MOdal logics and behavioural equivalences are very

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1 WU AND DENG: DISTRIBUTION-BASED BEHAVIOURAL DISTANCE FOR NONDETERMINISTIC FUZZY TRANSITION SYSTEMS 1 Distribution-Bsed Behviourl Distnce for Nondeterministic Fuzzy Trnsition Systems Hengyng Wu nd Yuxin Deng Abstrct Modl logics nd behviourl equivlences ply n importnt role in the specifiction nd verifiction of concurrent systems. In this pper, we first present new notion of bisimultion for nondeterministic fuzzy trnsition systems, which is distribution-bsed nd corser thn stte-bsed bisimultion ppered in the literture. Then, we define distribution-bsed bisimilrity metric s the lest fixed point of suitble monotonic function on complete lttice, which is behviourl distnce nd is more robust wy of formlising behviourl similrity between sttes thn bisimultions. We lso propose n on-the-fly lgorithm for computing the bisimilrity metric. Moreover, we present fuzzy modl logic nd provide sound nd complete chrcteriztion of the bisimilrity metric. Interestingly, this chrcteriztion holds for clss of fuzzy modl logics. In ddition, we show the non-expnsiveness of typicl prllel composition opertor with respect to the bisimilrity metric, which mkes compositionl verifiction possible. Index Terms Fuzzy trnsition system, Bisimultion, Behviourl distnce, Modl logic, Non-expnsiveness. I. INTRODUCTION MOdl logics nd behviourl equivlences re very importnt for the specifiction nd verifiction of concurrent systems. The former cn be used for model checking, prticulrly for specifying the properties to be verified. The ltter cn be used for stte-ggregtion lgorithms tht compress models by merging bisimilr sttes but gurntee tht the required properties re preserved. Recently, bisimultions hve been investigted in fuzzy systems. For exmple, Co et l. [1], [2] considered bisimultions for deterministic (resp. nondeterministic) fuzzy trnsition systems, bbrevited s FTSs (resp. NFTSs); Ćirić et l. [3] investigted bisimultions for fuzzy utomt; Qiu nd Deng [4], nd Xing et l. [5] studied (bi)simultions for fuzzy discrete event systems; Fn [6] defined bisimultions for fuzzy Kripke structures. For more informtion bout fuzzy (bi)simultions, we lso refer to [7] [14]. The bisimultion proposed by Co et l. [2] is stte-bsed nd represents brnching-time semntics. For instnce, it cn distinguish sttes s nd t in Fig.1. However, if we re minly interested in the mximl possibilities of the occurrences This work is prtilly supported by the Ntionl Nturl Science Foundtion of Chin ( , , , ), Shnghi Municipl Nturl Science Foundtion (16ZR ), nd ANR 12IS02001 PACE. H. Wu is with MoE Engineering Center for Softwre/Hrdwre Co-Design Technology nd Appliction, Est Chin Norml University, Shnghi , P.R. Chin. Emil: wuhengy 1974@liyun.com. Y. Deng (corresponding uthor) is with Shnghi Key Lbortory of Trustworthy Computing, MOE Interntionl Joint Lb of Trustworthy Softwre nd Interntionl Reserch Center of Trustworthy Softwre, Est Chin Norml University, Shnghi , P.R. Chin. Emil: yxdeng@sei.ecnu.edu.cn. of some events, then the two sttes should be intuitively identified. After performing the sme ction, both s nd t cn rech some stte with mximl possibility 0.8; fter performing the ction immeditely followed byb, both s nd t cn rech some stte with mximl possibility 0.6. Bsed on this observtion, we introduce notion of distributionbsed bisimultion tht represents liner-time semntics. This bisimultion is close to the clssicl bisimultion [20], which is bsed on trnsitions between (bre) sttes. We view distributions s generlized sttes nd thus define trnsitions between distributions. Two bisimilr distributions re required to hve the sme height, which shows the sme mximl possibility of reching some sttes, nd fter some mtching trnsitions, the two successor distributions still preserve the sme height. s s 1 s 2 b 0.8 t t 1 t 2 b 0.6 Fig.1 s nd t re distribution-bisimilr but not stte-bisimilr Built on bisimultions, notion of behviourl ultrmetric [2] hs been proposed to mesure the similrity of sttes in n NFTS. It is more robust wy of formlising behviourl similrity between fuzzy systems thn bisimultions. The smller the behviourl distnce, the more similrly the sttes behve. In prticulr, the behviourl distnce between two sttes is 0 if nd only if they re exctly bisimilr. In this pper, we generlize behviourl ultrmetrics nd propose notion of tringulr-conorm-bsed metric, clled -metric, to mesure the distnce between possibility distributions in n NFTS. Techniclly speking, we define distribution-bsed bisimilrity metric s the lest fixed point of suitble monotonic function, which cptures the similrity of the behviour of possibility distributions. We lso show tht prllel composition in the style of communicting sequentil processes (CSP) [15] is non-expnsive with respect to distribution-bsed bisimilrity metric, which mkes compositionl verifiction possible. Our metric differs from the ultrmetric of Co et l. [2] in two spects: (1) the former is defined on F(S), while the ltter is defined on S; (2) the former is bsed on clss of t 3

2 WU AND DENG: DISTRIBUTION-BASED BEHAVIOURAL DISTANCE FOR NONDETERMINISTIC FUZZY TRANSITION SYSTEMS 2 tringulr-conorms, while the lter is only bsed on -conorm, which is specil cse of ours. This generliztion llows us, ccording to the properties of different systems, to compute behviourl distnces by choosing pproprite opertors. Modl logics nd behviourl equivlences re closely relted. Whenever new equivlence is proposed, the quest strts for the ssocited logic such tht two systems (or sttes) re behviourlly equivlent if nd only if they stisfy the sme set of modl formule. Along this line, gret mount of work hs ppered tht chrcterizes vrious kinds of clssicl (or probbilistic) bisimultions by pproprite logics, e.g. [16] [24]. Although bisimultions hve been investigted extensively in fuzzy systems, there is little work bout the connection between bisimultions nd the corresponding modl logics. Fn [6] chrcterized (fuzzy) bisimultions for fuzzy Kripke structures in terms of Gödel modl logic; Wu nd Deng [14] chrcterized bisimultions for FTSs by using fuzzy style Hennessy-Milner logic. In this pper, we present fuzzy modl logic, which is directly interpreted on distributions. This logic chrcterizes not only distribution-bsed bisimilrity but lso distribution-bsed bisimilrity metric for NFTSs soundly nd completely. We show tht the distnce between two distributions is 0 if nd only if they hve the sme vlue on ech formul. Moreover, this chrcteriztion holds for clss of fuzzy modl logics, by instntiting tringulr-norm (tnorm) s Łuksiewicz t-norm, Gödel t-norm, product t-norm, or nilpotent minimum t-norm. The rest of this pper is structured s follows. We briefly review some bsic concepts on fuzzy sets in Section II nd introduce fuzzy logic used in this pper. In Section III, we define new bisimultion for NFTSs, which is distribution-bsed. Section IV embrks upon the development of distributionbsed bisimilrity metric. We first define tringulr-conormbsed metric, clled -metric. Then we give the notion of distribution-bsed bisimultion metric nd discuss the monotonicity of function on behviourl metrics. Thnks to Trski s fixed point theorem, we cn obtin the lest fixed point of the function nd define it s the distribution-bsed bisimilrity metric. Moreover, we present n on-the-fly lgorithm to compute the bisimilrity metric. In Section V, we give fuzzy modl logic, which chrcterizes distribution-bsed bisimilrity metric soundly nd completely. In Section VI, we define prllel composition nd show tht it is non-expnsive with respect to the distribution-bsed bisimilrity metric nd preserves distribution-bsed bisimilrity. As n exmple, we discuss the potentil ppliction of our results in Section VII nd review some relted work in Section VIII. Finlly, we conclude in Section IX with some future work. A. Fuzzy Set II. PRELIMINARIES We first briefly recll some bsic concepts on fuzzy sets. Let S be n ordinry set. A fuzzy set µ of S is function