Encoding Timed Models as Uniform Labeled Transition Systems

Transcription

1 Encoding Timed Models s Uniform Lbeled Trnsition Systems Mrco Bernrdo 1 Luc Tesei 2 1 Diprtimento di Scienze di Bse e Fondmenti Università di Urbino Itly 2 Scuol di Scienze e Tecnologie Università di Cmerino Itly Abstrct. We provide unifying view of timed models such s timed utomt, probbilistic timed utomt, nd Mrkov utomt. The timed models nd their bisimultion semntics re encoded in the frmework of uniform lbeled trnsition systems. In this unifying frmework, we show tht the timed bisimilrities present in the literture cn be reobtined nd tht new bisimilrity, of which we exhibit the modl logic chrcteriztion, cn be introduced for timed models including probbilities. We finlly highlight similrities nd differences mong the models. 1 Introduction Severl extensions of clssicl utomt hve been proposed in the lst twenty yers to model timed spects of the behvior of systems nd to support the verifiction of hrd nd soft rel-time constrints. The first of these extensions is given by timed utomt (TA) [1]. They re equipped with clock vribles tht mesure the pssge of time within sttes, while trnsitions re instntneous, my be subject to clock-bsed gurds, nd my reset the vlue of some clocks. A subsequent extension is tht of probbilistic timed utomt (PTA) [12]. They re TA where the destintion of every trnsition is function tht ssocites with ech stte the probbility of being the trget stte. This llows for the representtion both of nondeterministic choices nd of probbilistic choices, nd enbles the investigtion of properties such s the probbility of executing certin ctivities within given dedline is not lower thn given threshold. The semntics of TA/PTA cn be defined in terms of vrint of lbeled trnsition system (LTS) [11] together with notion of bisimultion [8]. The chrcteristic of the underlying vrint of LTS is tht of hving uncountbly mny sttes, s ny of these sttes essentilly corresponds to pir composed of TA/PTA stte nd vector of clock vlues ech tken from R 0. A more recent extension is constituted by Mrkov utomt (MA) [7], in which the probbilistic flvor of PTA trnsitions is retined, while temporl spects re described through exponentilly distributed rndom vribles rther thn deterministic quntities. Since exponentil distributions enjoy the memoryless property, n MA no longer needs clocks nd hence cn be directly viewed s vrint of LTS whose sttes correspond to the MA sttes. Work prtilly supported by the MIUR-PRIN Project CINA nd the Europen Commission FP7-ICT-FET Proctive Project TOPDRIM (grnt greement no ).

2 In order to emphsize similrities nd differences mong the vrious models, in this pper we provide unifying view of TA, PTA, nd MA by encoding ll of them s uniform lbeled trnsition systems (ULTrS) [4]. This is recently developed frmework tht hs proven to be well suited for uniformly representing different models rnging from LTS models to discrete-/continuous-time Mrkov chins nd Mrkov decision processes without/with internl nondeterminism together with their behviorl equivlences. The pper is orgnized s follows. In Sect. 2, we recll the notion of ULTrS nd we extend it in order to del with uncountble stte spces. In Sect. 3, we encode s ULTrS the vrint of LTS underlying TA nd we show tht the corresponding bisimilrity in [16, 20] coincides with suitble instnce of the bisimilrity for ULTrS. In Sect. 4, we reuse the sme encoding to hndle TA. In Sect. 5, we encode s ULTrS the vrint of LTS underlying PTA nd we show tht two different bisimilrities cn be defined: the one in [19] nd new one for which we exhibit modl logic chrcteriztion. In Sects. 6 nd 7, we reuse lmost the sme encoding to hndle PTA nd MA, respectively. Finlly, in Sect. 8 we drw some conclusions bout the considered timed models. 2 Revisiting the Definition of ULTrS The definition of ULTrS given in [4] ws bsed on set of sttes nd set of trnsition-lbeling ctions tht re t most countble. When deling with TA nd PTA models whose time domin is R 0, the underlying LTS models turn out to hve uncountbly mny sttes nd ctions. Therefore, we need to extend the definition of ULTrS by dmitting uncountble sets of sttes nd ctions, in wy tht preserves the results in [4]. Every ULTrS is prmeterized with respect to set D, whose vlues re interpreted s different degrees of one-step rechbility, nd preorder D equipped with minimum D, which denotes unrechbility. In this pper, we consider the set [S D] cs of countble-support functions from set S to D, i.e., the set of functions D : S D whose support supp(d) = {s S D(s) D } is t most countble. As in [4], when S is set of sttes, every element D of [S D] cs is interpreted s next-stte distribution function nd supp(d) represents the set of rechble sttes. Definition 1. Let (D, D, D ) be preordered set equipped with minimum. A uniform lbeled trnsition system on (D, D, D ), or D-ULTrS for short, is triple U = (S, A, ) where S is possibly uncountble set of sttes, A is possibly uncountble set of ctions, nd S A [S D] cs is trnsition reltion. We sy tht the D-ULTrS U is functionl iff is totl function from S A to [S D] cs. Every trnsition (s,, D) is written s D, where D(s ) is D-vlue quntifying the degree of rechbility of s from s vi tht trnsition nd D(s ) = D mens tht s is not rechble with tht trnsition. If the D-ULTrS is functionl, we shll write D s, (s ) to denote the sme D-vlue.

3 A D-ULTrS cn be depicted s directed grph-like structure in which vertices represent sttes nd ction-lbeled edges represent ction-lbeled trnsitions. Given trnsition s D, the corresponding -lbeled edge goes from the vertex representing s to set of vertices linked by dshed line, ech of which represents stte s supp(d) nd is lbeled with D(s ). Should D(s ) = D for ll sttes s which my hppen when the considered D-ULTrS is functionl the trnsition would not be depicted t ll. A B-ULTrS is shown on the righthnd side of Fig. 1, where B = {, } is the support set of the Boolen lgebr, (flse) denotes unrechbility, (true) denotes rechbility, nd B. In [4], vrious equivlences were defined over ULTrS nd shown to coincide in most cses with those ppered in the literture of nondeterministic, probbilistic, stochstic, nd mixed models. Since in this pper we focus on bisimilrity, we shll recll only the definition of bisimilrity for ULTrS. This definition, like the one of the other equivlences, is prmeterized with respect to mesure function tht expresses the degree of multi-step rechbility of set of sttes in terms of vlues tken from preordered set equipped with minimum. In the following, we cll trce n element α of A nd we denote by ε the empty trce, by the opertion tht computes the length of trce, nd by the opertion tht conctentes two trces. Definition 2. Let U =(S, A, ) be D-ULTrS, n N, s i S for 0 i n, 1 2 nd i A for 1 i n. We sy tht s 0 s 1 s 2... s n 1 s n is computtion of U of length n tht goes from s 0 to s n nd is lbeled with trce n iff for ll i = 1,..., n there exists trnsition s i i 1 Di such tht s i supp(d i ). Definition 3. Let U = (S, A, ) be D-ULTrS nd (M, M, M ) be preordered set equipped with minimum. A mesure function on (M, M, M ) for U, or M-mesure function for U, is function M M : S A 2 S M such tht the vlue of M M (s, α, S ) is defined by induction on α nd depends only on the rechbility of stte in S from stte s through computtions lbeled with trce α. Definition 4. Let U = (S, A, ) be D-ULTrS nd M M be n M-mesure function for U. An equivlence reltion B over S is n M M -bisimultion iff, whenever (s 1, s 2 ) B, then for ll ctions A nd groups of equivlence clsses G 2 S/B it holds tht: M M (s 1,, G) = M M (s 2,, G) where G is the union of ll the equivlence clsses in G. We sy tht s 1, s 2 S re M M -bisimilr, written s 1 B,MM s 2, iff there exists n M M -bisimultion B over S such tht (s 1, s 2 ) B. The preordered structure (M, M, M ) for multi-step rechbility used in the definition of the equivlence does not necessrily coincide with the preordered structure (D, D, D ) for one-step rechbility used in the definition of the model. In [4], vrious cses were illustrted tht demonstrte the necessity of keeping the two structures seprte to retrieve certin equivlences. n

