Computation Tree Logic with Deadlock Detection

Transcription

1 Comuttion Tree Logic with Dedlock Detection Rob vn Glbbeek 1,2, Bs Luttik 3,4 & Nikol Trčk 3 1 Ntionl ICT Austrli, Sydney, Austrli 2 School of Comuter Science nd Engineering, Univ. of New South Wles, Sydney 3 Det. of Mth. & Com. Sc., Technische Universiteit Eindhoven, The Netherlnds 4 CWI, The Netherlnds Abstrct. We study the equivlence reltion on sttes of lbelled trnsition systems of stisfying the sme formuls in Comuttion Tree Logic without the next stte modlity (CTL X ). This reltion is obtined by De Nicol & Vndrger by trnslting lbelled trnsition systems to Krike structures, while lifting the totlity restriction on the ltter. They chrcterised it s divergence sensitive brnching bisimultion equivlence. We find tht this equivlence fils to be congruence for interleving rllel comosition. The reson is tht the roosed liction of CTL X to non-totl Krike structures lcks the exressiveness to coe with dedlock roerties tht re imortnt in the context of rllel comosition. We roose n extension of CTL X, or n lterntive tretment of nontotlity, tht fills this hitus. The equivlence induced by our extension is chrcterised s brnching bisimultion equivlence with exlicit divergence, which is, moreover, shown to be the corsest congruence contined in divergence sensitive brnching bisimultion equivlence. 1 Introduction CTL [5] is owerful temorl logic combining liner time nd brnching time modlities; it generlises the brnching time temorl logic CTL [4]. CTL is interreted in terms of Krike structures. As the next stte modlity X is incomtible with bstrction of the notion of stte, it is often excluded in high-level secifictions. By CTL X we denote CTL without this modlity. To chrcterise the equivlence induced on sttes of Krike structures by vlidity of CTL X formuls, Browne, Clrke & Grumberg [2] defined the notion of stuttering equivlence. They roved tht two sttes in finite Krike structure re stuttering equivlent if nd only if they stisfy the sme CTL X formuls, nd moreover, they estblished tht this is lredy the cse if nd only if the two sttes stisfy the sme CTL X formuls. There is n intuitive corresondence between the notions of stuttering equivlence on Krike structures nd brnching bisimultion equivlence on lbelled trnsition systems (LTSs) s defined in [8]. De Nicol & Vndrger [3] hve rovided frmework for constructing nturl trnsltions between LTSs nd Krike structures in which this corresondence cn be formlised. Stuttering equivlence corresonds in their frmework to divergence sensitive vrint

2 2 Rob vn Glbbeek, Bs Luttik & Nikol Trčk of brnching bisimultion equivlence, nd conversely, brnching bisimultion equivlence corresonds to divergence blind vrint of stuttering equivlence. The ltter chrcterises the equivlence induced on sttes of Krike structures by divergence blind vrint of vlidity of CTL X formuls. In [4,5, 2] nd other work on CTL, Krike structures re required to be totl, mening tht every stte hs n outgoing trnsition. These corresond with LTSs tht re dedlock-free. In the world of LTSs requiring dedlock-freeness is considered serious limittion, s dedlock is introduced by useful rocess lgebric oertors like the restriction of CCS nd the synchronous rllel comosition of CSP. Concetully, dedlock my rise s the result of n unsuccessful synchronistion ttemt between rllel comonents, nd often one wnts to verify tht the result of rllel comosition is dedlock-free. This is ossible only when working in model of concurrency where dedlocks cn be exressed. Through the trnsltions of [3] it is ossible to define the vlidity of CTL X formuls on sttes of LTSs. To ly CTL X-formuls to LTSs tht my contin dedlocks we consider Krike structures with dedlocks s well, nd hence lift the requirement of totlity. Following [3], we do so by using mximl ths insted of infinite ths in the definition of vlidity of CTL X formuls. Brring further chnges, this mounts to the ddition of self loo to every dedlock stte. As consequence, CTL X formuls cnnot see the difference between stte without outgoing trnsitions ( dedlock) nd one whose only outgoing trnsition constitutes self loo ( livelock), nd ccordingly dedlock stte is stuttering equivlent to livelock stte tht stisfies the sme tomic roositions. This er will chllenge the wisdom of this set-u. We observe tht for systems with dedlock, the divergence sensitive brnching bisimultion equivlence of [3] fils to be congruence for interleving oertors. We chrcterise the corsest congruence contined in divergence sensitive brnching bisimultion equivlence s the brnching bisimultion equivlence with exlicit divergence, introduced in [8]. This equivlence differs from divergence sensitive brnching bisimultion equivlence in tht it distinguishes dedlock nd livelock. For dedlock-free systems the equivlences coincide. Tht divergence sensitive brnching bisimultion equivlence is not congruence for interleving oertors mens tht there re roerties of concurrent systems, ertining to their dedlock behviour, tht cnnot be exressed in CTL X, but tht cn be exressed in terms of the vlidity of CTL X formul on the result of utting these systems in given context involving n interleving oertor. We find this unstisfctory, nd therefore roose n extension of CTL X in which this tye of roerty cn be exressed directly. We obtin tht two sttes re brnching bisimultion equivlent with exlicit divergence if nd only if they stisfy the sme formuls in the resulting logic. Treting CTL X in the sme wy leds either to n extension of CTL X or, equivlently, to modifiction of its semntics. The new semntics we roose for CTL X is vlid extension of the originl semntics [4] to non-totl Krike structures. It slightly differs from the semntics of [3] nd it is rgubly better suited to del with dedlock behviour.

3 Comuttion Tree Logic with Dedlock Detection 3 Insted of extending CTL X or modifying CTL X we lso chieve the sme effect by mending the trnsltion from LTSs to Krike structures in such wy tht every LTS ms to totl Krike structure. 2 CTL X nd stuttering equivlence We resuose set AP of tomic roositions. A Krike structure is tule (S, L, ) consisting of set of sttes S, lbelling function L : S 2 AP nd trnsition reltion S S. For the reminder of the section we fix Krike structure (S, L, ). A finite th from s is finite sequence of sttes s 0,..., s n such tht s = s 0 nd s k s k+1 for ll 0 k < n. An infinite th from s is n infinite sequence of sttes s 0, s 1, s 2,... such tht s = s 0 nd s k s k+1 for ll k ω. A th is finite or infinite th. A mximl th is n infinite th or finite th s 0,..., s n such tht s. s n s. We write π π if the th π is suffix of the th π, nd π π if π π nd π π. Temorl roerties of sttes in S re defined using CTL X formuls. Definition 1. The clsses Φ of CTL X stte formuls nd Ψ of CTL X th formuls re generted by the following grmmr: ϕ ::= ϕ Φ ψ ψ ::= ϕ ψ Ψ ψ U ψ with AP, ϕ Φ, Φ Φ, ψ Ψ nd Ψ Ψ. In cse the crdinlity of the set of sttes of our Krike structure is less thn some infinite crdinl κ, 1 we my require tht Φ < κ nd Ψ < κ in conjunctions, thus obtining set of formuls rther thn roer clss. Normlly, S is required to be finite, nd ccordingly CTL X dmits finite conjunctions only. Definition 2. We define when CTL X stte formul ϕ is vlid in stte s (nottion: s = ϕ) nd when CTL X th formul ψ is vlid on mximl th π (nottion: π = ψ) by simultneous induction s follows: s = iff L(s); s = ϕ iff s = ϕ; s = Φ iff s = ϕ for ll ϕ Φ ; s = ψ iff there exists mximl th π from s such tht π = ψ; π = ϕ iff s is the first stte of π nd s = ϕ; π = ψ iff π = ψ; π = Ψ iff π = ψ for ll ψ Ψ ; nd π = ψ U ψ iff there exists suffix π of π such tht π = ψ, nd π = ψ for ll π π π. A formul ψ U ψ sys tht, long given th, ψ holds until ψ holds. One writes for the emty conjunction (which is lwys vlid), Fψ for U ψ ( ψ will hold eventully ) nd Gψ for F ψ ( ψ holds lwys (long th) ). 1 In fct it suffices to require tht for every stte s the crdinlity of the set of sttes rechble from s is less thn κ.

