ∈ ω such th at t − ։ t 0 , t ℓ τ − → t ℓ +1 and t ℓ | = ϕ for all ℓ ∈ ω . It follo ws that t ℓ 6∈ 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 so me k ∈ ω . Thereb y , also con dition (D 4 ) is establ ished.  W e already mention ed in Section 2 the result of Laroussi nie, Pinchin at & Schnoebelen [8] that the modal logic with negation, bin ary con junction and h ˆ a i and the logic w ith neg ation, binary conjunct ion and a are equally ex pressi ve. Below , we ada pt their method to show tha t replacin g a by h ˆ a i or h a i in the modal logic for BB ∆ propo sed in [4] also yields an equally exp ressi ve logic. Hencefor th we deno te by Φ ∆ u the set of formulas generated by the gra mmar that is obtaine d when replac ing ϕ a ϕ by ϕ h a i ϕ in the grammar for Φ ∆ jb (see (1) in S ection 2 and (2) at the beginni ng of this sectio n). The central idea, from [8], is that any formula in Φ ∆ jb can be w ritten as a Boolean combination of formulas that propaga te either upwards or do wnwards along a path of τ -transitio ns. A formula ϕ that propag ates upwar ds, i.e., w ith the property that if s − ։ t and s | = ϕ , then also t | = ϕ , we shall call an upwar d formula . A formu la ϕ that prop agates downwa rds, i.e., with the property that if s − ։ t and t | = ϕ , then also s | = ϕ , we shall call a downwar d formula . Lemma 6.1. Every ϕ ∈ Φ ∆ jb is equi vale nt with a formula of the f orm W Φ , where each formula in Φ is a conjun ction of an upwar d and a downwa rd formula. Pro of Note that ψ a χ and ∆ ψ are downw ard formulas and that the negation of a downwa rd formula is an upward formula. Furthermore, a conjunction of upward formulas is again an upward formula and a conjunct ion of do wnward formulas is again a downwa rd formula . It fo llo ws, by the st andard la ws of Boolean algebra , that the formula ϕ is equiv alent to a formula of the desired shape.  The proof that for ev ery formula ϕ ∈ Φ ∆ u there exists an equiv alent formula ϕ ′ ∈ Φ ∆ jb procee ds by induct ion on the structure of ϕ , and the only nontri vial case is when ϕ = ψ h a i χ . Acc ording to the inducti on hypothesis, for ψ and χ there exist equiv alent formulas in Φ ∆ jb , so, by Lemma 6.1, ψ is equi v alent to a disjunctio n of conjunc tions of upward and do wnward formulas. The proof in [8] then relies on th ese disjunctio ns being finite. T o general ise it to infinite disju nctions , we shall use the follo wing lemma. Lemma 6.2. Let Φ be a set of formulas and let ϕ be a formula. Then ( W Φ) h a i ϕ ⇚ ⇛ W { ( W Φ ′ ) h a i ϕ | Φ ′ a finite subs et of Φ } . Pro of ( ⇛ ) Suppose s | = ( W Φ) h a i ϕ . Then there exi st states s 0 , . . . , s n , s n +1 such that s = s 0 τ − → · · · τ − → s n ( a ) − − → s n +1 , s i | = W Φ for al l i = 0 , . . . , n and s n +1 | = ϕ . Since s i | = W Φ , we can associa te with e very s i ( i = 0 , . . . , n ) a formula ϕ i ∈ Φ such th at s i | = ϕ i . L et Φ ′ = { ϕ i | i = 0 , . . . , n } ; then Φ ′ is a finite subset of Φ such th at s i | = W Φ ′ for eve ry i = 0 , . . . , n . It follo ws that s | = ( W Φ ′ ) h a i ϕ , and hence s | = W { ( W Φ ′ ) h a i ϕ | Φ ′ a finite subset of Φ } . R.J. va n Glabbeek, B. Luttik, N. T r ˇ cka / Branc hing bisimilarity with explicit diver gence 19 ( ⇚ ) If s | = W { ( W Φ ′ ) h a i ϕ | Φ ′ a finite subset of Φ } , then s | = ( W Φ ′ ) h a i ϕ for some finite subs et Φ ′ of Φ . So there exist states s 0 , . . . , s n , s n +1 such that s = s 0 τ − → · · · τ − → s n ( a ) − − → s n +1 , s i | = W Φ ′ for all i = 0 , . . . , n and s n +1 | = ϕ . Since s i | = W Φ ′ implies s i | = W Φ for all i = 0 , . . . , n , it follo ws that s | = ( W Φ) h a i ϕ .  