4 The definition of bisimilrity is given in the style of [14], i.e., it requires bisimultion to be n equivlence reltion. However, it dels with rbitrry groups of equivlence clsses rther thn only with individul equivlence clsses. As shown in [4], working with groups of equivlence clsses provides n dequte support to models in which nondeterminism nd quntittive spects coexist. In prticulr, it gives rise to new probbilistic bisimultion equivlences tht hve interesting logicl chrcteriztions (see the references in [4]). 3 Encoding Timed LTS Models Timed processes cn be represented s models enriched with timing informtion. Following the orthogonl-time pproch 1 of [16], we consider n extension of LTS clled timed lbeled trnsition system (TLTS). In this model, functionl spects (i.e., process ctivities ssumed to be instntneous) re seprte from temporl spects (i.e., time pssing) by mens of two distinct trnsition reltions: one lbeled with ctions nd the other lbeled with mounts of time. Since we re interested in TLTS models obtined from TA, we shll consider R 0 s time domin nd llow for uncountbly mny sttes nd ctions. Definition 5. A timed lbeled trnsition system (TLTS) is qudruple (S, A,, ) where S is possibly uncountble set of sttes, A is possibly uncountble set of ctions, nd: S A S is n ction-trnsition reltion such tht for ll s S nd A it holds tht {s S (s,, s ) } is t most countble. S R 0 S is time-trnsition reltion stisfying (s, 0, s) [0-dely], (s, t, s 1) (s, t, s 2) = s 1 = s 2 [time determinism], nd (s, t 1, s ) (s, t 2, s ) = (s, t 1 + t 2, s ) [time dditivity]. Every ction-trnsition (s,, s ) is written s s nd mens tht s cn rech s by executing ction, whilst every time-trnsition (s, t, s ) is written s t s nd mens tht s cn evolve into s fter n mount of time equl to t. Following [20], we cn merge the two trnsition reltions into single one by dding specil time-elpsing ction ɛ(t) for every t R 0. With this in mind, it is immedite to see tht TLTS cn be encoded s functionl B-ULTrS. Definition 6. Let (S, A,, ) be TLTS. Its corresponding functionl B-ULTrS U = (S, A U, U ) is defined by letting: A U = A {ɛ(t) t R 0 }. s U D s, for ll s S nd A U. { D s, (s ) = if A nd s s, or = ɛ(t) nd s t s otherwise for ll s S. 1 As opposed to the integrted-time pproch, in which process ctivities re ssumed to be durtionl: see [6, 3] for n overview of both pproches in different settings.

5 TLTS corresponding ULT RAS t 2 ε( t ) b c t d e b c ε( t ) d e Fig. 1. Trnsltion of TLTS exhibiting both externl nd internl nondeterminism If TLTS stte hs certin number of differently lbeled outgoing ctiontrnsitions, then those trnsitions re retined in the corresponding functionl B-ULTrS. In other words, externl nondeterminism in the originl model is preserved by the resulting model. In contrst, internl nondeterminism is encoded within the trget countble-support functions of the trnsitions of the resulting model. Indeed, if TLTS stte hs severl identiclly lbeled outgoing ction-trnsitions, then single trnsition is generted in the corresponding functionl B-ULTrS, in which severl sttes re ssigned s rechbility vlue. The encoding of both forms of nondeterminism is exemplified in Fig. 1. A notion of bisimilrity for timed processes ws introduced in [16, 20], where the congruence property nd n equtionl chrcteriztion were lso studied. The decidbility of timed bisimilrity ws estblished in [5]. Definition 7. Let (S, A,, ) be TLTS. A reltion B over S is timed bisimultion iff, whenever (s 1, s 2 ) B, then for ll ctions A nd mounts of time t R 0 it holds tht: For ech s 1 s 1 (resp. s 2 s 2) there exists s 2 s 2 (resp. s 1 s 1) such tht (s 1, s 2) B. t For ech s 1 s t 1 (resp. s 2 s t 2) there exists s 2 s t 2 (resp. s 1 s 1) such tht (s 1, s 2) B. We sy tht s 1, s 2 S re timed bisimilr, written s 1 TB s 2, iff there exists timed bisimultion B over S such tht (s 1, s 2 ) B. Timed bisimilrity over TLTS models is cptured by B,MB over the corresponding functionl B-ULTrS models, where mesure function M B is defined in Tble 1. When α = α, the mesure function considers ech possible next stte s by exmining whether it is rechble from s vi (D s, (s )) nd it cn rech stte in S vi α (M B (s, α, S )). If this is the cse for t lest one of the possible next sttes s, then M B (s, α, S ) =, otherwise M B (s, α, S ) =. Note tht, for TLTS models, the preordered structure of the corresponding ULTrS models coincides with the preordered structure of the mesure function. Theorem 1. Let (S, A,, ) be TLTS nd U = (S, A U, U ) be the corresponding functionl B-ULTrS. For ll s 1, s 2 S: s 1 TB s 2 s 1 B,MB s 2