4 4 Rob vn Glbbeek, Bs Luttik & Nikol Trčk The bove is the stndrd, divergence sensitive interrettion of CTL X [5,2], but extended to Krike structures tht re not required to be totl. Following [3], this is chieved by using mximl ths in the definition of vlidity of CTL X formuls, insted of the trditionl use of infinite ths [5, 2]. The resulting definition generlises the trditionl one, becuse for totl Krike structures th is mximl iff it is infinite. An equivlent wy of thinking of this generlistion of CTL X to non-totl Krike structures is by mens of trnsformtion tht mkes Krike structure K totl by the ddition of self loo s s to every dedlock stte s, together with the convention tht formul is vlid in stte of K iff it is vlid in the sme stte of the totl Krike structure obtined by this trnsformtion. It is not hrd to check tht this yields the sme notion of vlidity s our Definition 2. The divergence blind interrettion of [3] (nottion: s = db ϕ nd π = db ψ) is obtined by droing the word mximl in the fourth cluse of Definition 2. Definition 3. A colouring is function C : S C, for C ny set of colours. Given colouring C nd (finite or infinite) th π = s 0, s 1, s 2,... from s, let C(π) be the sequence of colours obtined from C(s 0 ), C(s 1 ), C(s 2 ),... by contrcting ll its (finite or infinite) mximl consecutive subsequences C, C, C,... to C. The sequence C(π) is clled C-coloured trce of s; it is comlete if π is mximl. A colouring C is [fully] consistent if two sttes of the sme colour lwys stisfy the sme tomic roositions nd hve the sme [comlete] C-coloured trces. Two sttes s nd t re divergence blind stuttering equivlent, nottion s dbs t, if there exists consistent colouring C such tht C(s) = C(t). They re (divergence sensitive) stuttering equivlent, nottion s s t, if there exists fully consistent colouring C such tht C(s) = C(t). The difference between dbs nd s is illustrted in the following exmle. Exmle 1. Consider the Krike structure nd its colouring deicted in Figure 1. This colouring is consistent, imlying s dbs t dbs u nd x dbs y, but it is not fully consistent becuse stte t hs comlete trce while u does not. Note tht t hs, due to the self loo, comlete coloured trce tht consists of just the colour of -lbelled stte, wheres the unique comlete coloured trce of u contins the colour of q-lbelled stte too. Since consistent colouring ssigns different colours to sttes with different lbels, every fully consistent colouring must ssign different colours to sttes t nd u, i.e. it must be tht t s u. One such colouring is given in Figure 1b. This colouring shows tht x s y. Lemm 1. Let C be colouring such tht two sttes with the sme colour stisfy the sme tomic roositions nd hve the sme C-coloured trces of length two. Then C is consistent. Proof. Suose C(s 0 ) = C(t 0 ) nd C 0, C 1, C 2,... is n infinite coloured trce of s 0. Then, for i > 0, there re sttes s i nd finite ths π i from s i 1 to s i, such tht C(π i ) = C i 1, C i. With induction on i, for i > 0 we find sttes t i with

5 Comuttion Tree Logic with Dedlock Detection 5 ) s b) s t u t u x q q y x q q y Fig.1. Difference between ) dbs nd b) s. C(s i ) = C(t i ) nd finite ths ρ i from t i 1 to t i such tht C(ρ i ) = C i 1, C i. Nmely, the ssumtion bout C llows us to find ρ i given t i 1, nd then t i is defined s the lst stte of ρ i. Conctenting ll the ths ρ i yields n infinite th ρ from t 0 with C(ρ) = C 0, C 1, C 2,.... The cse tht C(s 0 ) = C(t 0 ) nd C 0,..., C n is finite coloured trce of s 0 goes likewise. Lemm 2. Let C be colouring such tht two sttes with the sme colour stisfy the sme tomic roositions nd hve the sme C-coloured trces of length two, nd the sme comlete C-coloured trces of length one. Then C is fully consistent. Proof. Suose C(s) = C(t) nd σ is comlete C-coloured trce of s. Then σ = C(π) for mximl th π from s. By Lemm 1, σ is lso C-coloured trce of t. It remins to show tht it is comlete C-coloured trce of t. Let ρ be th from t with C(ρ) = σ. If ρ is infinite, we re done. Otherwise, let t be the lst stte of ρ. Then C(t ) is the lst colour of σ. Therefore, there is stte s on π such tht the suffix π of π strting from s is mximl th with C(π ) = C(s ) = C(t ). By the ssumtion bout C, C(t ) must lso be comlete C-coloured trce of t, i.e. there is mximl th ρ from t with C(ρ ) = C(t ). Conctenting ρ nd ρ yields mximl th ρ from t with C(ρ ) = σ. The following two theorems were roved in [3] nd [2], resectively, for sttes s nd t in finite Krike structure. Here we dro the finiteness restriction. Theorem 1. s dbs t iff s = db ϕ t = db ϕ for ll CTL X stte formuls ϕ. Proof. Only if : Let C be consistent colouring. With structurl induction on ϕ nd ψ we show tht C(s) = C(t) (s = db ϕ t = db ϕ) nd C(π) = C(ρ) (π = db ψ ρ = db ψ). The cse ϕ = for AP follows immeditely from Definition 3. The cses ϕ = ϕ nd ϕ = Φ follow immeditely from the induction hyothesis. Suose C(s) = C(t) nd s = db ψ. Then there exists th π from s such tht π = db ψ. C(π) is coloured trce of s, nd hence of t. Thus there must be th ρ from t with C(π) = C(ρ). By induction, ρ = db ψ. Hence, t = db ψ.