W e no w adap t the metho d in [8] and sho w that repl acing a by h ˆ a i or h a i in the modal log ic for BB ∆ propo sed in [4] yields an equa lly exp ressi ve logic. Theor em 6.2. For e very formul a ϕ ∈ Φ ∆ u there exist s an equiv alent formula ϕ ′ ∈ Φ ∆ jb . Pro of The proof is by s tructural induct ion on ϕ ; the onl y nontri vial case is when ϕ = ψ h a i χ . By the induct ion hypothesis there ex ist formulas ψ ′ , χ ′ ∈ Φ ∆ jb such th at ψ ⇚ ⇛ ψ ′ and χ ⇚ ⇛ χ ′ . By Lemma 6.1, ψ ′ ⇚ ⇛ W Ψ , where each formula in Ψ is a conjun ction of an upward and a downwar d formula. Hence, by the e vident congru ence propert y of ⇚ ⇛ and L emma 6.2, ϕ ⇚ ⇛ W { ( W Ψ ′ ) h a i χ ′ | Ψ ′ a finite subset of Ψ } . Clearly , it no w suffices to establish that ( W Ψ ′ ) h a i χ ′ is equiv alent to a formula in Φ ∆ jb , for all finite subset s Ψ ′ of Ψ . R ecall that Ψ consists of conjun ctions of an upw ard and a do wnward formula, so we can assume that Ψ ′ = { ψ u i ∧ ψ d i | i = 1 , . . . , n } ; we proceed by inductio n on the cardinalit y of Ψ ′ . If | Ψ ′ | = 0 , then  _ Ψ ′  h a i χ ′ ⇚ ⇛ ⊥ , and ⊥ ∈ Φ ∆ jb . Suppose | Ψ ′ | > 0 . B y the induct ion hypothesis there exis ts, for eve ry i = 1 , . . . , n , a formula ϕ ′ i ∈ Φ ∆ jb such that  _ Ψ ′ − { ψ u i ∧ ψ d i }  h a i χ ′ ⇚ ⇛ ϕ ′ i . Then, it is easy to v erify that  _ Ψ ′  h a i χ ′ ⇚ ⇛ n _ i =1  ψ u i ∧  ψ d i a χ ′ ∨ ψ d i τ ϕ ′ i  , and the righ t-hand side formula is in Φ ∆ jb . Some intu ition for this last step is offere d in [8].  In the same va in, there is also an ob vious stren gthenin g of the div er gence modality ∆ . Let b ∆ be the unary di ver gence modality with the follo wing definitio n: (i v ′ ) s | = b ∆ ϕ iff there exists an infinite sequen ce ( s k ) k ∈ ω of states such that s = s 0 , s k τ − → s k +1 and s k | = ϕ for all k ∈ ω . W e deno te by Φ b ∆ jb the set of formulas genera ted by the grammar in (1) with ∆ ϕ repla ced by b ∆ ϕ . Note that the modalit y ∆ can be express ed in terms of b ∆ : ∆ ϕ ⇚ ⇛ ⊤ τ b ∆ ϕ . 20 R.J. va n Glabbeek, B. Luttik, N. T r ˇ cka / Branc hing bisimilarity with explicit diver gence s 1 s 2 s 3 τ τ τ τ s = s 0 a 2 a 3 a 4 a 1 b 0 b 1 b 2 b 3 t 1 t 2 t 3 t 0 u 0 u 1 u 2 u 3 Figure 11. A divergence. A crucia l step in our adaptation of the method of L arouss inie, P inchin at & Schnoebele n ab ov e con- sisted of sho wing that infinite disjunc tions in the left ar gument of h a i can be av oided. If infinite dis- juncti ons cou ld also be av oided as an arg ument of b ∆ , then a furthe r ada ptation of the method would be possib le, showin g that replacin g ∆ by b ∆ in the modal logic for BB ∆ would yield a logic w ith equal exp ressi vity . Howe ver , the follo wing examp le suggests that infinite disjun ctions under b ∆ cannot always be av oided. Example 6.1. Let a 1 , a 2 , a 3 , . . . and b 0 , b 1 , b 2 , . . . be infinite sequence s of distinct actions and consider the formula ϕ = b ∆ ∞ _ i =0 ( ¬ ( ⊤ h a i i ⊤ ) ∧ ( ⊤ h b i i ⊤ )) ! . The formula ϕ holds in a state iff there exists an infinite τ -path such that in ev ery state there is an i ≥ 0 such that the action b i is still pos sible, whereas th e acti on a i is not. Note th at ϕ holds in the state s of