6 8 W D s,(s >< ) M B (s, α, S ) if α = α s S M B (s, α, S ) = if α = ε nd s S >: if α = ε nd s / S Tble 1. Mesure function for functionl B-ULTrS models representing TLTS models 4 Encoding Timed Automt Timed utomt (TA) [1] extend clssicl utomt by introducing clock vribles, or simply clocks, tht mesure the pssge of time. They ll dvnce t the sme speed nd tke vlues in R 0. A clock vlution ν V X over finite set of clocks X is totl function from X to R 0. Given vlution ν nd dely t R 0, we let ν + t denote the vlution mpping ech clock x X into ν(x) + t. A reset γ is set of clocks in X whose vlue is set bck to zero. For vlution ν nd reset γ, we let ν\γ(x) = 0 if x γ nd ν\γ(x) = ν(x) if x γ. In TA, time elpses in sttes, clled loctions, s long s invrint conditions ssocited with the loctions themselves hold. These re constrints on the vlues of the clocks through which notions such s urgency or lziness of ctions cn be expressed [9]. In contrst, the execution of n ction trnsition is considered instntneous. Trnsitions re gurded, i.e., enbled/disbled, by constrints on the vlues of the clocks, nd cn reset the vlue of some clocks. The set Ψ X of clock constrints over finite set of clocks X is defined by the following grmmr: ψ ::= x # c ψ ψ where x X, c N, nd # {<, >,,, =}. Clock constrints re ssessed over clock vlutions. The stisfction of clock constrint ψ by vlution ν, denoted by ν = ψ, is defined s follows: (i) ν = x # c iff ν(x) # c, (ii) ν = ψ 1 ψ 2 iff ν = ψ 1 nd ν = ψ 2. The given syntx for constrints is miniml; the so-clled digonl clock constrints of the form x y # c cn be simulted by using more loctions nd the constrints of the given form [2]. An invrint condition is clock constrint with the property of being pst-closed, i.e., for ll vlutions ν nd delys t R 0 it holds tht ν + t = ψ = ν = ψ. Definition 8. A timed utomton (TA) is tuple (L, A, X, I, ) where L is finite set of loctions, A is set of ctions, X is finite set of clocks, I is function mpping ech loction into n invrint condition, nd L Ψ X A 2 X L is trnsition reltion. Every trnsition is written l ψ,,γ l where l is the source loction, ψ is the gurd, is the ction lbel, γ is the clock reset, nd l is the trget loction. The semntics of TA is given in terms of TLTS. Thus, it is nturl to encode TA s functionl B-ULTrS generted by using the sme conditions defining the TA semntics. Definition 9. Let (L, A, X, I, ) be TA. Its corresponding functionl B-ULTrS U = (S, A U, U ) is defined by letting:

7 S = {(l, ν) L V X ν = I(l)}. A U = A {ɛ(t) t R 0 }. (l, ν) U D (l,ν), for ll (l, ν) S nd A U. if A, l ψ,,γ l, ν = ψ, ν = ν\γ, ν = I(l ) D (l,ν), (l, ν ) = if = ɛ(t), l = l, ν = ν + t, ν = I(l ) otherwise for ll (l, ν ) S. Timed bisimilrity over TA models is defined in terms of the underlying TLTS models. Therefore, we cn reuse both Def. 7 nd Tble 1, so tht Thm. 1 lso pplies to functionl B-ULTrS models corresponding to TA models. 5 Encoding Probbilistic Timed LTS Models A probbilistic extension of the TLTS model (PTLTS) ws introduced in [19]. Following the simple probbilistic utomton model of [18], the ction-trnsition reltion is trnsformed into probbilistic ction-trnsition reltion. This mens tht PTLTS ction trnsition, insted of leding to single trget stte, hs probbility distribution over trget sttes ssigning ech such stte the probbility of being reched. Therefore, the choice mong severl outgoing ction trnsitions from the sme stte is nondeterministic, wheres the choice of the trget stte for the selected trnsition is probbilistic. Given possibly uncountble set S, we denote by Distr cs (S) the set of probbility distributions D over S whose support supp(d) = {s S D(s) > 0} is t most countble. While in [19] there is single trnsition reltion nd hence ech trnsition is lso lbeled with the durtion of the corresponding ction, here we stick to the orthogonl-time pproch nd hence keep using two trnsition reltions: probbilistic one lbeled with ctions nd deterministic one lbeled with mounts of time. We prefer to do so for two resons. Firstly, this llows us to use consistent nottion nd model structure throughout the pper. Secondly, seprting functionl spects from time spects simplifies the development of wek behviorl equivlences, s hs been shown in the deterministic time cse [20, 17, 13] nd in the stochstic time cse [10, 7]. Definition 10. A probbilistic timed lbeled trnsition system (PTLTS) is qudruple (S, A,, ) where S is possibly uncountble set of sttes, A is possibly uncountble set of ctions, nd: S A Distr cs (S) is probbilistic ction-trnsition reltion. S R 0 S is time-trnsition reltion stisfying 0-dely, time determinism, nd time dditivity. Every ction-trnsition (s,, D) is written s D which is lredy in the ULTrS trnsition formt whilst every time-trnsition (s, t, s ) is written s t s. As in the TLTS cse, we cn merge the two trnsition reltions into single one by dding specil time-elpsing ction ɛ(t) for every t R 0, such tht

8 PTLTS corresponding ULT RAS t 2 1 ε( t ) b b c t d e e b b c ε( t ) d e e 1 Fig. 2. Trnsltion of PTLTS exhibiting both externl nd internl nondeterminism the trget distributions of the trnsitions lbeled with such ctions concentrte ll the probbility mss into single stte. At this point, it is strightforwrd to encode PTLTS s n R [0,1] -ULTrS, which relies on the usul ordering for rel numbers with 0 denoting unrechbility nd is not necessrily functionl due to the coexistence of probbility nd internl nondeterminism [4]. In the following, given s S we denote by δ s the Dirc distribution for s, where δ s (s) = 1 nd δ s (s ) = 0 for ll s S \ {s}. Definition 11. Let (S, A,, ) be PTLTS. Its corresponding R [0,1] -ULTrS U = (S, A U, U ) is defined by letting: A U = A {ɛ(t) t R 0 }. s U D for ech s D. s ɛ(t) U δ s for ech s t s. Different from the TLTS encoding, both externl nondeterminism nd internl nondeterminism in the originl PTLTS re preserved in the corresponding R [0,1] -ULTrS. This is exemplified in Fig. 2. A notion of bisimilrity for probbilistic timed processes ws introduced in [19], where modl logic chrcteriztion nd decision procedure were lso studied. Below, we reformulte the definition in the orthogonl-time frmework nd we let D(C) = s C D(s) for D Distr cs(s) nd C S. Definition 12. Let (S, A,, ) be PTLTS. An equivlence reltion B over S is probbilistic timed bisimultion iff, whenever (s 1, s 2 ) B, then for ll ctions A nd mounts of time t R 0 it holds tht: For ech s 1 D 1 there exists s 2 D 2 such tht for ll equivlence clsses C S/B it holds tht D 1 (C) = D 2 (C). t For ech s 1 s t 1 there exists s 2 s 2 such tht (s 1, s 2) B. We sy tht s 1, s 2 S re probbilistic timed bisimilr, written s 1 PTB s 2, iff there exists probbilistic timed bisimultion B over S such tht (s 1, s 2 ) B. It is reltively esy to see tht the reltion PTB coincides with the following bisimultion equivlence defined over R [0,1] -ULTrS models corresponding to