6 6 Rob vn Glbbeek, Bs Luttik & Nikol Trčk The cse ϕ Φ follows since the first sttes of two ths with the sme colour lso hve the sme colour. The cses ψ = ψ nd ψ = Ψ follow immeditely from the induction hyothesis. Finlly, suose C(π) = C(ρ) nd π = db ψ U ψ. Then there exists suffix π of π such tht π = db ψ nd π = db ψ for ll π π π. As C(π) = C(ρ), there must be suffix ρ of ρ such tht C(π ) = C(ρ ) nd for every th ρ such tht ρ ρ ρ there exists th π with π π π such tht C(π ) = C(ρ ). By induction, this imlies ρ = db ψ nd ρ = db ψ for ll ρ ρ ρ. Hence ρ = db ψ U ψ. If : Let C be the colouring given by C(s) = {ϕ Φ s = db ϕ}. It suffices to show tht C is consistent. So suose C(s) = C(t). Trivilly, s nd t stisfy the sme tomic roositions. By Lemm 1 it remins to show tht s nd t hve the sme coloured trces of length two. Suose s hs coloured trce C, D. Let s 0,...,s k be th from s such tht C(s i ) = C for 0 i < k nd C(s k ) = D C. Let U = {u there is th from t to u nd C(u) C}, V = {v there is th from t to v nd C(v) D}. For every u U ick CTL X formul ϕ u C C(u) (using negtion on formul in C(u) C if needed), nd for every v V ick CTL X formul ϕ v D C(v). Now s = db ( u U ϕ u) U ( v V ϕ v) nd, s C(s) = C(t), lso t = db ( u U ϕ u ) U ( v V ϕ v ). Thus, there is th t 0,...,t l from t such tht t l = db v V ϕ v nd t j = db u U ϕ u for 0 j < l. It follows tht t l V nd t j U for 0 j < l. Hence C(t l ) = D nd C(t j ) = C for 0 j < l, so C, D is lso coloured trce of t. Theorem 2. s s t iff s = ϕ t = ϕ for ll CTL X stte formuls ϕ. Proof. Only if goes exctly s in the revious roof, reding = for = db, but requiring C to be fully consistent nd, in the second rgrh, the ths π nd ρ to be mximl nd C(π) to be comlete coloured trce of s nd t. If goes s in the revious roof, but this time we hve to show tht C is fully consistent. Thus, lying Lemm 2, nd ssuming C(s) = C(t), we dditionlly hve to show tht s nd t hve the sme comlete coloured trces of length one. Let π be mximl th from s with C(π) = C. Let U = {u there is th from t to u nd C(u) C}. For every u U ick CTL X formul ϕ u C C(u). Now s = G( u U ϕ u) nd, s C(s) = C(t), lso t = G( u U ϕ u ). Thus, there is mximl th ρ from t such tht t = u U ϕ u for ll sttes t in ρ. It follows tht t U. Hence C(t ) = C nd thus C(ρ) = C. Corollry 1. dbs nd s re equivlence reltions. Note tht, for ny colouring C, the C-coloured trces of stte s re comletely determined by the comlete C-coloured trces of s, nmely s their refixes.

7 Comuttion Tree Logic with Dedlock Detection 7 Hence, ny colouring tht is fully consistent is certinly consistent, nd thus s is finer (i.e. smller, more discriminting) equivlence reltion thn dbs. Above, the divergence blind interrettion of CTL X is defined by using ths insted of mximl ths. It cn equivlently be defined in terms of trnsformtion on Krike structures, nmely the ddition of self loo s s for every stte s. Now s dbs t holds in certin Krike structure iff s s t holds in the Krike structure obtined by dding ll these self loos. This is becuse the colour of th doesn t chnge when self-loos re dded to it, nd u to self loos ny th is mximl. Likewise, s = db ϕ in the originl Krike structure iff s = ϕ in the modified one. Just like dbs cn be exressed in terms of s by mens of trnsformtion on Krike structures, by mens of different trnsformtion, t lest for finite Krike structures, s cn be exressed in terms of dbs. This is done in [3], Definitions nd Brnching bisimultion in terms of coloured trces We resuose set A of ctions with secil element A. A lbelled trnsition system (LTS) is structure (S, ) consisting of set of sttes S nd trnsition reltion S A S. For the reminder of the section we fix n LTS (S, ). We write s s for (s,, s ). A th from s is n lternting sequence s 0, 1, s 1, 2, s 2,... of sttes nd ctions, ending with stte if the sequence is finite, such tht s = s 0 nd s k k 1 sk for ll relevnt k > 0. A mximl th is n infinite th or finite th s 0, 1, s 1, 2,..., n, s n such tht, s. s n s. We write π π if the th π is suffix of the th π, nd π π if π π nd π π. Definition 4. A colouring is function C : S C, for C ny set of colours. For π = s 0, 1, s 1, 2, s 2,... th from s, let C(π) be the lternting sequence of colours nd ctions obtined from C(s 0 ), 1, C(s 1 ), 2, C(s 2 ),... by contrcting ll (finite or infinite) mximl consecutive subsequences C,, C,, C,... to C. The sequence C(π) is clled C-coloured trce of s; it is comlete if π is mximl; it is divergent if it is finite whilst π is infinite. A colouring C is [fully] consistent if two sttes of the sme colour lwys hve the sme [comlete] C-coloured trces. Two sttes s nd t re (divergence blind) brnching bisimultion equivlent, nottion s b t, if there exists consistent colouring C such tht C(s) = C(t). They re divergence sensitive brnching bisimultion equivlent, nottion s λ b t, if there exists fully consistent colouring C such tht C(s) = C(t). A consistent colouring reserves divergence if two sttes of the sme colour lwys hve the sme divergent C-coloured trces. Two sttes s nd t re brnching bisimultion equivlent with exlicit divergence, nottion s b t, if there exists consistent, divergence reserving colouring C with C(s) = C(t). The difference between b, λ b, nd b is illustrted in the following exmle.

8 8 Rob vn Glbbeek, Bs Luttik & Nikol Trčk ) s t u v b) s t u v c) s t u v x y z x y z x y z Fig.2. Difference between ) b, b) λ b, nd c) b. Exmle 2. Consider first the LTS nd its colouring deicted in Figure 2. This colouring is consistent nd we hve s b t b u b v nd x b y b z. It is not fully consistent becuse stte s hs comlete trce wheres t hs not. It is esy to see tht every fully consistent colouring must ssign different colours to sttes t nd u, nd so tht t λ b u. One such colouring is given in Figure 2b nd it shows tht u λ b v nd x λ b y λ b z. Note, however, tht this colouring, lthough fully consistent, does not reserve divergence. Stte v hs divergent trce wheres u hs not, nd similrly stte z hs divergent trce wheres y hs not. Any colouring tht reserves divergence must ssign different colours to sttes u nd v nd to sttes y nd z, mening tht u b v nd y b z. One such colouring is given in Figure 2c. It shows tht x b y. In fct, these re the only two (different) sttes tht re brnching bisimultion equivlent with exlicit divergence. In the definition of b bove, consistency nd reservtion of divergence er s two serte roerties of colourings. Insted we could hve integrted them by dding n extr bit ( ) t the end of those finite coloured trces tht stem from infinite ths. Likewise, λ b could hve been defined by dding such n extr bit t the end of those finite coloured trces tht stem from mximl ths. Lemms 1 nd 2 bout colourings on Krike structures ly to lbelled trnsition systems s well. The roofs re essentilly the sme. Lemm 3. Let C be colouring such tht two sttes with the sme colour hve the sme C-coloured trces of length three (i.e. colour - ction - colour). Then C is consistent. Lemm 4. Let C be consistent colouring such tht two sttes with the sme colour hve the sme comlete C-coloured trces of length one. Then C is fully consistent. Lemm 5. Let C be consistent colouring such tht two sttes with the sme colour hve the sme divergent C-coloured trces of length one. Then C reserves divergence.