tht ssigns ech element s of S with vlue µ(s) in the unit intervl [0,1]. The support of µ, written supp(µ), is the set {s S µ(s) > 0}. We denote by F(S) the set of ll fuzzy sets in S. For convenience, whenever supp(µ) is finite set, sy {s 1,s 2,,s n }, then fuzzy set µ is written s µ = r1 s 1 + r2 s rn s n, where r i (0,1] nd r i = µ(s i ) with 1 i n. With slight buse of nottions, we sometimes write possibility distribution to men fuzzy set 1. For ny µ,ν F(S), we sy tht µ is contined in ν (or ν contins µ), denoted by µ ν, if µ(s) ν(s) for ll s S. Notice tht µ = ν if both µ ν nd ν µ. We use to denote the empty fuzzy set with (s) = 0 for ll s S. For ny s S, we write s for the point distribution, stisfying s(s ) = 1 if s = s nd 0 otherwise. For ny p,q [0,1], we write p q nd p q to men mx(p,q) nd min(p,q), respectively, write i I p i for the supremum of {p i i I}, nd i I p i for the infimum, where {p i i I} is fmily of elements in [0,1]. In prticulr, = 0 nd = 1. For ny c [0,1] nd µ F(S), the sclr multipliction c µ of c nd µ is defined by letting (c µ)(s) = c µ(s), for ech s S. Let {µ i i I} be fmily of elements of F(S), the union of µ i, denoted µ i, is given by ( i I µ i)(s) = i I µ i(s). The height of possibility distribution µ, written ht(µ), is defined s s S µ(s). Throughout this pper, we ssume n ordinry finite set S nd let it denote the set of sttes of fuzzy trnsition system. B. Fuzzy Logic Let us introduce n lgebric structure, which will ply key role in defining the bisimilrity metric nd in its logicl chrcteriztion. Definition 1. [26] Let (C,, 0, 1) be bounded linerly ordered set. A function n : C C is clled n involution or strong negtion function if it is order-reversing with n(1) = 0 nd n(n(x)) = x for ll x C. When C is tken s the unit intervl [0,1], the involutive negtion is usully interpreted by the stndrd negtion: n(x) = 1 x for ny x [0,1]. Definition 2. We consider the unit intervl [0,1] s n lgebr endowed with the opertions min nd mx, fixed leftcontinuous t-norm nd its residum, s well s the stndrd negtion n. The ordering is s usul. This lgebr will be denoted by L. Notice tht left-continuity is necessry nd sufficient condition for t-norm nd its residuum, defined s x y = sup{z [0,1] x z y}, to verify the soclled residution property: x y z iff x y z. In this cse, we sy (, ) is n djoint pir [26]. For the bsic properties of t-norms we refer to [27], [28]. There re four commonly used t-norms: Łuksiewicz t-norm (x y = mx(x+y 1, 0)), Gödel t-norm (x y = min(x, y)), product t-norm (x y = x y), nd nilpotent minimum t-norm [29] (x y = min(x,y) if x+y > 1 nd0otherwise). The first three re continuous, wheres the fourth is left-continuous. The corresponding lgebrs re denoted by L L, LG, LP, nd LN, respectively. For convenience, we sometimes use the symbols 1 Strictly speking possibility distribution is different from fuzzy set, though the former cn be viewed s the generlized chrcteristic function of the ltter. See [25] for more detiled discussion.

3 WU AND DENG: DISTRIBUTION-BASED BEHAVIOURAL DISTANCE FOR NONDETERMINISTIC FUZZY TRANSITION SYSTEMS 3 L, G, P nd N to denote the implictions in the lgebrs L L, L G, L P, L N, respectively, which cn be found in [28], [29]. In generl, n importnt im of introducing n lgebric structure is to estblish the completeness of forml system. Exmples include the BL-lgebr for the bsic logic introduced by Hájek [28] nd the MTL-lgebr for the monoidl t-norm bsed logic introduced by Estev nd Godo [26]. Nevertheless, our motivtion is to estblish n bstrct metric (see Section IV) nd to chrcterize it (see Section V). Below we discuss the resons of choosing L by compring it with some relevnt lgebrs. The stndrd semntics of BL (resp. MTL) uses the unit intervl [0,1] s the set of truth vlues nd continuous (resp. left-continuous) t-norm s n interprettion of the strong conjunction &. The impliction is interpreted by the residuum of. It is well known tht ([0, 1], min, mx,,, 0, 1) is n MTL-lgebr [26], nd it is BL-lgebr provided tht is continuous [28]. In BL nd MTL-lgebrs, the negtion x = x 0 is in generl not involutive. Thus, BL nd MTL s well s some of their xiomtic extensions lck strong disjunction dul to the strong conjunction. Attempts hve been mde to generlize them. For exmple, n IMTLlgebr [26], where I stnds for involution, is obtined by dding x = x to n MTL-lgebr. However, in IMTL the involution depends on the t-norm. As consequence, IMTL dmits only those left-continuous t- norms which yield n involutive negtion, but rules out opertors like Gödel nd product t-norms. In order to fill such gp, n MTL n -lgebr 2 is obtined by dding n independent involutive negtion n nd δ opertion to MTL; see [30] for more detils. The extensions of BL-lgebr cn be found in, for exmple, [31], [32]. Obviously, our L is like the MTL n -lgebr giving up the opertor δ. In L, we cn obtin t-conorm by letting x y = 1 [(1 x) (1 y)], which is dul to nd (,,n) is clled De Morgn triple. In this pper, the opertor n is used for defining ; the opertor is used for producing the djoint pir (, ) nd the De Morgn triple(,,n). It is not used to interpret the strong conjunction & s BL nd MTL-lgebrs do, since ϕ&ψ in generl is not formul in our fuzzy modl logic; the induced opertor is used for defining n bstrct metric nd for modeling the strong disjunction (see Definition 6). Note tht it is not used to interpret the strong disjunctive connective between formule, since ϕ ψ in generl is not formul in our fuzzy modl logic. This point is very different from [30], [32]. The induced biimpliction opertor helps us to model equivlence nd is given by x y = (x y) (y x) (see [3], [27]), which lso devites from BL nd MTL-lgebrs, where it is defined s x y = (x y) (y x). For more differences between L nd the bove lgebrs, plese see Section V. 2 We useninsted of becuse this notion will be reserved for bisimilrity. The De Morgn triple (,, n) nd the djoint pir (, ) gurntee the min results of this pper to hold. The following properties will be useful. Proposition 1. Under L, for ny x,y,z,x 1,x 2,y 1,y 2,y i in the intervl [0,1], where i I. The following properties hold. (1) x y = 1 if nd only if x y, (2) 1 x = x, (3) x (x y) y, (4) (x y) (y z) x z, (5) (x 1 y 1 ) (x 2 y 2 ) (x 1 x 2 ) (y 1 y 2 ), (6) x 1 x 2 (x 1 y) (x 2 y), (7) x 1 x 2 (y x 1 ) (y x 2 ), (8) x ( i I y i) i I (x y i), (9) x ( i I y i) = i I (x y i). Proof. The first five items cn be found in [27], [28], [33]. Items (6) nd (7) cn be proven by djoint condition. As n exmple, we choose to prove (6). Let t x 1 x 2. Then t x 1 x 2 ndt x 2 x 1. It follows by djoint condition tht t x 1 x 2 nd t x 2 x 1. Further, we hve tht t x 2 (x 1 y) x 1 (x 1 y) y by (3). Agin, by djoint condition, we hve tht t (x 1 y) x 2 y, nd then t (x 1 y) (x 2 y). In similr wy, we cn get t (x 2 y) (x 1 y). So t (x 1 y) (x 2 y). Thus, we obtin (x 1 x 2 ) (x 1 y) (x 2 y). Item (8) is trivil since is monotonic. Item (9) cn be obtined by using the dulity between nd nd the leftcontinuity of. III. DISTRIBUTION-BASED BISIMULATION In this section, we introduce the notion of distribution-bsed bisimultion for nondeterministic fuzzy trnsition systems. Definition 3. [34] A nondeterministic fuzzy trnsition system (NFTS) is triple M = (S,A, ), where S is set of sttes, A is set of ctions, nd S A F(S) is the trnsition reltion. For ech stte s, more thn one possibility distribution my be reched by performing ction becuse of nondeterminism. We often write s µ for (s,,µ). We write s if there exists no µ such tht s µ. Let us define trnsition reltion between distributions in n NFTS. Definition 4. Given n NFTS M = (S,A, ), the trnsition reltion is lifted to d F(S) A F(S) by tking the smllest reltion stisfying: if s µ then s d µ; if s then s d ; if s d ν s for ech s supp(µ), then µ d µ(s) ν s. s supp(µ) We often buse the nottion nd d is lso denoted by. We define T (µ) s the set {ν F(S) µ ν}. An NFTS (S,A, ) is imge-finite if for ech distribution µ nd ech ction, the set T (µ) is finite. The NFTS is finitry if it is imge-finite nd hs finitely mny sttes.