9 8 if α = α >< nd there exists s D M 2Ṛ[0,1] (s, α, S ) = {1} if α = ε nd s S >: S { P s D s S D(s ) p s p s M 2Ṛ[0,1] (s, α, S )} {0} if α = α nd there is no s D or α = ε nd s / S Tble 2. Mesure function for R [0,1] -ULTrS models representing PTLTS models PTLTS models. The equivlence below is clled group-distribution bisimilrity becuse it compres entire distributions of reching groups of equivlence clsses. Given two relted sttes, for ech trnsition of one of the two sttes there must exist n eqully lbeled trnsition of the other stte such tht, for every group of equivlence clsses, the two trnsitions hve the sme probbility of reching stte in tht group. In other words, the two trnsitions must be fully mtching, i.e., they must mtch with respect to ll groups. Definition 13. Let U = (S, A U, U ) be the R [0,1] -ULTrS corresponding to PTLTS (S, A,, ). An equivlence reltion B over S is probbilistic timed group-distribution bisimultion iff, whenever (s 1, s 2 ) B, then for ll ctions A U it holds tht: For ech s 1 U D 1 there exists s 2 U D 2 such tht for ll groups of equivlence clsses G 2 S/B it holds tht D 1 ( G) = D 2 ( G). We sy tht s 1, s 2 S re probbilistic timed group-distribution bisimilr, written s 1 PTB,dis s 2, iff there exists probbilistic timed group-distribution bisimultion B over S such tht (s 1, s 2 ) B. Theorem 2. Let (S, A,, ) be PTLTS nd U = (S, A U, U ) be the corresponding R [0,1] -ULTrS. For ll s 1, s 2 S: s 1 PTB s 2 s 1 PTB,dis s 2 The reltion PTB over PTLTS models hs been expressed s PTB,dis in the ULTrS setting, but cnnot be cptured by ny instntition of the generl bisimilrity for ULTrS given in Def. 4. In the cse of probbilistic timed processes, nturl mesure function is the one defined in Tble 2. Denoting by 2 Ṛ [0,1] the set of nonempty subsets of R [0,1], this mesure function ssocites suitble element of 2 Ṛ [0,1] with every triple composed of source stte s, trce α, nd set of destintion sttes S. The set M 2Ṛ[0,1] (s, α, S ) contins for ech possible wy of resolving nondeterminism the probbility of performing computtion tht is lbeled with trce α nd leds to stte in S from stte s. It is worth pointing out tht, while the considered ULTrS models re bsed on the preordered structure (R [0,1],, 0), the mesure function relies on the different preordered structure (2 Ṛ [0,1],, {0}) where R1 R 2 mens inf R 1 inf R 2 nd R 1 R 2 (the ltter condition ensures {0} being the minimum).

10 s 1 ~PTB,dis s 2 ~PTB,gbg ε( t 1 ) ε( t 2 ) ε( t 1 ) ε( t 3 ) ε( t 2 ) ε( t 3 ) ε( t 1 ) ε( t 2 ) ε( t 1 ) ε( t 3 ) ε( t 2 ) ε( t 3 ) Fig. 3. Counterexmple showing tht PTB,gbg is strictly corser thn PTB,dis The resulting bisimilrity B,M2Ṛ[0,1] cptures the following equivlence tht we cll group-by-group bisimilrity becuse it considers single group of equivlence clsses t time. Techniclly speking, this mounts to nticipting the quntifiction over groups (underlined in Def. 13) with respect to the quntifiction over trnsitions. In this wy, trnsition deprting from one of two relted sttes is llowed to be mtched, with respect to the probbilities of reching different groups, by severl different trnsitions deprting from the other stte. In other words, prtilly mtching trnsitions re llowed. Definition 14. Let U = (S, A U, U ) be the R [0,1] -ULTrS corresponding to PTLTS (S, A,, ). An equivlence reltion B over S is probbilistic timed group-by-group bisimultion iff, whenever (s 1, s 2 ) B, then for ll ctions A U nd for ll groups of equivlence clsses G 2 S/B it holds tht: For ech s 1 U D 1 there exists s 2 U D 2 such tht D 1 ( G) = D 2 ( G). We sy tht s 1, s 2 S re probbilistic timed group-by-group bisimilr, written s 1 PTB,gbg s 2, iff there exists probbilistic timed group-by-group bisimultion B over S such tht (s 1, s 2 ) B. Theorem 3. Let U = (S, A U, U ) be the R [0,1] -ULTrS corresponding to PTLTS (S, A,, ). For ll s 1, s 2 S: s 1 PTB,gbg s 2 s 1 B,M2Ṛ[0,1] s 2 In presence of internl nondeterminism, PTB,gbg strictly contins PTB,dis, s shown in Fig. 3. Indicting sttes with the ctions they enble, it holds tht s 1 PTB,dis s 2 becuse the group distribution of the leftmost -trnsition of s 1 which ssigns probbility 1 to ech group contining both the ɛ(t 1 )- stte nd the ɛ(t 2 )-stte, probbility 0.4 to ech group contining the ɛ(t 1 )-stte but not the ɛ(t 2 )-stte, probbility 0.6 to ech group contining the ɛ(t 2 )-stte but not the ɛ(t 1 )-stte, nd probbility 0 to ny other group is not mtched by the group distribution of ny of the three -trnsitions of s 2. In contrst, s 1 PTB,gbg s 2. For instnce, the leftmost -trnsition of s 1 is mtched by the leftmost -trnsition of s 2 with respect to every group contining both the ɛ(t 1 )- stte nd the ɛ(t 2 )-stte, the centrl -trnsition of s 2 with respect to every group contining the ɛ(t 1 )-stte but not the ɛ(t 2 )-stte, nd the rightmost - trnsition of s 2 with respect to every group contining the ɛ(t 2 )-stte but not the ɛ(t 1 )-stte.