9 Comuttion Tree Logic with Dedlock Detection 9 Proof. Exctly like the roof of Lemm 2, but letting σ be divergent C-coloured trce of s; π, π infinite ths; C(t ) divergent C-coloured trce of t ; nd ρ, ρ infinite ths. Brnching bisimultion equivlence nd brnching bisimultion equivlence with exlicit divergence were originlly defined in Vn Glbbeek & Weijlnd [8]. There, only finite coloured trces re considered, nd consistent colouring ws defined s colouring with the roerty tht two sttes hve the sme colour only if they hve the sme finite coloured trces. By Lemm 3 this yields the sme concet of consistent colouring s Definition 4 bove. In [8], consistent colouring is sid to reserve divergence if no divergent stte hs the sme colour s nondivergent stte. Here stte s is divergent if it is the strting oint of n infinite th of which ll nodes hve the sme colour. This is the cse if s hs divergent coloured trce of length one. Now Lemm 5 sys tht the definition of reservtion of divergence from [8] grees with the one roosed bove. Hence the concets of brnching bisimultion nd brnching bisimultion with exlicit divergence of [8] gree with ours. Theorem 3. b, λ b nd b re equivlence reltions. Proof. We show the roof for b ; the other two cses roceed likewise. We will regrd ny equivlence reltion on S s colouring, the colour of stte being its equivlence clss. Conversely, ny colouring cn be considered s n equivlence reltion on sttes. The digonl on S (i.e., the binry reltion {(s, s) s S}) is consistent colouring, so b is reflexive. Tht b is symmetric is immedite from the required symmetry of colourings. To rove tht b is trnsitive, suose s b t nd t b u. So there exist consistent colourings C nd D with C(s) = C(t) nd D(t) = D(u). Let E be the finest equivlence reltion contining C nd D. Then E(s) = E(t) = E(u). It suffices to show tht E is consistent. First of ll note tht the E-colour of stte is comletely determined by its C-colour, s well s by its D-colour: C() = C(q) E() = E(q) nd D() = D(q) E() = E(q) for ll, q S. Thus, if two sttes hve the sme sets of C-coloured trces or the sme sets of D-coloured trces, they must lso hve the sme sets of E-coloured trces. Suose E() = E(q). Then there must be sequence of sttes ( i ) 0 i n such tht = 0, q = n nd for 0 i < n we hve either C( i ) = C( i+1 ) or D( i ) = D( i+1 ). As C nd D re consistent colourings, i nd i+1 hve the sme C-coloured trces or the sme D-coloured trces. In either cse they lso hve the sme E-coloured trces. This holds for 0 i < n, so nd q hve the sme E-coloured trces. Thus E is consistent. Lemm 6. Let C be consistent colouring nd s S. Then the comlete C- coloured trces of s consist of the C-coloured trces of s tht re infinite, divergent, or mximl, in the sense tht they cnnot be extended.

10 10 Rob vn Glbbeek, Bs Luttik & Nikol Trčk Proof. By definition, infinite nd divergent C-coloured trces of s re comlete. Let σ be mximl C-coloured trce of s, nd let π be th from s such tht C(π) = σ. Let π be n extension of π to mximl th. As σ is mximl C-coloured trce, in the sense tht it cnnot be extended, we hve C(π ) = σ. Hence σ is comlete C-coloured trce of s. Now let σ be comlete C-coloured trce of s tht is not infinite, nor divergent C-coloured trce of s. In tht cse σ = C(π) for π finite mximl th from s. Let t be the lst stte of π. We hve, t. t t. Suose, towrds contrdiction, tht σ is not mximl, i.e. there is th π from s such tht C(π ) is roer extension of σ. Then there must be stte u on π with C(u) = C(t), such tht u hs coloured trce σ of length > 1, which is suffix of C(π ). As C is consistent, σ is lso coloured trce of t, contrdicting, t. t t. As for Krike structures, for ny colouring C, the C-coloured trces of stte s re the refixes of the comlete C-coloured trces of s. Moreover, Lemm 6 sys tht the comlete C-coloured trces of stte s re comletely determined by the C-coloured trces of s together with the divergent C-coloured trces of s. Hence, ny colouring tht is consistent nd reserves divergence is lso fully consistent. Therefore, b is finer thn λ b, which is finer thn b. The difference between λ b nd b is tht only the ltter sees the difference between those mximl finite coloured trces tht stem from finite ths (ending in dedlock) nd those tht stem from infinite ths (ending in livelock). For dedlock-free LTSs (hving no sttes s with, s. s s ) the equivlences λ b nd b coincide. 4 Trnslting between Krike structures nd lbelled trnsition systems We resuose set A of ctions with secil element A, nd set AP of tomic roositions. A doubly lbelled trnsition system (L 2 TS) is structure (S, L, ) tht consists of set of sttes S, lbelling function L : S 2 AP nd lbelled trnsition reltion S A S. From n L 2 TS one nturlly obtins n LTS by omitting the lbelling function L, nd Krike structure by relcing the lbelled trnsition reltion by one from which the lbels re omitted. We cll these the LTS or Krike structure ssocited to the L 2 TS. An L 2 TS (S, L, ) is consistent if it stisfies the following three conditions: (i) if s t, then (L(s) = L(t) iff = ); (ii) if s t, s t nd L(s) = L(s ), then L(t) = L(t ); nd (iii) if s t, s b t, L(s) = L(s ) nd L(t) = L(t ), then = b. Consistent L 2 TSs were introduced in De Nicol & Vndrger [3] for studying reltionshis between notions defined for Krike structures nd notions defined for LTSs. Condition (iii) sys tht the lbel of trnsition is fully determined by the lbels of its source nd trget stte, wheres condition (ii) entils tht