4 WU AND DENG: DISTRIBUTION-BASED BEHAVIOURAL DISTANCE FOR NONDETERMINISTIC FUZZY TRANSITION SYSTEMS 4 Note tht if µ ν then some (not necessrily ll) sttes in the support of µ cn perform ction. For exmple, consider b the two sttes s 1 nd s 2 in Figure 1. Since s nd s 2 cnnot perform ction b, the distribution µ = 0.6 s s 2 cn mke the trnsition µ b 0.6 to rech the distribution 0.6. Definition 5 (Distribution-bsed bisimultion). A symmetric reltion R F(S) F(S) is distribution-bsed bisimultion on n NFTS (S,A, ) if for ll µ,ν F(S), whenever µrν then: (i) ht(µ) = ht(ν) nd (ii) if µ µ, then there exists some ν F(S) such tht ν ν nd µ Rν. Two distributions µ,ν F(S) re bisimilr, written µ ν, if there exists distribution-bsed bisimultion R with µrν. It is esy to see tht is the gretest distribution-bsed bisimultion since it is the union of ll distribution-bsed bisimultions. Moreover, it is n equivlence reltion on F(S). IV. DISTRIBUTION-BASED BISIMILARITY METRIC Before defining the distribution-bsed bisimilrity metric, we first introduce tringulr conorm-bsed metric. A. Tringulr Conorm-Bsed Metric Recll tht the similrity reltion on S proposed by Zdeh in [35] is fuzzy reltion S on S, i.e., function of type S S [0,1], tht stisfies the following three conditions: (1) reflexive, i.e., S(s,s) = 1 for ll s S, (2) symmetric, i.e., S(s,t) = S(t,s) for ll s,t S, (3) mx-str trnsitive, i.e., S S S, or, more explicitly, S(s,s ) S(s,t) S(s,t) s S where is ssocitive nd monotone non-decresing in ech of its rguments. Now, we tke to be t-norm nd define, for ll s,t S, d(s,t) = 1 S(s,t). It turns out tht, for ll s,s,t S, we hve d(s,t) d(s,s ) d(s,t). Moreover, d(s,s) = 0 nd d(s,t) = d(t,s). We cll the distnce function d -metric. Next, we generlize the -metric from S to F(S). Definition 6. Under L, function d : F(S) F(S) [0,1] is clled -metric on F(S) if for ll µ,ν,η F(S), (P1) d(µ,µ) = 0, (P2) d(µ,ν) = d(ν,µ), (P3) d(µ,η) d(µ,ν) d(ν,η). We cll (F(S),d) -metric spce. This definition generlizes the notion of the pseudoultrmetric given by Co et l. in [2] in two spects: (1) our metric is directly defined on distributions, wheres Co s is defined on sttes; (2) we use insted of in Co s definition, i.e, the mx-disjunction is replced by the strong disjunction. The following proposition indictes tht -metric lwys exists. Proposition 2. Given t-conorm from L, the -metric lwys exists. Proof. Let (µ,ν) = 1 (ht(µ) ht(ν)) for ny distributions µ,ν F(S). Then it is esy to see tht stisfies P1 nd P2. Now, we verify P3. For ny µ,ν,η F(S), we hve (µ,ν) (ν,η) = [1 (ht(µ) ht(ν))] [1 (ht(ν) ht(η))] = 1 (ht(µ) ht(ν)) (ht(ν) ht(η)) 1 (ht(µ) ht(η)) (by Proposition 1 (4)) = (µ,η) s expected. We cn understnd (µ, ν) s how fr distributions µ nd ν re from the sme height bsed on the -metric. If is tken from L L, then we hve (µ,ν) = ht(µ) ht(ν), which gives pseudometric bounded by 1 often used in probbilistic systems. If is tken from L G, then (µ,ν) = 0 when ht(µ) = ht(ν), nd mx(1 ht(µ), 1 ht(ν)) otherwise, which is pseudo-ultrmetric. In the sme wy, one cn get (µ,ν) when is tken from L N nd LP, respectively. A pseudo-ultrmetric is -metric becuse is the minimum t-conorm, but the inverse is not necessrily true, s witnessed by when is tken from L L, L N, nd L P, respectively. Note tht is importnt for defining our bisimilrity metric nd we use it s specil nottion in the sequel. Let D F(S) be the set of ll -metrics onf(s). The order on D F(S) is defined by letting d 1 d 2 if for ll µ,ν F(S) we hve d 1 (µ,ν) d 2 (µ,ν). It is esy to check tht is indeed prtil order on D F(S). Recll tht prtilly ordered set (X, ) is clled complete lttice if every subset of X hs supremum nd n infimum in X. We now show tht D F(S) endowed with the bove order forms complete lttice. Lemm 1. Under L, (D F(S), ) is complete lttice. Proof. For ny X D F(S), we define X by letting ( X)(µ,ν) = {d(µ,ν) d X} nd X by letting X = {d DF(S) d X,d d }. We need to verify tht X nd X re the supremum nd infimum of X, respectively. We only prove tht X is the supremum; the infimum cn be proven similrly. We first show tht X DF(S). It is obvious tht ( X)(µ,µ) = 0 nd ( X)(µ,ν) = ( X)(ν,µ). For (P3), we hve tht ( X)(µ,ν) = d D F(S) d(µ,ν) d D F(S) (d(µ,η) d(η,ν)) d D F(S) d(µ,η) d D F(S) d(η,ν) = [( X)(µ,η)] [( X)(η,ν)] for ny µ,ν,η F(S). Hence X is -metric. In ddition, it is not hrd to prove tht X is the supremum of X. Hence (D F(S), ) is complete lttice. Husdorff distnce mesures how fr two subsets of metric spce re from ech other, which llows us to lift -metric on F(S) to -metric on P(F(S)).

5 WU AND DENG: DISTRIBUTION-BASED BEHAVIOURAL DISTANCE FOR NONDETERMINISTIC FUZZY TRANSITION SYSTEMS 5 Definition 7. Let (F(S), d) be -metric spce. For ny µ F(S) nd finite subset A of F(S), define d(µ,a) = d(µ,ν). ν A Furthermore, given pir of finite subsets A,B F(S), the Husdorff distnce induced by d is defined s H d (A,B) = [ d(µ,b)] [ d(ν,a)]. µ A ν B As expected, H d hs the following property. Lemm 2. Under L, if d is -metric on F(S), then H d is -metric on P f (F(S)), where P f (F(S)) stnds for the set of ll finite subsets of F(S). Proof. Firstly, it is esy to see tht H d (A,A) = 0 nd H d (A,B) = H d (B,A). Secondly, H d (A,B) H d (B,C) [ µ A d(µ,b)] [ ν B d(ν,c)] µ A ν B [d(µ,b) d(ν,c)] (by Proposition 1 (8)) µ A ν B [d(µ,ν ) d(ν,c)] (for some ν B) µ A [d(µ,ν ) d(ν,c)] = µ A [d(µ,ν ) ( η C d(ν,η))] = µ A [ η C (d(µ,ν ) d(ν,η))] µ A η C d(µ,η) = µ A d(µ,c). Similrly, we hve H d (A,B) H d (B,C) η C d(η,a). Hence, H d (A,B) H d (B,C) [ µ A d(µ,c)] [ η C d(η,a)] = H d (A,C). This completes the proof. The following lemm shows tht the distnce between distribution nd finite set of distributions is preserved by the sup of n incresing -metric chin. Lemm 3. Let (F(S),d) be -metric spce, {d n n N} be n incresing chin, i.e., d 0 d 1 d 2, nd B be finite subset of F(S). Then for ny µ F(S), we hve ( d n )(µ,b) = n (µ,b). n N n Nd Proof. It is sufficient to prove tht inf sup d n (µ,ν) = sup inf d n(µ,ν). ν B ν B n N n N First, for ny n N, it is cler tht inf d n(µ,ν) inf ν B sup ν B n N d n (µ,ν). This shows tht inf ν B sup n N d n (µ,ν) is n upper bound of the set {inf ν B d n (µ,ν) n N}. Second, for ny ν B, we consider sup n N d n (µ,ν). For ny ǫ > 0, there exists some n ν such tht d nν (µ,ν) > supd n (µ,ν) ǫ. n N Since B is finite nd d n (n N) is n incresing chin, there exists d such tht d(µ,ν) > sup n N d n (µ,ν) ǫ for ny ν B. Hence, inf d(µ,ν) > inf (sup d n (µ,ν) ǫ) = inf ν B ν B n N sup ν B n N d n (µ,ν) ǫ. Thus, inf ν B sup n N d n (µ,ν) is the supremum of the set {inf ν B d n (µ,ν) n N}. We hve completed the proof. An immedite corollry is tht the Husdorff distnce between two finite sets of possibility distributions is preserved by the sup of n incresing -metric chin. Corollry 1. Let (F(S),d) be -metric spce, A,B be finite subsets of F(S), nd {d n n N} be n incresing chin. Then H n N dn(a,b) = n N H d n (A,B). Proof. Strightforwrd by using Lemm 3. B. Distribution-Bsed Bisimultion Metric Inspired by [36], we define bisimilrity metric tht cn mesure the behviourl distnce between two distributions. Definition 8. Given n NFTS (S,A, ) nd L, -metric d on F(S) is clled distribution-bsed bisimultion metric if for ll µ,ν F(S), (i) d nd (ii) d(µ,ν) < 1 implies tht whenever µ µ then there exists trnsition ν ν with d(µ,ν ) d(µ,ν). The first condition sys tht d should be t lest. The second condition requires the mtching of trnsitions nd the distnce between the two successor distributions should not exceed the distnce between the originl distributions. Below we im to give fixed-point chrcteriztion of bisimultion metric. Definition 9. Given n imge-finite NFTS (S, A, ), the functionl : D F(S) D F(S) is defined s follows: for ny d D F(S) nd µ,ν F(S), (d)(µ,ν) = mx( H d (T (µ),t (ν)), (µ,ν)). A Now, we check tht is well-defined. Proposition 3. Given n imge-finite NFTS (S, A, ), then for ny d D F(S), we hve (d) D F(S). Proof. It is esy to see tht (d) stisfiesp1 ndp2. Now, we verify P3. For ny µ,ν,η F(S), it is obvious tht, for ny A, (d)(µ,ν) H d (T (µ),t (ν)) nd symmetriclly (d)(ν,η) H d (T (ν),t (η)). Hence, for ny A, Consequently, (d)(µ,ν) (d)(ν,η) H d (T (µ),t (ν)) H d (T (ν),t (η)) H d (T (µ),t (η)) (by Lemm 2). (d)(µ,ν) (d)(ν,η) H d (T (µ),t (η)). On the other hnd, we lso hve A (d)(µ,ν) (µ,ν) nd (d)(ν,η) (ν,η).