11 Theorem 4. Let U = (S, A U, U ) be the R [0,1] -ULTrS corresponding to PTLTS (S, A,, ). For ll s 1, s 2 S: s 1 PTB,dis s 2 = s 1 PTB,gbg s 2 We conclude by exhibiting modl logic chrcteriztion of PTB,gbg (nd hence of B,M2Ṛ[0,1] ). Unlike the chrcteriztion of PTB (i.e., PTB,dis ) provided in [19], which relies on n expressive probbilistic extension of HML [8] interpreted over stte distributions, here it is sufficient to consider the intervlbsed vrint IPML of the probbilistic modl logic in [14] with the following syntx: φ ::= true φ φ φ [p1,p 2]φ where A U nd p 1, p 2 R [0,1] such tht p 1 p 2. A stte s S belongs to the set M[ [p1,p 2]φ] of sttes stisfying [p1,p 2]φ iff there exists s U D such tht p 1 D(M[φ]) p 2. Theorem 5. Let U = (S, A U, U ) be the R [0,1] -ULTrS corresponding to PTLTS (S, A,, ). For ll s 1, s 2 S it holds tht s 1 PTB,gbg s 2 iff s 1 nd s 2 stisfy the sme formule of IPML. 6 Encoding Probbilistic Timed Automt Probbilistic timed utomt (PTA) [12] extend TA with probbilities. While the pssge of time remins deterministic, the trget of ech ction trnsition becomes probbility distribution. The pproch is exctly the one described in Sect. 5 for moving from TLTS models to PTLTS models. Definition 15. A probbilistic timed utomton (PTA) is tuple (L, A, X, I, ) where L is finite set of loctions, A is set of ctions, X is finite set of clocks, I is function mpping ech loction into n invrint condition, nd L Ψ X A Distr cs (2 X L) is trnsition reltion. Every trnsition is written l ψ, D where D is the probbility distribution ssigning ech pir (γ, l ) the probbility of being reched vi tht trnsition. Like for TA, the semntics of PTA is given in terms of PTLTS. Following the sme pproch used in Sect. 4, we thus encode PTA s n R [0,1] -ULTrS generted by using the sme conditions defining the PTA semntics. Definition 16. Let (L, A, X, I, ) be PTA. Its corresponding R [0,1] -ULTrS U = (S, A U, U ) is defined by letting: S = {(l, ν) L V X ν = I(l)}. A U = A {ɛ(t) t R 0 }. (l, ν) U D for ech l ψ, D such tht ν = ψ, where for ll (l, ν ) S D(l, ν ) = γ reset(ν,ν ) D (γ, l ) with reset(ν, ν ) = {γ 2 X ν\γ = ν }. (l, ν) ɛ(t) U δ (l,ν ) for l = l, ν = ν + t, ν = I(l ). Similr to TA models, probbilistic timed bisimilrity over PTA models is defined in terms of the underlying PTLTS models. Therefore, we cn reuse Defs. 12, 13, nd 14 s well s Tble 2, so tht Thms. 2, 3, 4, nd 5 lso pply to R [0,1] -ULTrS corresponding to PTA models.

12 7 Encoding Mrkov Automt So fr, we hve considered timed models in which temporl spects re described s fixed mounts of time. In other words, in these models the pssge of time is represented deterministiclly. However, in mny situtions there re fluctutions in the time tht elpses between instntneous ctivities. When these fluctutions re quntifible, the pssge of time cn be represented stochsticlly. Due to the simplicity of their mthemticl tretment, exponentilly distributed rndom vribles re mostly used for stochstic representtion of time. Given one such vrible X with prmeter λ R >0, the probbility tht durtion smpled from X is t most t R 0 is given by Pr{X t} = 1 e λ t. The prmeter λ is sid the rte of X; its reciprocl is the expected vlue of X. If severl lterntive exponentilly distributed delys cn elpse from stte, the rce policy is dopted; the dely tht elpses is the one smpling the lest durtion. It cn be shown tht the following property RP holds in tht stte: the sojourn time is exponentilly distributed with rte given by the sum of the rtes of the vrious delys, with the probbility of selecting ech such dely being proportionl to its rte. The recently introduced model of Mrkov utomt (MA) [7] cn be viewed s vrint of PTA models in which time pssing is described through exponentilly distributed rndom vribles. An importnt property of ny of these vribles is tht it enjoys the memoryless property; even if we know tht certin mount of time hs lredy elpsed, the residul time is still quntified by the sme exponentilly distributed rndom vrible. As consequence, in this setting there is no need for clocks nd hence Mrkov utomt cn ctully be viewed s vrint of PTLTS models, with exponentilly distributed delys (uniquely identified by their rtes) in plce of deterministic delys. Definition 17. A Mrkov utomton (MA) is qudruple (S, A,, ) where S is possibly uncountble set of sttes, A is possibly uncountble set of ctions, nd: S A Distr cs (S) is probbilistic ction-trnsition reltion. S R >0 S is bounded time-trnsition reltion, i.e., for ll s S it holds tht {s S λ R >0. (s, λ, s ) } is t most countble nd (s,λ,s ) λ <. Similr to the PTLTS cse, every ction-trnsition (s,, D) is written s D, every time-trnsition (s, λ, s ) is written s λ s, nd we cn merge the two trnsition reltions into single one by dding specil time-elpsing ction ɛ(λ) for every λ R >0. Following the trnsformtion sketched in [7], it is strightforwrd to encode n MA s not necessrily functionl R [0,1] -ULTrS, in which the rce policy is represented bsed on RP. For ech stte hving outgoing timetrnsitions, we generte single time-elpsing trnsition insted of one such trnsition for ech originl dely such tht its rte λ is the sum of the rtes identifying the originl delys nd its trget distribution ssigns to every stte probbility proportionl to the rte t which tht stte cn be reched.

13 MA corresponding ULT RAS λ 1 λ ε( λ) p1 p b b c λ 1 λ 2 d e e b b c ε( λ 1 ) ε( λ 2 ) d e e Fig. 4. Trnsltion of n MA (λ = λ 1 + λ 2, p 1 = λ 1/λ, p 2 = λ 2/λ) Definition 18. Let (S, A,, ) be n MA. Its corresponding R [0,1] -ULTrS U = (S, A U, U ) is defined by letting: A U = A {ɛ(λ) λ R >0 }. s U D for ech s D. s ɛ(λ) U D for ll s S hving outgoing time-trnsitions, where λ = s λ s λ nd D(s ) = s λ s λ /λ for ll s S. Nondeterministic choices over ctions, probbilistic choices over sttes, nd the rce policy for exponentilly distributed delys in the originl MA re preserved in the corresponding R [0,1] -ULTrS. This is exemplified in Fig. 4. A notion of bisimilrity for probbilistic exponentilly-timed processes ws introduced in [7]. Below, we reformulte the definition in terms of the two distinct trnsition reltions. Definition 19. Let (S, A,, ) be n MA. An equivlence reltion B over S is probbilistic exponentilly-timed bisimultion iff, whenever (s 1, s 2 ) B, then for ll ctions A nd rtes λ R >0 it holds tht: For ech s 1 D 1 there exists s 2 D 2 such tht for ll equivlence clsses C S/B it holds tht D 1 (C) = D 2 (C). If s 1 hs outgoing time-trnsitions, then s 2 hs outgoing time-trnsitions too nd for ll equivlence clsses C S/B it holds tht: λ = λ λ s 1 s λ 1 C s 2 s 2 C We sy tht s 1, s 2 S re probbilistic exponentilly-timed bisimilr, written s 1 PEB s 2, iff there exists probbilistic exponentilly-timed bisimultion B over S such tht (s 1, s 2 ) B. The reltion PEB over MA models coincides with the reltion PTB,dis over R [0,1] -ULTrS models given in Def. 13. As consequence, ll the subsequent definitions nd results in Sect. 5 lso pply to R [0,1] -ULTrS models corresponding to MA models. Theorem 6. Let (S, A,, ) be n MA nd U = (S, A U, U ) be the corresponding R [0,1] -ULTrS. For ll s 1, s 2 S: s 1 PEB s 2 s 1 PTB,dis s 2