11 Comuttion Tree Logic with Dedlock Detection 11 the lbel of stte t rechble from stte s is comletely determined by the lbel of s nd the sequence of lbels of the trnsitions leding from s to t. Exmle 3. The three L 2 TSs from Figure 3 re not consistent becuse they violte condition (i), (ii), nd (iii) resectively; the L 2 TS in Figure 3b is consistent. ) q r q b q b) b q q r Fig.3. ) Three inconsistent L 2 TSs nd b) consistent L 2 TS. Mny semntic equivlences on LTSs, such s b, λ b nd b, re considered in the literture; for n overview see [6]. Definition 5. Any semntic equivlence on LTSs extends to L 2 TSs by declring, for ll sttes s nd t in n L 2 TS, tht s t iff L(s) = L(t) nd s t in the ssocited LTS. Any semntic equivlence on Krike structures extends to L 2 TSs by declring, for ll sttes s nd t in n L 2 TS, tht s t iff s t in the ssocited Krike structure. The following theorem ws roved in [3] for finite consistent L 2 TSs. Here we dro the finiteness restriction. Theorem 4. On consistent L 2 TS, dbs equls b, nd s equls λ b. Proof. Suose s dbs t [or s s t]. Then there is colouring C on the sttes of the L 2 TS tht is [fully] consistent on the ssocited Krike structure K nd stisfies C(s) = C(t). By definition, this entils L(s) = L(t). It remins to show tht C is [fully] consistent on the ssocited LTS L. So let C() = C(q), nd let σ be [comlete] coloured trce of in L. Using symmetry, it suffices to show tht σ is lso [comlete] coloured trce of q in L. Let ρ be obtined by omitting ll ctions from the lternting sequence of sttes nd ctions σ. Using direction only if of cluse (i) in the definition of consistent L 2 TS, ρ must be [comlete] coloured trce of in K. As C is [fully] consistent on K, ρ must lso be [comlete] coloured trce of q in K. Finlly, using cluses (i) only if nd (iii), σ must be [comlete] coloured trce of q in L. Now suose s b t [or s λ b t]. Then L(s) = L(t) nd there is colouring C on the sttes of the L 2 TS, with C(s) = C(t), tht is [fully] consistent on L. Let D be the colouring given by D() := (C(), L()) for ll S, so tht we hve D() = D(q) [C() = C(q) L() = L(q)]. It suffices to show tht D is [fully]

12 12 Rob vn Glbbeek, Bs Luttik & Nikol Trčk consistent on K. The requirement D() = D(q) L() = L(q) is built into the definition of D. So let D() = D(q), nd let ν be [comlete] D-coloured trce of in K. Using symmetry, it suffices to show tht ν is lso [comlete] D-coloured trce of q in K. Using cluse (i) only if, there must be [comlete] D-coloured trce ρ of in L such tht ν is obtined from ρ by omitting its ctions. Let σ be the [comlete] C-coloured trce of s in K obtined from ρ by omitting the second comonent of ech D-colour of stte. As C() = C(q) nd C is [fully] consistent on L, σ must lso be [comlete] C-coloured trce of q in L. By lying cluses (i) if nd (ii) one derives tht ρ is [comlete] D-coloured trce of q in L. Therefore, gin using cluse (i) only if, ν must be [comlete] D-coloured trce of q in K. Observtion 1. For every Krike structure K there exists consistent L 2 TS D such tht K is the Krike structure ssocited to D. One wy to obtin D is to lbel ny trnsition s t by the ir (L(s), L(t)) (or simly by L(t)) when L(s) L(t), or when L(s) = L(t). An lterntive is the lbel (L(s) L(t), L(t) L(s)), where (, ) is identified with. Unlike the sitution for Krike structures (Observtion 1) it is not the cse tht every LTS cn be obtined s the LTS ssocited to consistent L 2 TS. A simle counterexmle is resented in [3]. Thus, in encoding LTSs s L 2 TSs, it is in generl not ossible to kee the set of sttes the sme. Definition 6. An LTS-to-L 2 TS trnsformtion η consist of function tking ny LTS L to consistent L 2 TS η(l), nd in ddition tking ny stte s in L to stte η(s) in η(l). Such trnsformtion should t lest stisfy s λ b t η(s) λ b η(t), tht is, it reserves ( ) nd reflects ( ) divergence sensitive brnching bisimultion equivlence, nd likewise for b, nd b. A common LTS-to-L 2 TS trnsformtion is resented in [3]. It tkes n LTS L = (S, ) to n L 2 TS η(l) by inserting new stte hlfwy ny trnsition s t with. This new stte is lbelled {}, nd ech of the two trnsitions relcing s t (from s to the new stte nd from there to t) is lbelled. Trnsitions s t re untouched. One tkes η(s) = s for s S nd ll such sttes from L re lbelled { } in η(l). (Consult [3] for the forml definition nd exmles.) In [3] it is shown tht this trnsformtion reserves nd reflects λ b ; the sme roof lies to b nd b. An LTS-to-L 2 TS trnsformtion η yields n LTS-to-Krike-structure trnsformtion tht we lso cll η, nmely the one trnsforming n LTS L into the Krike structure ssocited to η(l). In fct, using Theorem 4 nd Observtion 1, ny LTS-to-Krike-structure trnsformtion η tht reserves nd reflects the required equivlences cn be obtined in this wy. An LTS-to-L 2 TS trnsformtion η mkes it ossible to define when stte s in n LTS stisfies CTL X formul ϕ. Nmely, one defines s =η ϕ iff η(s) = ϕ. This wy, CTL X cn be used s temorl logic on LTSs. Theorem 5. Let s nd t be sttes in n LTS, nd let η be n LTS-to-L 2 TS trnsformtion. Then

13 Comuttion Tree Logic with Dedlock Detection 13 s b t iff s = η db ϕ t =η db ϕ for ll CTL X stte formuls ϕ s λ b t iff s =η ϕ t = η ϕ for ll CTL X stte formuls ϕ. Proof. This is n immedite consequence of the requirement tht η reserves nd reflects b nd λ b, in combintion with Theorems 1, 2 nd 4. 5 Prllel comosition For behviourl equivlence to be useful in rocess lgebric setting, it is essentil tht it is congruence for the oertions under considertion. In this section we rove tht b nd b re congruences for the merge or interleving oertor. This oertor is often used to reresent (synchronous) rllel comosition. However, λ b fils to be congruence for. We chrcterise the lest discriminting congruence tht mkes ll the distinctions of λ b s b. Definition 7. A binry oertion on the sttes of n LTS is merge if for ll s, t, u S nd for ll A it holds tht s t u iff there exists s S such tht s s nd u = s t; or there exists t S such tht t t nd u = s t. Using well known techniques from structurl oertionl semntics [1] ny LTS cn be ugmented with merge. Moreover, n LTS with merge is generted by the structurl oertionl semntics of ny rocess clculus tht includes n oertion for ure interleving. Henceforth we del with LTSs with merge. Theorem 6. The reltion b is congruence for, i.e., if s b t nd u b v, then s u b t v. Proof. Let R be the reflexive nd trnsitive closure of the reltion {( q, q ) b & q b q }. Let C be the function tht ssigns to ech stte its equivlence clss with resect to R. It suffices to rove tht C is consistent divergence reserving colouring. So suose C(r) = C(r ). Using Lemms 3 nd 5 it suffices to show tht r nd r hve the sme C-coloured trces of length three nd the sme divergent C- coloured trces of length one. It is strightforwrd, but nottionlly cumbersome, to estblish this in the secil cse tht r = q nd r = q with b nd q b q. The generl cse then follows by induction on the length of chin of irs from the reltion dislyed bove showing tht the ir (r, r ) is in its reflexive nd trnsitive closure. A similr roof shows tht lso b is congruence for. However, λ b is not. Exmle 4. Consider n LTS with merge tht contins stte 0 without outgoing trnsitions, stte 0 with -loo (n outgoing -lbelled trnsition to itself) nd no other outgoing trnsitions, nd stte with 0 nd no other outgoing trnsitions. (Note tht such n LTS is, e.g., generted by the