the t ransition sys tem in Fig ure 11; each of th e disju ncts ¬ ( ⊤ h a i i ⊤ ) ∧ ( ⊤ h b i i ⊤ ) holds in precise ly one state. W e conjectur e that the formula of E xample 6.1 is not equi va lent to a formula in Φ ∆ jb , and that, henc e, replac ing ∆ by b ∆ in the modal logic for BB ∆ yields a strictly more expre ssi ve logic. W e conclude the paper with a pro of that the equi v alence ≈ b ∆ ⊆ S × S induce d on states by va lidity of formulas in Φ b ∆ jb , defined by s ≈ b ∆ t iff ∀ ϕ ∈ Φ b ∆ jb . s | = ϕ ⇔ t | = ϕ , ne vert heless also coincides w ith ↔ ∆ b . Theor em 6.3. For all states s and t : s ↔ ∆ b t iff s ≈ b ∆ t . Pro of For the implicati on from left to right, we prov e by struc tural inducti on on ϕ that if s ↔ ∆ b t and s | = ϕ , then t | = ϕ . W e only treat the case ϕ = b ∆ ψ , for the cases ϕ = ¬ ψ , ϕ = V Ψ and ϕ = ψ a χ are alread y treated in the proo f of Theorem 6.1. So, suppose ϕ = b ∆ ψ and s | = ϕ . Then there e xists an infinite se quence ( s k ) k ∈ ω of states such that s = s 0 , s k τ − → s k +1 and s k | = ψ for all k ∈ ω . F rom R.J. va n Glabbeek, B. Luttik, N. T r ˇ cka / Branc hing bisimilarity with explicit diver gence 21 Corollary 4.3 it follo ws that ↔ ∆ b satisfies (D 3 ), so ther e exi st an infinite seque nce of states ( t ℓ ) ℓ ∈ ω and a m apping σ : ω → ω such th at t = t 0 , t ℓ τ − → t ℓ +1 and s σ ( ℓ ) ↔ ∆ b t ℓ for all ℓ ∈ ω . By the inducti on hypot hesis t ℓ | = ψ for all ℓ ∈ ω , and hence t | = ϕ . T o est ablish the impli cation from right t o left, no te that if s ≈ b ∆ t , then, since e very fo rmula in Φ ∆ jb is equi valen t to a formula in Φ b ∆ jb , also s ≈ ∆ t , so by T heore m 6.1 it follo ws that s ↔ ∆ b t .  Comment on Definition 2.3 I f in Definition 2.3 we had used a notion of equiv alence between moda l formulas ϕ and ψ that merely require s that s | = ϕ ⇔ s | = ψ for all states s in the presuppo sed labelled transit ion system, rather than quantifyin g ov er all labelled transitio n systems, the resulti ng concept of equall y expressi ve logics would be much weaker , an d the logics Φ ∆ jb and Φ b ∆ jb would be equally expressi ve. In general, let ∼ be an equ iv alence on the set of states S , and con sider two logics L 1 and L 2 that both ha ve neg ation a nd arbitrary infinite conjunct ion, and both c haracteri se ∼ . For ev ery pair of st ates s, t ∈ S with s 6∼ t take a formula ϕ s,t from L 1 such that s | = ϕ s,t b ut t 6| = ϕ s,t . Then χ s = V { ϕ s,t | t 6∼ s } is called a char acteristic formul a of s : one has t | = χ s if f t ∼ s . Now let ψ be a formula from L 2 . Then W { χ s | s | = ψ } is equi valen t to ψ , in the sense that t | = ψ ⇔ t | = W { χ s | s | = ψ } for all states t ∈ S . This pro ves that the two lo gics are equally express iv e. Similar reas oning using the notion of eq ui va lence from Definition 2.3 wou ld break do wn, because one canno t tak e conjunctio ns of a proper class of formula. Refer ences [1] T . Basten (199 6): Branching bisimilarity is an equivalen ce indeed! Inform ation Processing Letters 58(3) , pp. 141–1 47. [2] R. De Nicola & F . W . V aandrag er ( 1995) : Three lo gics for br anching bisimulation. Journal of the A CM 42(2) , pp. 458–48 7. [3] R. Gerth, R. Kuiper, D. Peled & W . Penczek (199 9): A p artial or d er appr oach to branching time logic model chec king. Inform ation and Computatio n 150(2 ), pp. 132–1 52. 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