6 WU AND DENG: DISTRIBUTION-BASED BEHAVIOURAL DISTANCE FOR NONDETERMINISTIC FUZZY TRANSITION SYSTEMS 6 Hence, (d)(µ,ν) (d)(ν,η) (µ,ν) (ν,η) (µ,η) by Proposition 2. It follows tht (d)(µ,ν) (d)(ν,η) mx( A H d(t (µ),t (η)), (µ,η)). Tht is, (d)(µ, ν) (d)(ν, η) (d)(µ, η) s desired. Moreover, is monotonic with respect to the order, i.e., if d 1 d 2, then we hve (d 1 ) (d 2 ). Recll tht the Knster-Trski fixed-point theorem [37] sys tht ech monotonic function on complete lttice hs lest fixed point. Hence the following proposition holds. Proposition 4. Given n imge-finite NFTS (S, A, ), the functionl : D F(S) D F(S) hs lest fixed point given by min = {d : (d) d}. Let us present n explicit chrcteriztion of bisimultion metrics in terms of pre-fixed points of. Lemm 4. Given n imge-finite NFTS (S,A, ), for ny d D F(S), d is pre-fixed points of if nd only if it is distribution-bsed bisimultion metric. Proof. Strightforwrd by the definitions of distribution-bsed bisimultion metric nd. Proposition 4 nd Lemm 4 imply the following theorem. Theorem 1. Given n imge-finite NFTS (S, A, ). The smllest distribution-bsed bisimultion metric exists, moreover it is just min. Let d b = min nd cll it distribution- def bsed bisimilrity metric. In wht follows, we use d L b, dg b, dp b, nd dn b to denote distribution-bsed bisimilrity metrics bsed on L L, L G, L P nd LN, respectively. Under the ssumption of imgefiniteness, Corollry 1 implies tht preserves the sup of n incresing chin of D F(S). Lemm 5. Given n imge-finite NFTS (S,A, ), we hve ( d n ) = n ) n N n N (d where d n (n N) is n incresing chin of D F(S). Combining Theorem 1 nd Lemm 5 gives wy to clculte the distribution-bsed bisimilrity metric for imgefinite NFTSs. Corollry 2. Let (S, A, ) be n imge-finite NFTS. We define 0 ( ) def = nd n+1 ( ) def = ( n ( )). Then d b = min = n N n ( ). Proof. The proof is stndrd; see e.g. Pge 183 in [37]. We give simple exmple below to illustrte the bove notions nd results. Exmple 1. Consider Figure 1 gin. Let µ 1 = 0.6 s s 2, µ 2 = 0.8, ν 1 = 0.6 t t 2 nd ν 2 = 0.6 t 3. We write d n for n ( ), clerly d i d i+1 for ny i N. Now, we compute d G b ( s, t) nd d G b ( s 1, t 2 ). For ny n 3, d n ( s, t) = (d n 1 )( s, t) = mx(d n 1 (µ 1,ν 1 ),1 (ht( s) G ht( t))) = d n 1 (µ 1,ν 1 ) = (d n 2 )(µ 1,ν 1 ) = mx(d n 2 ( 0.6, 0.6 t 3 ),1 (ht(µ 1 ) G ht(ν 1 ))) = d n 2 ( 0.6, 0.6 t 3 ) = (d n 3 )( 0.6, 0.6 t 3 ) = mx(d n 3 (, ),1 (ht( 0.6 ) G ht( 0.6 t 3 ))) = 0. Hence d G b ( s, t) = 0. In the sme wy, we cn clculte tht d G b ( s 1, t 2 ) = 0.4. For ny pir of possibility distributions µ nd ν, we observe tht the smller the vlue of d b (µ,ν), the more similr the distributions. Corollry 3. Given n imge-finite NFTS (S,A, ). For ny s,t S, let S(s,t) def = 1 d b ( s, t). Then S is similrity reltion on S. The bove corollry reltes distribution-bsed bisimilrity metric to Zdeh s similrity reltion. We cn see tht plys key role in computing the distribution-bsed bisimilrity metric, which is obtined by n itertion of strting from. This pper uses the lest fixed point to estblish the distribution-bsed bisimilrity metric, while Co et l. [2] used the gretest fixed point to estblish the behviourl distnce. However, there is no essentil difference becuse in [2] the prtil order on D F(S) is reverse, i.e., d 1 d 2 if d 1 (s,t) d 2 (s,t) for ll s,t S. C. An Algorithm for Computing d b First of ll, let us fix finitry NFTS M = (S,A, ). Given ny pir of possibility distributions µ, ν F(S), we present n on-the-fly lgorithm to compute their distnce. In order to compute the distnce d b (µ,ν), we need to successively compute d n (µ,ν) = n ( )(µ,ν) by Corollry 2, until n N ppers such tht d N (µ,ν) = d N+1 (µ,ν). Then d b (µ,ν) = d N (µ,ν). The tricky point is tht there my be severl n stisfying d n (µ,ν) = d n+1 (µ,ν) nd we cnnot simply choose ny of them to be N. We hve to mke sure tht similr condition d N (µ,ν ) = d N+1 (µ,ν ) lso holds for ny µ nd ν rechble from µ nd ν, respectively. Algorithm 1 gives the detils. The min procedure is distnce(µ, ν), which first uses the procedure generte to generte ll the generlized sttes rechble from µ nd ν, then repetedly clls dist(n,µ,ν ) for computing d n (µ,ν ). The procedure dist(n, µ, ν) fithfully implements the inductive definition n ( )(µ,ν) for n 0. When sufficiently lrge number n is obtined such tht d n (µ,ν ) = d b (µ,ν ) for ny µ nd ν rechble from µ nd ν, respectively, the lgorithm termintes. Therefore, the correctness of the lgorithm is ensured by the fixed point chrcteriztion of the bisimilrity metric given in Corollry 2.