14 14 Rob vn Glbbeek, Bs Luttik & Nikol Trčk structurl oertionl semntics of CCS with recursion.) Then 0 λ b 0. Figure 4 shows the frgment consisting of the sttes 0, 0 nd of the LTS under considertion. Figure 4b shows frgment where the merge is lied. Observe tht 0 λ b 0. The reson is tht 0 hs mximl th tht stys in its initil stte, wheres 0 hs not. This roblem does not ly to b becuse 0 b 0. It does not ly to b becuse 0 b 0. ) b) 0 λ b 0 0 λ b Fig.4. λ b is not congruence for rllel comosition The exmle bove involves dedlock stte, nmely 0. This is unvoidble, s on LTSs without dedlock λ b coincides with b (cf. Section 3) nd hence is congruence for. The stndrd solution to the roblem of n equivlence filing to be congruence for desirble oertor O is to relce it by the corsest congruence for O tht is included in [10]. Alying this technique to the current sitution, the corsest congruence for included in λ b turns out to be b. Theorem 7. b is the corsest congruence for tht is included in λ b.2 Proof. We hve lredy seen tht b is congruence for, nd tht it is included in λ b. To show tht it is the corsest, it suffices to show tht whenever s b t, we cn find stte u such tht s u λ b t u. In fct, we cn lwys tke u to be the stte from Exmle 4, for n ction tht does not occur in ny th from s or t. Nmely, suose tht s λ b t. Let C be fully consistent colouring with C(s ) = C(t ). Define the colouring D by D() = C( ) for stte 2 Strictly seking, we merely show tht b is the corsest congruence for tht is included in λ b nd stisfies the Fresh Atom Princile (FAP). This rincile, described in [7], is stisfied by semntic equivlence on LTSs when on n LTS L cn lwys be obtined s the restriction of on ny given lrger LTS of which L is sublts, nd whose trnsition lbels my be drwn from lrger set of ctions thn those of L. FAP llows us to use the stte tht figures in the roof of Theorem 7, regrdless of whether such stte, or the fresh ction, occurs in the given LTS or not. FAP is stisfied by virtully ll semntic equivlences documented in the literture, nd cn be used s snity check for meningful equivlences [7].

15 Comuttion Tree Logic with Dedlock Detection 15 rechble from s or t, nd D() = otherwise. Then D(s) = D(t). It suffices to show tht D is consistent nd reserves divergence, imlying s b t. So suose D() = D(q) with q. Then C( ) = C(q ). First we show tht nd q hve the sme D-coloured trces. Let σ be D-coloured trce of. Then σ is lso C-coloured trce of. As nd q hve the sme comlete C-coloured trces, they surely hve the sme C-coloured trces (for the coloured trces of stte re the refixes of its comlete coloured trces). Hence σ is C-coloured trce of q. As is rechble from s or t, the ction cnnot occur in σ. Therefore, σ must lso be D-coloured trce of q. By symmetry, ny D-coloured trce of q is lso D-coloured trce of, nd hence nd q hve the sme D-coloured trces. Next, we show tht nd q hve the sme divergent D-coloured trces. So let σ be divergent D-coloured trce of. Then σ is lso divergent C-coloured trce of. Hence σ is comlete C-coloured trce of nd thus lso of q. As the ction cnnot occur in σ, it is not ossible tht σ stems from finite mximl th from q. Therefore, σ must be divergent C-coloured trce of q, nd hence divergent D-coloured trce of q. Agin invoking symmetry, nd q hve the sme divergent D-coloured trces. It follows tht D is consistent nd reserves divergence; thus s b t. So if one is in serch of semntics such tht, for s nd t sttes in n LTS, if there is CTL X stte formul ϕ such tht s =η ϕ but t = η ϕ, then s nd t should be distinguished, if s nd t cn be distinguished fter lcing them in context u for some u, then they should be distinguished to strt with nd no two sttes should be distinguished unless this is required by the revious two conditions, then brnching bisimultion semntics with exlicit divergence is the nswer, for s b t iff for ll u nd ll ϕ Φ we hve s u =η ϕ t u = η ϕ. 6 Adding dedlock detection to CTL X We sw bove tht there re imortnt roerties of sttes s in n LTS tht cn be exressed in terms of context u nd CTL X formul ϕ, nmely s s u = η ϕ, but tht cnnot be directly exressed in terms of CTL X. This is somewht unstisfctory, nd therefore we roose n extension of CTL X in which this tye of roerty cn be exressed directly. We dd th modlity tht is vlid on th π iff π is infinite. This th modlity, or ctully n eqully exressive one, ws studied rior by Kivol & Vlmri [9] in the context of Liner Temorl Logic without the next stte oertor see Section 9. Definition 8. The syntx of CTL is given by ϕ ::= ϕ Φ ψ ψ ::= ϕ ψ Ψ ψ U ψ with AP, ϕ Φ, Φ Φ, ψ Ψ nd Ψ Ψ. Vlidity is defined s in Definition 2, but dding the cluse

16 16 Rob vn Glbbeek, Bs Luttik & Nikol Trčk π = iff the th π is infinite. We write ψ for ( ψ); this formul holds in stte s if there exists n infinite th π from s such tht π = ψ. Likewise ψ = ( ψ) holds in s if for ll infinite ths π from s we hve tht s = ψ. These constructs re dul, in the sense tht s = ψ iff s = ψ. The negtion of holds for mximl th π iff π is finite, nd hence ends in dedlock. It is temting to simly extend CTL X with stte formul δ such tht s = δ iff s. s s. This would mke it ossible to exress s Fδ. However, this would mke the resulting logic too exressive: the two sttes in the Krike structure (with the emty lbelling) re brnching bisimultion equivlent with exlicit divergence, yet they would be distinguished by this extension of CTL X, s only the lst stte stisfies δ. CTL is n extension of CTL X. There is no need for similr extension of CTL, for δ cn be exressed s X. In rticulr, CTL is not more exressive thn CTL. The definition of brnching bisimultion equivlence with exlicit divergence lifts esily to Krike structures: s b t, for s nd t sttes in Krike structure, iff there exists consistent nd divergence reserving colouring C such tht C(s) = C(t). Here divergence reserving is defined s in Section 3; by Lemm 5, this time lied to Krike structures, consistent colouring reserves divergence iff, for ny sttes s nd t, C(s) = C(t) imlies for ny infinite th π from s with C(π) = C(s) there is n infinite th ρ from t with C(ρ) = C(t). Theorem 8. s b t iff s = ϕ t = ϕ for ll CTL stte formuls ϕ. Proof. Only if goes s in the roof of Theorem 1, reding = for = db, requiring C to be consistent nd divergence reserving, nd, in the second rgrh, requiring the ths π nd ρ to be mximl nd C(π) to be comlete coloured trce of s nd t. Here we use tht if colouring is consistent nd divergence reserving, then two sttes with the sme colour must lso hve the sme comlete coloured trces. This follows from Lemm 6, this time lied to Krike structures. There is one extr cse to check. Suose C(π) = C(ρ) nd π =, but ρ =. Then the lst stte t of ρ hs the sme colour C(t) s one of the sttes s of π. Let π be the (infinite) suffix of π strting t s. Then C(π ) = C(s) = C(t), yet there is no infinite th from t, contrdicting tht C is divergence reserving. If goes s in the roof of Theorem 1, but this time we hve lso to show tht C reserves divergence. So let s nd t be sttes nd π n infinite th from s with C(π) = C(s) = C(t) = C. Let U = {u there is th from t to u nd C(u) C}. For every u U ick CTL formul ϕ u C C(u). Now s = G( u U ϕ u) nd, s C(s) = C(t), lso t = G( u U ϕ u ). Thus, there is n infinite th ρ from t such tht t = u U ϕ u for ll sttes t in ρ. It follows tht t U. Hence C(t ) = C nd thus C(ρ) = C.