14 8 Discussion nd Conclusions In this pper, widely used timed models such s TA [1] (together with their underlying semntic model TLTS), PTA [12] (with their underlying PTLTS), nd MA [7] hve been put in unifying view by encoding them in the ULTrS frmework [4] nd by exmining their bisimultion semntics [16, 20, 19, 7]. As immedite results of this work, we hve been ble to re-obtin the lredy existing timed bisimilrities nd, most notbly, to give new contributions. In prticulr, by nturlly instntiting the ULTrS generl bisimilrity definition to the cse of deterministiclly timed models i.e., TLTS nd TA we hve retrieved the sme timed bisimilrity introduced in the literture (Thm. 1). Insted, when time is mixed with probbility i.e., for PTLTS, PTA, nd MA models we hve found tht the bisimilrities present in the literture, lthough expressible within the ULTrS frmework (Thms. 2 nd 6), re different from the one tht cn be nturlly obtined from ULTrS. This hs led us to introduce new bisimilrity for those models (Def. 14 nd Thm. 3), which we hve clled group-by-group nd shown to be corser thn the originl one (Thm. 4). Moreover, we hve exhibited modl logic chrcteriztion for the group-bygroup bisimilrity by using n intervl-bsed vrint of the logic in [14] (Thm. 5), while the originl bisimilrity needs much more expressive logic [19]. The ULTrS-bsed encodings permit lso more generl considertions bout the studied models. Firstly, the trnsition reltion of the ULTrS corresponding to TA is functionl nd bsed on B, whilst in the cse of PTA/MA it is not function (becuse internl nondeterminism cnnot be mixed with probbilities in the trget stte distributions of trnsitions) nd it is necessrily bsed on R [0,1]. This stresses the higher expressivity of PTA/MA compred to TA with regrd to describing stte rechbility. Furthermore, it evidences structurl nlogy between PTA nd MA tht hs not been ddressed so fr in the literture. Secondly, the quntittive informtion relted to time in TA/PTA/MA cn be mde dispper to lrge extent, while quntittive informtion relted to probbilities in PTA/MA cnnot be bstrcted. This underlines n importnt difference between time nd probbility. Time elpses independent of the occurrence of events nd hence its pssge cn be viewed s n event in its own, which cn thus be represented like the other events. Probbilities, insted, re inherently ssocited with the occurrence of events nd must therefore be explicitly represented s event ttributes. Indeed, in our encodings time pssing hs been represented through specil ctions tht encompss the durtion/rte of delys. A purely qulittive representtion of time bsed on single specil ction ɛ is lso possible nd ws used, for instnce, in the construction of the region/zone grph nd in the notion of time-bstrct bisimilrity [15]. This supports compct description of the stte spce of the ULTrS corresponding to TA/PTA, which is uncountble while this is not necessrily the cse for n MA. The reson is the inherent difference between deterministic time, which needs the concrete representtion of ll possible delys, nd exponentilly distributed time, for which symbolic representtion bsed on rtes is sufficient thnks to the memoryless property.

15 A nturl continution of our work is to investigte trce nd testing equivlences by pplying their generl definitions in [4] to the considered timed models. References 1. R. Alur nd D.L. Dill. A theory of timed utomt. Theoreticl Computer Science, 126: , B. Bérrd, A. Petit, V. Diekert, nd P. Gstin. Chrcteriztion of the expressive power of silent trnsitions in timed utomt. Fundment Informtice, 36: , M. Bernrdo. On the expressiveness of Mrkovin process clculi with durtionl nd durtionless ctions. In Proc. of GANDALF 2010, volume 25 of EPTCS, pges , M. Bernrdo, R. De Nicol, nd M. Loreti. A uniform frmework for modeling nondeterministic, probbilistic, stochstic, or mixed processes nd their behviorl equivlences. Informtion nd Computtion, 225:29 82, K. Cerns. Decidbility of bisimultion equivlences for prllel timer processes. In Proc. of CAV 1992, volume 663 of LNCS, pges Springer, F. Corrdini. Absolute versus reltive time in process lgebrs. Informtion nd Computtion, 156: , C. Eisentrut, H. Hermnns, nd L. Zhng. On probbilistic utomt in continuous time. In Proc. of LICS 2010, pges IEEE-CS Press, M. Hennessy nd R. Milner. Algebric lws for nondeterminism nd concurrency. Journl of the ACM, 32: , T.A. Henzinger, X. Nicollin, J. Sifkis, nd S. Yovine. Symbolic model checking for rel-time systems. Informtion nd Computtion, 111: , H. Hermnns. Interctive Mrkov Chins. Springer, Volume 2428 of LNCS. 11. R.M. Keller. Forml verifiction of prllel progrms. Communictions of the ACM, 19: , M. Kwitkowsk, G. Normn, R. Segl, nd J. Sproston. Automtic verifiction of rel-time systems with discrete probbility distributions. Theoreticl Computer Science, 282: , R. Lnotte, A. Mggiolo-Schettini, nd A. Troin. Wek bisimultion for probbilistic timed utomt. Theoreticl Computer Science, 411: , K.G. Lrsen nd A. Skou. Bisimultion through probbilistic testing. Informtion nd Computtion, 94:1 28, K.G. Lrsen nd W. Yi. Time bstrcted bisimultion: Implicit specifictions nd decidbility. In Proc. of MFPS 1993, volume 802 of LNCS, pges Springer, F. Moller nd C. Tofts. A temporl clculus of communicting systems. In Proc. of CONCUR 1990, volume 458 of LNCS, pges Springer, F. Moller nd C. Tofts. Behviourl bstrction in TCCS. In Proc. of ICALP 1992, volume 623 of LNCS, pges Springer, R. Segl. Modeling nd Verifiction of Rndomized Distributed Rel-Time Systems. PhD Thesis, J. Sproston nd A. Troin. Simultion nd bisimultion for probbilistic timed utomt. In Proc. of FORMATS 2010, volume 6246 of LNCS, pges Springer, W. Yi. CCS + time = n interleving model for rel time systems. In Proc. of ICALP 1991, volume 510 of LNCS, pges Springer, 1991.

16 Appendix: Proofs of Results Proof of Thm. 1. Let s 1, s 2 S. Assume tht s 1 TB s 2 due to some timed bisimultion B over S such tht (s 1, s 2 ) B. This mens tht, whenever (s 1, s 2) B, then for ll A nd t R 0 it holds tht: For ech s 1 s 1 (resp. s 2 s 2) there exists s 2 s 2 (resp. s 1 s 1) such tht (s 1, s 2) B. For ech s t 1 s 1 (resp. s t 2 s 2) there exists s t 2 s 2 (resp. s t 1 s 1) such tht (s 1, s 2) B. Without loss of generlity, we cn suppose tht B is n equivlence reltion; should this not be the cse, it suffices to tke the reflexive, symmetric, nd trnsitive closure of B s this is still timed bisimultion. As consequence, the ssumption is equivlent to hving tht, whenever (s 1, s 2) B, then for ll A, t R 0, nd C S/B it holds tht: For ech s 1 s 1 such tht s 1 C there exists s 2 s 2 such tht s 2 C, nd vice vers. For ech s t 1 s 1 such tht s 1 C there exists s t 2 s 2 such tht s 2 C, nd vice vers. In turn, this is equivlent to hving tht, whenever (s 1, s 2) B, then for ll A, t R 0, nd C S/B it holds tht: There exists s 1 C such tht s 1 s 1 iff there exists s 2 C such tht s 2 s 2. There exists s 1 C such tht s t 1 s 1 iff there exists s 2 C such tht s t 2 s 2. Since for ll s S, A, t R 0, nd G 2 S/B it holds tht the existence of s G such tht s s (resp. s t s ) corresponds to the existence of s C such tht s s (resp. s t s ) for some C G, we immeditely derive tht the ssumption is equivlent to hving tht, whenever (s 1, s 2) B, then for ll A, t R 0, nd G 2 S/B : There exists s 1 G such tht s 1 s 1 iff there exists s 2 G such tht s 2 s 2. There exists s 1 G such tht s t 1 s 1 iff there exists s 2 G such tht s t 2 s 2. Since for ll s S, A U, nd G 2 S/B it holds tht: M B (s,, G) = { D s S s, (s ( s G. s s ) if A ) = G ( s G. s t s ) if = ɛ(t) we further derive tht the ssumption is equivlent to hving tht, whenever