7 WU AND DENG: DISTRIBUTION-BASED BEHAVIOURAL DISTANCE FOR NONDETERMINISTIC FUZZY TRANSITION SYSTEMS 7 Proposition 5. For ny infinite sequence of trnsitions µ µ1 3 µ2 there exists some i 0 such tht for ll j > i we hve ht(µ j ) = ht(µ i ). Proof. Suppose for contrction tht i, j > i with ht(µ j ) ht(µ i ). So in prticulr, for i = 0, j 1 > 0 such tht ht(µ j1 ) ht(µ 0 ). Then for j 1, j 2 > j 1 such tht ht(µ j2 ) ht(µ j1 ). Continuing this process, we will get tht ht(µ jn ) ht(µ jn 1 ) for ny n 2. Note tht Definition 4 implies tht ht(µ) ht(ν) for ny µ ν with A. Hence, we get tht ht(µ j1 ) > ht(µ j2 ) >, i.e., n infinite descending chin of non-zero vlues of ht(µ jn ). However, this is impossible. The reson is s follows. Let X = {µ 0 (s ) s S} nd Y be the set defined by {ν(s ) s S, s S,s ν}. Note tht the two sets re finite becuse S is finite nd the trnsitions re finite. Now, by Definition 4, µ 1 = s supp(µ 0) µ 0(s) ν s, where the distribution ν s is obtined by s in supp(µ 0 ) performing the ction. For ny s S, µ 1 (s ) only tkes vlues in the set X or Y, implying tht ht(µ 1 ) only tkes vlues in these two sets. In the sme wy, ech ht(µ jn )(n 1) only tkes vlue in these two sets, which is in contrdiction with the fct tht {ht(µ jn ) n 1} hs n infinite descending chin. This proposition shows tht for ny given distribution µ, fter finitely mny trnsitions, the height of the resulting distributions will not chnge ny more, which is importnt for the termintion of the lgorithm tht we will soon present. Theorem 2. Under the L, the lgorithm termintes nd is EXPTIME. Proof. We first show the lgorithm termintes. Proposition 5 sys tht for ny chin of trnsitions µ b µ 1 µ2, the height of distributions in this chin will not chnge ny more fter finite number of steps. The proof of tht proposition lso sys tht the trnsition systems between distributions re finitry, which is importnt for the nlysis below. For ny µ,ν F(S) nd n 0, we know tht d n+1 (µ,ν) = mx( H dn (T (µ),t (ν)), (µ,ν)). (1) A If µ nd ν re fixed, the choices on the right hnd side re finite: d n+1 (µ,ν) is determined by either (µ,ν) or d n (µ i,ν j ) with µ µ i nd ν ν j, where there re only finitely mny outgoing trnsitions from µ nd ν. However, (1) holds for ny n 0. It must be the cse tht either or d n+1 (µ,ν) = (µ,ν), (2) d n+1 (µ,ν) = d n (µ i,ν j ) (3) for some fixed trnsitions µ µ i nd ν ν j, but infinitely mny different n s. In other words, the distnce between µ nd ν is either fixed number (µ,ν) or determined by pir of their successor distributions. In the ltter cse, the sme remrk cn be mde gin. Eventully, the computtion of dist(n, µ, ν) (n 0) boils down to the computtions Algorithm 1 Compute the distnce between two distributions Input: A finite NFTS, nd µ,ν F(S). Output: d b (µ,ν). procedure distnce(µ, ν) n 1; for ll µ i generte(µ) do for ll ν j generte(ν) do n ij 1; while dist(n ij 1,µ i,ν j ) dist(n ij,µ i,ν j ) do n ij n ij +1; end while if n ij > n then n n ij end if return dist(n, µ, ν); end procedure procedure generte(µ) D {µ} repet for ll A do for ll µ D do D D T (µ ) until D = D return D procedure dist(n, µ, ν) if n = 0 then return 1 (ht(µ ) ht(ν )); else for ll A do for ll µ i T (µ) do for ll ν j T (ν) do d ij dist(n 1,µ i,ν j ); return (( i end if end procedure j d ij) ( j i d ij)) dist(0,µ,ν); of (µ k,ν l ), where µ k nd ν l re elements in the chins generted by µ nd ν, respectively. Hence, when the heights of distributions do not chnge, the vlues of (µ k,ν l ) do not chnge either. Consequently, there exists some N such tht dist(n,µ,ν ) = dist(n + 1,µ,ν ) for ny µ nd ν rechble from µ nd ν, respectively, hence the lgorithm will terminte. We now consider the time complexity. Let (S,A, ) be the NFTS under considertion. Assume tht the size of the stte spce is n = S nd the number of edges (viewing the

8 WU AND DENG: DISTRIBUTION-BASED BEHAVIOURAL DISTANCE FOR NONDETERMINISTIC FUZZY TRANSITION SYSTEMS 8 NFTS s directed grph) lbelled with possibility vlues is m. Suppose the density of the NFTS is k. Tht is, from ny stte s there re t most k outgoing trnsitions s µ i with the sme lbel, for ny A. In the worst cse, the support of µ is S. By Definition 4, from µ there re t most k n outgoing trnsitions with the sme lbel. Similrly for ν. Therefore, in the procedure dist(l, µ, ν), we will compute dist(l 1,µ i,ν j ) t most A k 2n times. For ny µ,ν F(S), let t l be the time of computing dist(l,µ,ν). We see tht t l = A k 2n t l 1. It follows tht t l is in O(( A k 2n ) l ). In the proof of Proposition 5, the sizes of the sets X nd Y re bounded by n nd m, respectively. Therefore, the procedure dist(l,µ,ν ) will stbilize fter n + m itertions. Note tht for ech distribution µ rechble from µ, its support is t most S nd the possibility µ (s) for ny s in its support must be possibility vlue tht hs lredy ppered nd lbelled some trnsition in the originl NFTS. So the totl number of those vlues is less thn the totl number m of edges of the grph of the originl NFTS. It follows tht the totl number of distributions rechble from µ or ν is t most n m. Therefore, the overll time consumed is thus in O(n m A n+m k 2n2 +2nm ), which mens tht the lgorithm tkes exponentil time. Remrk 1. Let us consider the lgorithm in three prticulr cses. 1) If n NFTS is deterministic, i.e. its density k is equl to 1, the bove lgorithm still tkes exponentil time. There re two time-consuming tsks: (1) to find some N such tht d N (µ,ν) stbilizes; (2) to gurntee tht d N is stble not only for µ nd ν, but lso for ny pir of distributions rechble from µ nd ν. This obstcle reminds us of the problem of computing the bisimilrity metric for probbilistic systems [38] [40], for which there does not seem to exist polynomil time lgorithms. For exmple, vn Breugel nd Worrell [40] point out tht the problem of computing the bisimilrity pseudometric on the stte spce of probbilistic utomton is in PPAD, which stnds for polynomil prity rgument in directed grph. It lies between the serch problem versions of P nd NP. 2) Given finitry NFTS without loops, we know tht it hs finite tree structure. Suppose the height of the tree is h. If we pply the bove lgorithm, it is esy to see tht fter t most h + 1 rounds of itertion, the pproximte computtion of the distnce between ny pir of distributions will stbilize. Tht is, dist(h,µ,ν) = dist(h + 1,µ,ν), nd similrly for ny pir of distributions rechble from µ nd ν. Moreover, the procedure distnce cn be simplified: there is no need to cll generte nd it suffices to require the while loop to iterte h+1 times. The overll time thus consumed is O( A h+1 k 2n(h+1) ). 3) If the NFTS in 2) is deterministic, then the running time of the lgorithm becomes O( A h+1 ). So in this very prticulr cse the lgorithm becomes polynomil. V. FUZZY MODAL LOGIC In this section, we introduce fuzzy modl logic tht cn chrcterize distribution-bsed bisimilrity metric. Let A be set of ctions rnged over by,b,, nd T be fixed proposition, which is tken from the clssicl Hennessy-Milner Logic [20]. The lnguge L of formule is the lest set generted by the following BNF grmmr: ψ ::= T ψ 1 ψ 2 ψ p p ψ ψ where p [0,1]. We write ϕ p for (ϕ p) ( p ϕ). Note tht the formule in L re defined inductively, so we cn only generte formule of finite depth such s T, T, b T, T p, p T, ( T p) ( r b T), etc. However, T T is not legitimte formul. Let us fix n NFTSM = (S,A, ). We define the semntic function [ ] : L F(S) [0,1] to clculte the degree for distribution to stisfy formul. Definition 10. A formul ψ L evlutes in µ F(S) s follows: [T](µ) def = ht(µ), [ψ 1 ψ 2 ](µ) def = min([ψ 1 ](µ), [ψ 2 ](µ)), [ψ p](µ) def = [ψ](µ) p, [ p ψ](µ) def = p [ψ](µ), [ ψ](µ) def = mx µ µ [ψ](µ ). The roles of the opertors in L will be clerer when connecting them with the fuzzy modl logic L. 1) In generl, for ny ϕ,ψ L, ϕ ψ,ϕ&ψ, nd ϕ ψ need not be fuzzy modl logicl formule. This point is completely different from BL nd MTL; 2) The opertor min is used to interpret the minconjunction s BL nd MTL-lgebrs do; 3) The opertor mx is used to interpret the formul ψ, which specifies the property for distribution to perform ction nd result in possible distribution to stisfy ψ. Due to nondeterminism, from µ there my be severl trnsitions lbelled by the sme ction, e.g. µ µ i with i I. We tke the optiml cse by letting [ ψ](µ) be the mximl [ψ](µ i ) when i I; 4) The opertor is used to interpret the formul ψ p, where p [0,1]. Definition 11. Let S = (S, A, ) be n NFTS, for ny distributions µ,ν F(S), the logicl metric between µ nd ν is defined by d l (µ,ν) = 1 inf ψ L ([ψ](µ) [ψ](ν)). By Proposition 1 (4), it is not difficult to verify tht d l is -metric on F(S). In the following, we use d L l, dg l, dp l, nd d N l to denote logicl metrics bsed on L L, LG, LP nd LN, respectively. The rest of this section is devoted to logicl chrcteriztion of d b. It turns out tht d l coincides with d b if we consider imge-finite NFTSs. We split the proof of this coincidence result into two prts, to show tht one metric is dominted by the other nd vice vers. Lemm 6. For imge-finite NFTSs, we hve d l d b.