17 Comuttion Tree Logic with Dedlock Detection 17 7 Adding dedlock detection to CTL X CTL X is the sublogic of CTL X tht only llows th formuls of the form ϕ Uϕ nd (ϕ Uϕ ), where ϕ nd ϕ re stte formuls. Equivlently, it cn be defined s only llowing th formuls of the form ϕ U ϕ nd Gϕ, for we hve s = Gϕ iff s = ( U ϕ) s = (ϕ U ϕ ) iff s = [( ϕ ) U (ϕ ϕ )] G ϕ. Theorems 1 nd 2 re lso vlid when using CTL X insted of CTL X, for their roofs use no other temorl constructs thn (ϕ U ϕ ) nd Gϕ. A nturl roosl for CTL would be to dd the th quntifier to CTL X, thus yielding the syntx ϕ ::= ϕ Φ (ϕ U ϕ) (ϕ U ϕ) Gϕ Gϕ. However, we cn economise on tht, for s = (ϕ U ϕ ) iff s = (ϕ U (ϕ G )) s = Gϕ iff s = Gϕ (ϕ U ( Gϕ)) where Gϕ is n bbrevition for ( U ϕ). Hence CTL cn be given by the syntx ϕ ::= ϕ Φ (ϕ U ϕ) Gϕ. It follows immeditely from the roof of Theorem 8 tht this lnguge is sufficiently exressive to chrcterise brnching bisimultion equivlence with exlicit divergence: Theorem 9. s b t iff s = ϕ t = ϕ for ll CTL formuls ϕ. It is temting to simly write G s G; tht is, to kee the sme syntx s for CTL X but define its semntics in such wy tht (ϕ U ϕ ) sks merely for finite th, wheres Gϕ sks for n infinite one. This dedlock sensitive interrettion of CTL X is n lterntive for the interrettion of [3]. It is consistent with the clssicl interrettion of CTL [5, 2], s for totl Krike structures there is no difference between nd. 8 The dedlock extension of Krike structures Following De Nicol & Vndrger [3] we hve lied CTL X to non-totl Krike structures by using mximl insted of infinite ths in the definition of vlidity. As remrked in Section 2, the sme effect cn be obtined by trnsforming non-totl Krike structure into totl one by dding self loo s s to every dedlock stte s, nd lying the stndrd CTL X semntics to the resulting totl Krike structure. However, the ltter does not ly to CTL, becuse the self loo s s invlidtes the formul tht holds in ny dedlock stte s. Here we define nother trnsformtion on Krike structures tht mkes ny Krike structure totl, nd llows encoding of CTL in terms of CTL X.

18 18 Rob vn Glbbeek, Bs Luttik & Nikol Trčk Definition 9. The dedlock extension D(K) of Krike structure K is obtined by the ddition of fresh stte s δ, lbelled by the fresh tomic roosition δ, together with trnsition from s δ nd from every dedlock stte in K to s δ. An exmle of this trnsformtion is deicted in Figure 5. K : q D(K) : q r r s δ δ Fig.5. Dedlock extension of Krike structure Theorem 10. Let K be Krike structure, with sttes s nd t. Then s b within the Krike structure K iff s b t within the Krike structure D(K). Proof. If : Let D be consistent nd divergence reserving colouring on D(K). Note tht D(s δ ) D(s) for ny stte s s δ in D(K). Let C be the restriction of D to the sttes of K. Then the C-coloured trces of stte s in K equl the D- coloured trces of s in D(K), but with the colour D(s δ ) omitted from the end of such trces. It follows tht C is consistent. It reserves divergence by Lemm 5. Only if : Let C be consistent nd divergence reserving colouring on K. Extend it to colouring D on D(K) by ssigning fresh colour δ to the extr stte s δ of D(K). It suffices to check tht D is consistent nd divergence reserving. Clim. From ny stte s in K with the sme colour s dedlock stte t in K there must be th π to dedlock stte such tht C(π) = C(t). Proof of clim. As t hs no C-coloured trces of length two, neither does s, nd s t hs no divergent C-coloured trces, neither does s. Thus, ll ths from s re finite nd only ss through sttes with colour C(t). Aliction of the clim. The D-coloured trces of length two of stte s s δ in D(K) re the C-coloured trces of length two of the stte s in K, together with the trce C(t)δ in cse s hs the sme colour s dedlock stte t in K. Thus D is consistent by Lemm 1, nd reserves divergence by Lemm 5. The if -direction of the theorem, with similr roof, lso lies to s nd dbs, but the only if -direction does not. As counterexmle, let K be Krike structure with dedlock stte d (hving no outgoing trnsitions) nd livelock stte l (with self-loo s its only one outgoing trnsition); neither stte stisfies ny tomic roositions. In K we hve d s l, nd hence d dbs l, but in D(K) we hve d dbs l, nd hence d s l. Considering tht Krike structures of the form D(K) re totl, nd tht on totl Krike structures s nd b coincide, it is in fct imossible to define trnsformtion like D for which Theorem 10 holds for both b nd s. t