17 (s 1, s 2) B, then for ll A U nd G 2 S/B it holds tht: M B (s 1,, G) = M B (s 2,, G) This mens tht B is n M B -bisimultion such tht (s 1, s 2 ) B. In other words, s 1 B,MB s 2. Proof of Thm. 2. It follows immeditely from the trnsltion in Def. 11 fter observing tht the discriminting power of PTB does not chnge if, in the definition of the equivlence (Def. 12), we replce equivlence clsses with groups of equivlence clsses. Proof of Thm. 3. Let s 1, s 2 S. Assume tht s 1 PTB,gbg s 2 due to some probbilistic timed group-by-group bisimultion B over S such tht (s 1, s 2 ) B. This mens tht, whenever (s 1, s 2) B, then for ll A nd G 2 S/B it holds tht: For ech s 1 D 1 there exists s 2 D 2 such tht D 1 ( G) = D 2 ( G). In other words, whenever (s 1, s 2) B, then for ll A nd G 2 S/B : {D 1 ( G)} {D 2 ( G)} s 1 D 1 s 2 D 2 {D 2 ( G)} {D 1 ( G)} s 2 D 2 s 1 D 1 or equivlently: {D 1 ( G)} = {D 2 ( G)} s 1 D 1 s 2 D 2 Since for ll s S, A, nd G 2 S/B it holds tht M 2Ṛ[0,1] (s,, G) = {0} when s hs no -trnsition, otherwise: M 2Ṛ[0,1] (s,, G) = { D(s )} = {D( G)} s D s S G s D we derive tht the ssumption is equivlent to hving tht, whenever (s 1, s 2) B, then for ll A nd G 2 S/B : M 2Ṛ[0,1] (s 1,, G) = M 2Ṛ[0,1] (s 2,, G) This mens tht B is n M 2Ṛ[0,1] -bisimultion such tht (s 1, s 2 ) B. In other words, s 1 B,M2Ṛ[0,1] s 2. Proof of Thm. 4. A strightforwrd consequence of the fct tht probbilistic timed group-distribution bisimultion is lso probbilistic timed group-bygroup bisimultion, s cn be esily seen by tking the sme fully mtching trnsitions considered in the group-distribution bisimultion. Proof of Thm. 5. The result holds when U is imge finite nd stisfies the miniml probbility ssumption. U is imge finite [8] iff for ll s S nd A U the set {D Distr cs (S) s U D} is finite. U stisfies the miniml probbility ssumption [14] iff there exists ɛ R >0 such tht, whenever s U D, then D(s ) ɛ for ll s supp(d); this implies tht supp(d) is finite becuse it cn hve t most 1/ɛ elements. The proof of the result is divided into two prts. In the first prt, we provide n lterntive chrcteriztion of PTB,gbg s the limit of sequence of equivlence reltions i PTB,gbg, i N, which re inductively defined s follows:

18 0 PTB,gbg = S S. i+1 PTB,gbg is the set of ll pirs (s 1, s 2 ) i PTB,gbg such tht for ll ctions A U nd groups of equivlence clsses G 2 S/ i PTB,gbg it holds tht for ech s 1 U D 1 there exists s 2 U D 2 such tht D 1 ( G) = D 2 ( G). Denoting by the reltion i N i PTB,gbg, we prove tht = PTB,gbg. Firstly, we observe wht follows: is n equivlence reltion becuse so is i PTB,gbg for ll i N. Given C S/ nd i N, there exists unique element C i in S/ i PTB,gbg such tht C i C, nd hence C = i N C i with C i1 C i2 for i 1 i 2. As consequence, given G 2 S/ nd i N, there exists unique element G i in 2 S/ i PTB,gbg such tht every clss in Gi contins some clss in G, nd hence G = i N ( G i ) with G i1 G i2 for i 1 i 2. Moreover, if s U D, then D( G) = inf i N D( G i ). In fct, observing tht for ll i N it holds tht D( G i ) D( G) becuse G i G, if we let p = inf i N D( G i ), then p D( G) becuse D( G) is lower bound of the sequence (D( G i )) i N nd p is the gretest lower bound of tht sequence. Suppose p > D( G) nd let δ = p D( G). Since δ > 0 nd the summtion D(S \ G) stisfies s S\ S G D(s) 1 nd hence converges, there exists finite subset X of S \ G such tht the rest D((S \ G) \ X) of the previously considered summtion stisfies s (S\ S G)\X D(s) < δ. Let Y = (S \ G) \ X. For ll i N, it holds tht: Gi = G (Y G i ) (X G i ) where the three sets on the right-hnd side re pirwise disjoint nd hence: D( G i ) = D( G) + D(Y G i ) + D(X G i ) D( G) + D(Y ) + D(X G i ) < D( G) + δ + D(X G i ) = p + D(X G i ) From the inequlity bove nd D( G i ) p, we derive tht: D(X G i ) > D( G i ) p 0 nd hence X G i for ll i N. As consequence, X G becuse X is finite nd G 0 G This contrdicts the fct tht X is subset of S \ G. Therefore, it must be p = D( G). Secondly, it holds tht PTB,gbg becuse i PTB,gbg PTB,gbg for ll i N s we now show by proceeding by induction on i: If i = 0, then i PTB,gbg = S S PTB,gbg. Let i be n element of N for which the result holds nd consider i + 1. If s 1, s 2 S stisfy s 1 PTB,gbg s 2, then: For ll A U nd G 2 S/ PTB,gbg, it holds tht for ech s 1 U D 1 there exists s 2 U D 2 such tht D 1 ( G) = D 2 ( G). s 1 i PTB,gbg s 2 becuse i PTB,gbg PTB,gbg by the induction hypothesis.