9 WU AND DENG: DISTRIBUTION-BASED BEHAVIOURAL DISTANCE FOR NONDETERMINISTIC FUZZY TRANSITION SYSTEMS 9 Proof. Let (S, A, ) be n imge-finite NFTS. For ny distributions µ, ν F(S), we need to show the inequlity d l (µ,ν) d b (µ,ν). It is sufficient to prove [ψ](µ) [ψ](ν) 1 d b (µ,ν) for ll ψ L. We proceed by structurl induction on formule. We first nlyze the structure of ψ L. ψ T. Then it is trivil to see tht [T](µ) [T](ν) = ht(µ) ht(ν) 1 d b (µ,ν) becuse (µ,ν) = 1 (ht(µ) ht(ν)) d b (µ,ν) nd d b is bisimultion metric. ψ ψ 1 ψ 2. We infer tht [ψ 1 ψ 2 ](µ) [ψ 1 ψ 2 ](ν) = min([ψ 1 ](µ), [ψ 2 ](µ)) min([ψ 1 ](ν), [ψ 2 ](ν)) min([ψ 1 ](µ) [ψ 1 ](ν), [ψ 2 ](µ) [ψ 2 ](ν)) 1 d b (µ,ν) (by induction on ψ 1 nd ψ 2 ). The second step holds becuse of Proposition 1 (5). ψ ψ 1 p. We infer tht [ψ 1 p](µ) [ψ 1 p](ν) = ([ψ 1 ](µ) p) ([ψ 1 ](ν) p) [ψ 1 ](µ) [ψ 1 ](ν) (by Proposition 1 (6)) 1 d b (µ,ν) (by induction on ψ 1 ). ψ p ψ 1. Similr to the lst cse by using Proposition 1 (7). ψ ψ. Let µ be the distribution such tht µ µ nd [ ψ ](µ) = [ψ ](µ ). Since d b is distributionbsed bisimultion metric, by Definition 8, there exists some ν such tht ν ν nd d b (µ,ν ) d b (µ,ν). Without loss of generlity, we ssume [ψ](µ) [ψ](ν). In this cse, [ψ ](µ ) [ψ ](ν ). We distinguish two possibilities: [ψ](µ) = [ψ](ν). Then [ψ](µ) > [ψ](ν). Then [ψ](µ) [ψ](ν) = 1 1 d b (µ,ν) [ψ](µ) [ψ](ν) = [ψ](µ) [ψ](ν) (by Proposition 1 (1)) = [ψ ](µ ) mx ν ν [ψ ](ν ) [ψ ](µ ) [ψ ](ν ) 1 d b (µ,ν ) (by induction on ψ ). 1 d b (µ,ν). Lemm 7. For imge-finite NFTSs, we hve d b d l. Proof. Since d b is the lest distribution-bsed bisimultion metric, it suffices to prove tht d l is distribution-bsed bisimultion metric. Let (S, A, ) be n imge-finite NFTS nd µ,ν F(S) be ny two distributions. First, we prove the inequlity d l. This holds becuse d l (µ,ν) = 1 inf ψ L ([ψ](µ) [ψ](ν)) 1 ([T](µ) [T](ν)) = 1 (ht(µ) ht(ν)) = (µ,ν). Now, ssume tht µ µ nd d l (µ,ν) = r < 1. We need to show tht there exists some trnsition ν ν with d l (µ,ν ) d l (µ,ν). Suppose for contrdiction tht no -trnsition from ν stisfies this condition. Then for ech ν i (i I) with ν ν i we hve d l (µ,ν i ) > r. There must exist some formulψ i L such tht1 ([ψ i ](µ ) [ψ i ](ν i )) > r, tht is, [ψ i ](µ ) [ψ i ](ν i ) < 1 r, for ech i I. Note tht by ssumption the NFTS is imge-finite, so the index set I is finite. Define ψ i = ψ i [ψ i ](µ ). Then for ll i I, [ψ i](µ ) = 1 nd [ψ i](ν i ) = [ψ i ](µ ) [ψ i ](ν i ) < 1 r. Let ψ = i I ψ i. Then we infer tht [ψ](µ) = mx µ µ [ i I ψ i ](µ ) [ i I ψ i ](µ ) = min i I [ψ i ](µ ) = 1. On the other hnd, we hve It follows tht [ψ](ν) = mx ν ν i [ j I ψ j ](ν i ) = mx ν ν i min j I [ψ j ](ν i ) mx ν ν i [ψ i ](ν i ) < 1 r. d l (µ,ν) = 1 inf ψ L([ψ ](µ) [ψ ](ν)) 1 [ψ](µ) [ψ](ν) = 1 [ψ](ν) (by Proposition 1 (2)) > 1 (1 r) = r which is in contrdiction to d l (µ,ν) = r. By combining the bove two lemms, we rrive t the following logicl chrcteriztion theorem. Theorem 3. For imge-finite NFTSs, we hve d l = d b. For ny µ,ν F(S) nd ny ψ L, the proof of Lemm 6 implies [ψ](µ) = [ψ](ν) when µ ν. It follows tht µ ν implies d l (µ,ν) = 0. On the other hnd, define the binry reltion R def = {(µ,ν) d l (µ,ν) = 0}. Then R is obvious n equivlence reltion. The proof of Lemm 7 implies tht R is distribution-bsed bisimultion. Hence d l (µ,ν) = 0 implies µ ν. Therefore, two distributions re bisimilr if nd only of the logicl distnce between them is 0. Corollry 4. Let µ nd ν be two distributions over n imgefinite NFTS. Then µ ν iff d l (µ,ν) = d b (µ,ν) = 0 iff [ψ](µ) = [ψ](ν) for ll ψ L. The bove property cn be used to judge whether two distributions re bisimilr.