19 Comuttion Tree Logic with Dedlock Detection 19 Now let η be n rbitrry LTS-to-L 2 TS-trnsformtion, yielding n LTS-to- Krike-structure trnsformtion tht is lso clled η (see Section 4). Then D η is not vlid LTS-to-Krike-structure trnsformtion s intended in [3], for it fils to reserve λ b / s nd b / dbs (cf. Definition 6). Yet, it stisfies s b t D η(s) s D η(t) (becuse s b t η(s) b η(t) D η(s) b D η(t) nd on totl Krike structures b nd s coincide), nd s such it is suitble trnsformtion for defining vlidity of CTL X formul on sttes in LTSs. We obtin: Corollry 2. Let s nd t be sttes in n LTS, nd let η be n LTS-to-L 2 TS trnsformtion. Then s b t iff s =D η ϕ t = D η ϕ for ll CTL X stte formuls ϕ. Thus, one wy to mke CTL X suitble for deling with dedlock behviour on LTSs is to stick to totl Krike structures nd trnslte LTSs to Krike structures by trnsltion D η insted of trnsformtion η s roosed in [3]. This wy brnching bisimultion equivlence with exlicit divergence becomes the nturl counterrt of stuttering equivlence on Krike structures, nd we hve the modl chrcteristion of Corollry 2. An lterntive is to stick to more nturl trnsformtions η meeting the criteri on Definition 6, ly the definition of vlidity of CTL X formuls to non-totl Krike structures s in [3], nd extend CTL X to CTL s indicted in Section 6. Below we show tht these solutions led to eqully exressive logics on LTSs. Definition 10. Given set of tomic roositions, let CTL δ be the logic CTL X extended with n extr tomic roosition δ. The mings D from CTL to CTL δ formul nd E from CTL to CTL δ formul re defined inductively by D() = E () = D( ϕ) = δ D(ϕ) E( ϕ) = E(ϕ) D( i I ϕ i) = i I D(ϕ i) E( i I ϕ i) = i I E(ϕ i) D( ψ) = D(ψ) E( ψ) = E(ψ) D( ψ) = δ D(ψ) E( ψ) = E(ψ) D( i I ψ i) = i I D(ψ i) E( i I ψ i) = i I E(ψ i) D(ψ U ψ ) = D(ψ) U D(ψ ) E(ψ U ψ ) = (E(ψ) U δ ψ ) (E(ψ) U E(ψ )) D( ) = Fδ E(δ) =. { δ if sδ = ψ Here δ ψ =, nd ψ U δ bbrevites Gψ. otherwise Theorem 11. Let K be Krike structure nd s stte in K. Then for ny CTL stte formul ϕ we hve s = ϕ in K iff s = D(ϕ) in D(K), nd for ny CTL δ stte formul ϕ we hve s = ϕ in D(K) iff s = E(ϕ) in K.

20 20 Rob vn Glbbeek, Bs Luttik & Nikol Trčk Proof. For stte formul ϕ, let [[ϕ]] K denote the set of sttes s in K with s = ϕ. Likewise, for th formul ψ, [[ψ]] K denotes the set of mximl ths π in K with π = ϕ. Note tht there is bijective corresondence between the mximl ths in K nd those in D(K) not strting in s δ. A strightforwrd structurl induction shows tht [[ϕ]] K = [[D(ϕ)]] D(K) for ny CTL stte formul ϕ nd, u to the forementioned bijective corresondence, [[ψ]] K = [[D(ψ)]] D(K) for ny CTL th formul ψ. For the second sttement, let π δ be the unique th in D(K) strting in s δ. A strightforwrd structurl induction shows tht [[ϕ]] D(K) {s δ } = [[E(ϕ)]] K for ny CTL δ stte formul ϕ nd, u to the bove bijective corresondence, [[ψ]] D(K) {π δ } = [[E(ψ)]] K for ny CTL δ th formul ψ. In CTL the th modlity is eqully exressive s the th modlity ψuδ of Definition 10, sying of th tht it is finite nd ll its suffixes stisfy ψ. This is becuse π = ψuδ π = Gψ nd π = π = Fδ π = Uδ. In this light, the encoding D of CTL into CTL δ merely dds conjunct δ here nd there. These conjuncts re not otionl; they enble, for instnce, the correct trnsltion of the CTL th formul G by the CTL δ formul δ G(δ ). Theorem 12. Also the logics CTL δ nd CTL re eqully exressive. Proof. This follows becuse D cn be restricted to ming from CTL to CTL δ formul nd E to ming from CTL to CTL δ formul. In rticulr, D( (ϕ U ϕ )) = (D(ϕ) U D(ϕ )) D( G ϕ) = G( δ D(ϕ)) { E( (ϕ U ϕ (E(ϕ) U E(ϕ )) = )) (E(ϕ) U ( G GE(ϕ))) if s δ = ϕ (E(ϕ) U E(ϕ )) otherwise { G nd E( Gϕ) = E(ϕ) if s δ = ϕ G E(ϕ) otherwise. 9 Concluding remrk: liner lemorl logic Liner Temorl Logic [11] (LTL) is the sublogic of CTL tht llows roositionl vribles AP but no other stte formuls to be used s th formuls. Pth formuls re lied to sttes by n imlicit universl quntifiction: s = π iff s = π. Kivol & Vlmri [9] study equivlences on LTSs with the roerty tht under ll lusible trnsformtions of LTSs into Krike structures two equivlent sttes (trnsformed into sttes of Krike structures) stisfy the sme formuls in either LTL X (LTL without the next stte modlity) or LTL (LTL X ugmented with n oertor tht is eqully exressive s our modlity ). They chrcterise the corsest such congruences for selection of stndrd rocess lgebr oertors including the merge s NDFD-equivlence (for LTL X ) nd CFFD-equivlence (for LTL ). There re two striking differences with the brnching time cse: Wheres b, the equivlence chrcterising vlidity in

21 Comuttion Tree Logic with Dedlock Detection 21 CTL, is congruence for the selected rocess lgebr oertors directly, in the cse of LTL the resulting equivlence becomes much finer s result of congruence closure. Furthermore, the ddition of to LTL X leds to distinctions tht re not mde by CFFD-equivlence. Kivol & Vlmri do not consider the question of rising the exressiveness of LTL X to the level where it chrcterises NDFD- or CFFD-equivlence directly. References 1. L. Aceto, W.J. Fokkink & C. Verhoef (2001): Structurl oertionl semntics. In J.A. Bergstr, A. Ponse & S.A. Smolk, editors: Hndbook of Process Algebr, Elsevier, M.C. Browne, E.M. Clrke & O. Grumberg (1988): Chrcterizing finite Krike structures in roositionl temorl logic. Theoreticl Comuter Science 59, R. De Nicol & F.W. Vndrger (1995): Three logics for brnching bisimultion. Journl of the ACM 42(2), E.A. Emerson & E.M. Clrke (1982): Using brnching time temorl logic to synthesize synchroniztion skeletons. Science of Comuter Progrmming 2(3), E.A. Emerson & J.Y. Hlern (1986): Sometimes nd Not Never revisited: on brnching time versus liner time temorl logic. Journl of the ACM 33(1), R.J. vn Glbbeek (1993): The liner time - brnching time sectrum II. In E. Best, editor: Proceedings of CONCUR 93, LNCS 715, Sringer, R.J. vn Glbbeek (2005): A chrcteristion of wek bisimultion congruence. In A. Middeldor, V. vn Oostrom, F. vn Rmsdonk & R. de Vrijer, editors: Processes, Terms nd Cycles: Stes on the Rod to Infinity: Essys Dedicted to Jn Willem Klo on the Occsion of His 60th Birthdy, LNCS 3838, Sringer, R.J. vn Glbbeek & W.P. Weijlnd (1996): Brnching time nd bstrction in bisimultion semntics. Journl of the ACM 43(3), R. Kivol & A. Vlmri (1992): The wekest comositionl semntic equivlence reserving Nexttime-less Liner Temorl Logic. In W.R. Clevelnd, editor: Proceedings of CONCUR 02, LNCS 630, R. Milner (1989): Communiction nd Concurrency. Prentice Hll, Englewood Cliffs. 11. A. Pnueli (1977): The Temorl Logic of Progrms. In: Proceedings of FOCS 77, IEEE Comuter Society Press,