19 Since every equivlence clss of i PTB,gbg is equl to the union of some equivlence clsses of PTB,gbg nd hence the union of equivlence clsses in every group G of i PTB,gbg is equl to the union of the equivlence clsses in some group G of PTB,gbg, we derive tht for ll A U nd G 2 S/ i PTB,gbg it holds tht for ech s 1 U D 1 there exists s 2 U D 2 such tht: D 1 ( G ) = D 1 ( G) = D 2 ( G) = D 2 ( G ) This mens tht s 1 i+1 PTB,gbg s 2. Thirdly, we prove tht PTB,gbg by showing tht is probbilistic timed group-by-group bisimultion. Suppose tht s 1, s 2 S stisfy s 1 s 2 nd, given A U nd G 2 S/, ssume tht s 1 U D 1. Then D 1 ( G) = inf i N D 1 ( G i ) where ech G i is the unique element in 2 S/ i PTB,gbg such tht every clss in Gi contins some clss in G. Observing tht 0 PTB,gbg induces single equivlence clss equl to S nd hence D 1 ( G 0 ) = D 1 (S) = 1, from s 1 s 2 nd s 1 U D 1 it follows tht for ll i N 1 there exists s 2 U D 2,i such tht D 1 ( G i 1 ) = D 2,i ( G i 1 ). Since U is imge finite, the set {D 2,i i N 1 } is finite nd we enumerte it s {D2, 1..., D2}. k For ech j {1,..., k}, we lso let I j be the set of indexes i N 1 such tht D 1 ( G i 1 ) = D j 2 ( G i 1 ). At lest one set in {I 1,..., I k } is infinite. Indeed, if every I j were finite, then there would exist n integer i such tht i / I j for ech j {1,..., k}. Hence, there would exists group G i 1 such tht D 1 ( G i 1 ) D j 2 ( G i 1 ) for ech j {1,..., k}. However, this implies s 1 i PTB,gbg s 2, which whould contrdict the ssumption s 1 s 2. Let j be such tht I j is infinite. We hve tht: D 1 (G) = inf i N D 1 ( G i ) = inf i Ij D 1 ( G i ) (1) = inf i Ij D j 2 ( G i ) = inf i N D j 2 ( G i ) = D j 2 (G) (2) where the equlities (1) nd (2) bove derive from the fct tht (D 1 (G i )) i I j nd (D j 2 (G i)) i I j re infinite subsequences of (D 1 (G i )) i N nd (D j 2 (G i)) i N, respectively, nd therefore they hve the sme infimum. In conclusion, we hve tht is probbilistic timed group-by-group bisimultion. In the second prt of the proof, we show tht s 1, s 2 S re equted by i PTB,gbg iff they stisfy the sme formule in F i IPML, which is the set of formule in IPML whose mximum number of nested dimond opertors is t most i. Denoting by FIPML i (s) the set of formule in Fi IPML stisfied by s S, we proceed by induction on i N. Let i = 0. Since 0 PTB,gbg = S S nd F IPML 0 (s) = {φ F0 IPML φ true} for ll s S, it trivilly holds tht: s 1 0 PTB,gbg s 2 FIPML 0 (s 1) = FIPML 0 (s 2) Let i N nd suppose tht for ll j = 0,..., i it holds tht: s 1 j PTB,gbg s 2 F j IPML (s 1) = F j IPML (s 2) We prove tht both implictions hold lso for i + 1 by resoning on their corresponding contrpositive sttements, i.e., we prove tht:

20 F i+1 IPML (s 1) F i+1 IPML (s 2) s 1 i+1 PTB,gbg s 2 (= ) If F i+1 IPML (s 1) F i+1 IPML (s 2), then there re two cses: If FIPML i (s 1) FIPML i (s 2), then by the induction hypothesis it holds tht s 1 i PTB,gbg s 2 nd hence s 1 i+1 PTB,gbg s 2. If FIPML i (s 1) = FIPML i (s 2), then from F i+1 IPML (s 1) F i+1 IPML (s 2) it follows tht there exists φ F i+1 IPML such tht s 1 M[φ] nd s 2 M[φ]. We now proceed by induction on the syntcticl structure of φ. Here we only consider the cse φ = [p1,p 2]φ becuse the other cses re routine. From s 1 M[ [p1,p 2]φ ] nd s 2 M[ [p1,p 2]φ ], it follows tht: p 1 D 1 (M[φ ]) p 2 for some D 1 such tht s 1 U D 1. D 2 (M[φ ]) < p 1 or D 2 (M[φ ]) > p 2 for ll D 2 such tht s 2 U D 2. Since φ F i IPML, by the induction hypothesis there exists G 2S/ i PTB,gbg such tht C G C = M[φ ]. Then: D 1 ( G) = q R [p1,p 2]. D 2 ( G) q for ll D 2 such tht s 2 U D 2. Therefore s 1 i+1 PTB,gbg s 2. ( =) If s 1 i+1 PTB,gbg s 2, then there re two cses: If s 1 i PTB,gbg s 2, then by the induction hypothesis it holds tht FIPML i (s 1) FIPML i (s 2) nd hence F i+1 IPML (s 1) F i+1 IPML (s 2). If s 1 i PTB,gbg s 2, then from s 1 i+1 PTB,gbg s 2 it follows tht there exist p R [0,1] nd G 2 S/ i PTB,gbg such tht: D 1 ( G) = p for some D 1 such tht s 1 U D 1. D 2 ( G) p for ll D 2 such tht s 2 U D 2. Let G 1 = {C S/ i PTB,gbg D 1(C) > 0} nd G 2 = {C S/ i PTB,gbg D 2. s 2 U D 2 D 2 (C) > 0}. Thnks to the ssumptions of imge finiteness nd miniml probbility, both G 1 nd G 2 re finite. By the induction hypothesis, there exists distinguishing formul φ <C1,C 2> F i IPML for ll C 1 nd C 2 in S/ i PTB,gbg such tht C 1 C 2, i.e.: C 1 M[φ <C1,C 2> ] C 2 M[φ <C1,C 2> ] = Then: φ G = ( ) φ <C,C1> C G C 1 G 1\{C} φ <C,C2> C 2 G 2\{C} where i I φ i = i I φ i for I finite nd i I φ i = true for I =, yields distinguishing formul for s 1 nd s 2 becuse: s 1 M[ [p,p] φ G ]. s 2 M[ [p,p] φ G ]. Since [p,p] φ G F i+1 i+1 IPML, we derive tht FIPML (s 1) F i+1 IPML (s 2).

21 Proof of Thm. 6. It follows immeditely from the trnsltion in Def. 18 fter observing tht the discriminting power of PEB does not chnge if, in the definition of the equivlence (Def. 19), we replce equivlence clsses with groups of equivlence clsses. In prticulr, note tht, if s 1 PEB s 2 nd both sttes hve outgoing timetrnsitions, then in the corresponding R [0,1] -ULTrS ech of the two sttes hs single outgoing trnsition representing time pssing. The lbel of this trnsition contins the sme rte for both sttes s cn be seen by tking G equl to the set of ll equivlence clsses so tht G = S in the second cluse of the group-bsed version of Def. 19.