10 WU AND DENG: DISTRIBUTION-BASED BEHAVIOURAL DISTANCE FOR NONDETERMINISTIC FUZZY TRANSITION SYSTEMS 10 Exmple 2. According to Corollry 4 nd the computtion results of Exmple 1, we cn see tht s nd t re bisimilr, wheres s 1 nd t 2 re not. In ddition, with Theorem 3 sometimes we cn esily clculte distribution-bsed bisimilrity metric in terms of d l. For instnce, consider Exmple 1 gin. The only relevnt formul is b T, nd it is direct to clculte tht [ b T]( s 1 ) = 0.8 nd [ b T]( t 2 ) = 0.6. Thus [ b T]( s 1 ) [ b T]( t 2 ) = 0.6 under the Gödel impliction. It follows from Theorem 3 tht d G b ( s 1, t 2 ) = d G l ( s 1, t 2 ) = = 0.4. VI. NON-EXPANSIVENESS In this section, we show the non-expnsiveness of our distribution-bsed bisimilrity metric with respect to prllel composition in the style of CSP [15]. It mens tht if the difference (with respect to bisimilrity metric) between µ i nd ν i is ǫ i, then the difference between f(µ 1,,µ n ) nd f(ν 1,,ν n ) is no more thn n i=1 ǫ i [2], where f is prllel composition opertor with n rguments. This nonexpnsiveness mkes compositionl verifiction possible. Given n NFTS (S, A, ) nd two possibility distributions µ,ν F(S), the key to define prllel composition is how to ppropritely define the fuzzy set represented by µ ν. For exmple, let µ = 0.3 s s 2 nd ν = 0.4. Intuitively, in their prllel composition, (s 1, ) ppers with possibility 0.3 nd (s 2, ) ppers with possibility of 0.4. Tht is, µ ν is the 0.3 fuzzy set (s ,) (s 2,). Hence, µ ν is given by (µ ν)(s,t) = µ(s) ν(t) for ll (s,t) S S. Clerly, we hve µ ν F(S S) nd in generl, (µ ν) nd (ν µ) re different fuzzy sets on F(S S). For instnce, (ν µ) = 0.3 (s ,s 1) (,s 2). Note tht ( s t)(s,t ) = 1 if (s,t ) = (s,t) nd 0 otherwise. The 0.3 bove fuzzy set µ ν cn be understood s s s 2. For 0.3 simplicity, we write s s 2 insted of s s 2. The prllel composition opertor we re going to define is synchronous in the sense tht the components cn either synchronize or ct independently. Given n NFTS (S, A, ), for ny µ,ν F(S), the events intended to synchronize t µ nd ν re listed in the set A µ A ν nd the rest of the events cn be performed independently, where A µ stnds for the set { A : µ,µ µ }. Formlly, the prllel composition µ ν cn perform ction in four different wys: {µ ν : µ T (µ),ν T (ν)} if A µ A ν {µ T (µ ν)= ν : µ T (µ)} if A µ \A ν {µ ν : ν T (ν)} if A ν \A µ otherwise. Let us present n exmple of the prllel composition nd compute the bisimilrity metric between distributions. Exmple 3. Let S = ({s 1,s 2,,s 4 },{}, ) be n NFTS, where is defined by the trnsitions s 1 { s 4 }, s 2 { s 4 }, nd { 0.4 s 4 }. Then, s 1 s 2 {µ : µ = s s s 4 s 4 } s 2 {ν : ν = 0.4 s s 4 s 4 } If we tke the fuzzy logic L L, then by direct computtion shown in Exmple 1, we cn obtin tht d L b ( s 1, s 2 ) = 0 nd d L b ( s 2, ) = 0.4. Now, for n 2, we hve tht d n ( s 1 s 2, s 2 ) = (d n 1 )( s 1 s 2, s 2 ) = mx(d n 1 (µ,ν),1 (ht( s 1 s 2 ) L ht( s 2 ))) = d n 1 (µ,ν) = (d n 2 )(µ,ν) = mx(d n 2 ( s 4 s 4, s 4 s 4 ),1 (ht(µ) L ht(ν))) = mx(0, 0.2) = 0.2. Hence d L b ( s 1 s 2, s 2 ) = 0.2. In similr wy, one cn compute this distnce under other fuzzy logics, which re shown in the following digrm. ( s 1, s 2 ) ( s 2, ) ( s 1 s 2, s 2 ) d L b d G b d P 1 b d N b It is obvious tht d b ( s 1 s 2, s 2 ) d b ( s 1, s 2 ) d b ( s 2, ), i.e., the prllel composition is non-expnsive. More generlly, we hve the following theorem. Theorem 4. Let (S,A, ) be n NFTS nd µ i,ν i F(S) with d b (µ i,ν i ) = ǫ i, where i = 1,2. Then we hve d b (µ 1 µ 2,ν 1 ν 2 ) ǫ 1 ǫ 2. Proof. Let F(S) F(S) = {µ ν µ,ν F(S)}. We define the function D : (F(S) F(S)) (F(S) F(S)) [0,1] by D(µ µ,ν ν ) def = d b (µ,ν) d b (µ,ν ) for ny µ µ, ν ν F(S) F(S). It is esy to check tht D is -metric on F(S) F(S). We clim tht D is pre-fixed point of the function : D F(S) F(S) D F(S) F(S). Let us verify it. For ny µ 1 µ 2,ν 1 ν 2 F(S) F(S), we first check tht 1 (ht(µ 1 µ 2 ) ht(ν 1 ν 2 )) D(µ 1 µ 2,ν 1 ν 2 ). It holds becuse D(µ 1 µ 2,ν 1 ν 2 ) = d b (µ 1,ν 1 ) d b (µ 2,ν 2 ) [1 (ht(µ 1 ) ht(ν 1 ))] [1 (ht(µ 2 ) ht(ν 2 ))] = 1 (ht(µ 1 ) ht(ν 1 )) (ht(µ 2 ) ht(ν 2 )) 1 (ht(µ 1 ) ht(µ 2 ) ht(ν 1 ) ht(ν 2 )) = 1 (ht(µ 1 µ 2 ) ht(ν 1 ν 2 )). The second step holds becuse d b is the bisimilrity metric, nd the fourth step holds becuse of Proposition 1 (5). The lst equlity holds becuse of the following fcts: there exist s 0,t 0 such tht ht(µ 1 ) = µ 1 (s 0 ) nd ht(µ 2 ) = µ 2 (t 0 ), then ht(µ 1 µ 2 ) = sup (s,t) S S µ 1 (s) µ 2 (t) = µ 1 (s 0 ) µ 2 (t 0 ) = ht(µ 1 ) ht(µ 2 ); similrly, ht(ν 1 ν 2 ) = ht(ν 1 ) ht(ν 2 ). Next, suppose tht µ 1 µ 2 µ 1 µ 2. We need to show tht there exists some ν 1 ν 2 such tht ν 1 ν 2 ν 1 ν 2 nd D(µ 1 µ 2,ν 1 ν 2) D(µ 1 µ 2,ν 1 ν 2 ). There re four

11 WU AND DENG: DISTRIBUTION-BASED BEHAVIOURAL DISTANCE FOR NONDETERMINISTIC FUZZY TRANSITION SYSTEMS 11 cses: A µ1 A µ2, A µ1 \A µ2, A µ2 \A µ1, nd A µ1 A µ2. We only give the detils for the first cse; the other cses re simpler nd cn be proved in similr wy. In the cse of A µ1 A µ2, we get by the definition of the prllel composition tht there re µ 1 µ 1 nd µ 2 µ 2. Since the lest fixed point of is pre-fixed point s well. By Lemm 4 there re ν 1 nd ν 2 such tht ν 1 ν 1 nd ν 2 ν 2 with d b (µ 1,ν 1 ) d b(µ 1,ν 1 ) nd d b (µ 2,ν 2 ) d b(µ 2,ν 2 ). Clerly, we hve ν 1 ν 2 ν 1 ν 2 nd it remins to verify tht D(µ 1 µ 2,ν 1 ν 2 ) D(µ 1 µ 2,ν 1 ν 2 ). This follows becuse it is esy to see tht d b (µ 1,ν 1) d b (µ 2,ν 2) d b (µ 1,ν 1 ) d b (µ 2,ν 2 ). Bsed on the clim, it follows from the definition of d b tht d b D, which implies d b (µ 1 µ 2,ν 1 ν 2 ) D(µ 1 µ 2,ν 1 ν 2 ) = d b (µ 1,ν 1 ) d b (µ 2,ν 2 ) = ǫ 1 ǫ 2. This completes the proof. As consequence, we cn show tht the prllel composition preserves distribution-bsed bisimilrity. Corollry 5. If µ 1 ν 1 nd µ 2 ν 2, then µ 1 µ 2 ν 1 ν 2. Proof. It is strightforwrd by using Theorem 4 nd Corollry 4. VII. AN ILLUSTRATIVE EXAMPLE We hve seen from Corollry 4 tht the problem of checking whether two distributions re bisimilr cn be converted into logicl judgment of whether the two distributions hve the sme vlues on the sme set of logicl formule, which cn benefit from trditionl logic theories nd be ssisted by some prcticl tools. In prticulr, it is very useful to judge the nonbisimilrity of two distributions becuse we only need to find formul tht witnesses the difference of the two distributions. As n ppliction, we consider n exmple relted to medicl dignosis nd tretment, s described by Co et l. [2] nd Qiu [41] (see lso, [5], [42]). We ssume tht there is n unknown bcteril infection. Bsed on his experience, physicin believes tht two drugs, sy 1 nd 2, my be useful for treting this disese. Three possible negtive symptoms, e.g., b 1, b 2, b 3, must lso be considered. A ptient s condition cn be in one of three rough types, e.g., poor, fir, nd excellent, which re denoted by the cpitl letters P, F, nd E, respectively. A tretment (or negtive symptom) my led to stte mong multiple possible sttes with certin degree. For exmple, the trnsition F 1[0.6] E mens tht the ptient s condition hs chnged from fir to excellent with possibility 0.6 fter using drug 1, wheres F b1[0.3] P mens tht the ptient s condition hs chnged from fir to poor with possibility 0.3 if the ptient hs negtive symptom b 1. The trnsition possibilities of these events re estimted by the physicin. Different physicl conditions of ptients my led to nondeterministic chnges even if the ptients re in the sme stte nd re given the sme tretment. For exmple, we my hve the following two trnsition: P 1 { 0.1 P F 1 } nd P { 0.6 F E }. So we obtin n NFTS (S,A, ), where S = {P,F,E}, A = { 1, 2,b 1,b 2,b 3 } nd F(S) A F(S). In order to compre with the work of Co et l., we revisit Exmple 4 in [2]. According to the physicin s estimtion, the trnsition possibilities of these events mong sttes re s follows. { 0.1 { 0.5 P 1 F 1 F b1 P F F E 2 }, P { }, F { 0.3 P F } F E } { 0.2 b2 }, F { 0.3 b3 }, E { 0.1 P P P F }. Suppose now tht there re two ptients: Alice nd Bob. The physicin describes the ptients sttes s S A = [0.8,0.2,0.1] nd S B = [0.8,0.3,0], respectively. The fuzzy vlue vector, sy S B, mens tht Bob is in poor stte with membership 0.8, nd in fir stte with membership 0.3. For simplicity, we ssume tht Alice nd Bob hve no negtive symptoms when the drugs re used for the first time in therpy. It turns out tht { 0.1 S A 1 S A 2 S B 1 S B 2 { 0.6 { 0.1 { 0.6 P F E } P F E } P F E } P F E } We first show by Corollry 4 tht the sttes of Alice nd Bob re not bisimilr, which is not esy to judge directly by the definition of distribution-bsed bisimultion. In fct, we cn find formul 1 b 3 T such tht [ 1 b 3 T](S A ) = 0.2 nd [ 1 b 3 T](S B ) = 0.3 nd hence the sttes of Alice nd Bob re not bisimilr. Furthermore, we cn obtin more quntittive informtion by mesuring the distnce between their sttes. Using L N, it is not difficult to see tht d N b (S A,S B ) = 0.3 by direct computtion shown in Exmple 1. In the sme wy, we cn lso obtin tht d N b (S A,S B ) = 0.5 where S A = [0.5,0.5,0] nd S B = [0,0.5,0.5]. By Corollry 3, S(S A,S B ) = 0.7 nd S(S A,S B ) = 0.5 in the bove two cses, respectively, which mesures the similrity between Alice s nd Bob s conditions in the progress of tretment. A little surprise is tht d N b (S A,S B ) is equl to d f (S A,S B ) in the bove two cses, respectively, where d f is the behviourl distnce of Co et l. [2]. However, Figure 1, Exmple 1, Exmple 2, Corollry 4 in this pper nd Theorem 4 in [2] tell us tht d N b is not lwys equl to d f for ny s,t S. On the other hnd, if we tke different fuzzy logic, then d b (S A,S B ) is in generl different. For exmple, underl P, we get tht d P b (S A,S B ) = 0.4 when S A = [0.5,0.5,0] nd S B = [0,0.5,0.5], but d P b (S A,S B ) = 1 3 when S A = [0.8,0.2,0.1] nd S B = [0.8,0.3,0], respectively. VIII. RELATED WORK Simultions nd bisimultions for fuzzy (or lttice-vlued) systems hve received much ttention in the field. Errico nd Loreti [9] proposed notion of bisimultion nd pplied it to

12 WU AND DENG: DISTRIBUTION-BASED BEHAVIOURAL DISTANCE FOR NONDETERMINISTIC FUZZY TRANSITION SYSTEMS 12 fuzzy resoning. Kupfermn nd Lusting [11] studied ltticed simultion between two lttice-vlued Kripke structures, which re pplied to ltticed gmes. P. Eleftheriou et l. [10] defined bisimultion for Heyting-vlued modl lnguges. Pn et l. [12] investigted simultion for lttice-vlued doubly lbeled trnsition systems. Co et l. [1] defined bisimultion reltion between two different FTSs by correltionl pir bsed on some reltion. Ćirić et l. [3] introduced four types of bisimultions (forwrd, bckwrd, forwrd-bckwrd, nd bckwrdforwrd) for fuzzy utomt. Sun et l. [13] investigted forwrd nd bckwrd bisimultions for fuzzy utomt. Qiu nd Deng [4], nd Xing et l. [5] ddressed the supervisory control of fuzzy discrete event systems by using simultion equivlence nd bisimultion equivlence, respectively. Fn [6] defined bisimultions for fuzzy Kripke structures. Dmljnović et l. [8] lso studied simultion nd bisimultion for weighted utomt in similr mnner, s lso described in [3], [7]. Wu nd Deng [14] considered bisimultions for FTSs. All these pproches cn be divided into two clsses. In the first clss, (bi)simultions re bsed on crisp reltion on the stte spce, nd thus one stte is either (bi)similr to nother stte or not. As [1], [5], [8] [10], [13], [14], the present study belongs to this clss. In the second clss, (bi)simultions re bsed on fuzzy reltion (or lttice-vlued reltion) on the stte spce, which shows the degree to which one stte is (bi)similr to nother. This pproch ws dpted in [3], [4], [6], [11], [12]. Recently, lttice-vlued nondeterministic fuzzy utomt hve lso been proposed [43]. This pper investigtes distribution-bsed bisimultion, which is crisp reltion nd to our knowledge, remins underexplored in fuzzy systems. We just compre our distributionbsed bisimultion with the stte-bsed bisimultion given by Co et l [2]. An equivlence reltion R S S is stte-bsed bisimultion if s R t implies tht whenever s µ, there exists some trnsition t ν with µ([s ]) = ν([s ]) for ll R-equivlence clsses [s ]. We cn prove tht if R is stte-bsed bisimultion, then the lifted reltion R (see Definition 8 in [2]) is distributionbsed bisimultion. This mens tht for ny s,t S, if s nd t re relted by stte-bsed bisimultion then s nd t re relted by distribution-bsed bisimultion. However, the inverse does not hold, s cn be seen from Figure 1 nd Exmple 2. In this sense, we sy tht distribution-bsed bisimilrity is corser thn stte-bsed bisimilrity. As modl logics, Wu nd Deng [14] gve fuzzy style Hennessy-Milner logic, which chrcterizes bisimultions for FTSs soundly nd completely. There the formul is interpreted on (bre) sttes nd is two-vlued, i.e., stte either stisfies formul or not. Fn [6] chrcterized her fuzzy bisimultion in terms of Gödel modl logic G( ) s follows: ϕ ::= p c ϕ 1 ϕ 2 ϕ 1 ϕ 2 ϕ ϕ. Its semntic interprettion is given in terms of (bre) sttes nd is quntittive, i.e., the semntic domin is the unit intervl [0, 1]. This logic is different from the logic in the current work both in the syntx nd in the semntics. In ddition, the formul ϕ p in our logic plys key role in proving Theorem 3. One cn sk whether we cn use ψ 1 ψ 2 insted of ψ p in our setting. The nswer is negtive. The reson is tht Lemm 6 does not hold for ψ ψ 1 ψ 2. In ddition, Fn [44] hs pointed out tht fuzzified Hennessy- Milner Theorem does not hold for modl logics bsed on Łuksiewicz t-norm nd the product t-norm. However, this theorem holds in our frmework under clss of fuzzy logics generted by t-norm including Łuksiewicz t-norm nd the product t-norm. Therefore, the results in this pper provide n nswer to the question of Fn [44]: to investigte lterntive definitions of fuzzy bisimultion tht cn stisfy the Hennessy- Milner Theorem with respect to Łuksiewicz nd product structures. In [10] Eleftheriou gve Heyting-vlued modl logics L H. The reltionship between LH nd G( ), nd the corresponding bisimultions hve been discussed in detil [6]. Another direction concerning modl logics is to investigte model checking of nondeterministic systems. For exmple, Li et l. [45], [46] studied quntittive computtion tree logic for model checking bsed on possibility mesures. IX. CONCLUSION AND FUTURE WORK We hve proposed new notion of bisimultion, which is distribution-bsed. We hve introduced n bstrct metric bsed on tringulr-conorm nd constructed distributionbsed bisimilrity metric for mesuring the behviourl distnce between generlized sttes in NFTSs. The bisimilrity metric is quntittive nlogue of bisimilrity. The smller the distnce, the more like the distributions re. In prticulr, two distributions re bisimilr if nd only if they hve distnce 0. Moreover, we hve given fuzzy modl logic to chrcterize distributions-bsed bisimilrity metric soundly nd completely. In ddition, we hve lso shown tht the CSP-style prllel composition with respect to bisimilrity metric is nonexpnsive. Unlike the exct notion of bisimultion, bisimilrity metric is more robust for fuzzy systems, which enbles us to investigte the pproximte equivlence of fuzzy systems. There re severl problems tht re worth further study. First, it would be interesting to find n pproprite logic to chrcterize the behviourl distnce d f of Co et l. [2]. In fct, we hve logic tht cn chrcterize the stte-bsed bisimultion of Co et l., but cnnot chrcterize d f. Second, it is uncler if our methodology cn be generlized to the generl frmework of lttices. One subtlety lies in Lemm 7. In the current setting, sup{x A x < r} < r holds when A is finite subset of the unit intervl [0,1]. However, this property fils if we del with generl lttice. Hence, it seems necessry to give new proof of Theorem 3 or to chnge the modl logic. Lst but not lest, we would like to figure out whether it is possible nd how to improve the lgorithm for computing the bisimilrity metric so tht they cn be used in prcticl nlysis of fuzzy systems. X. ACKNOWLEDGMENT The uthors would like to thnk the nonymous reviewers for their invluble suggestions nd comments.

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Fuzzy Syst., vol. 21, no. 5, pp , Oct [46] Y. M. Li nd Z. Y. M, Quntittive computtion tree logic model checking bsed on generlized possibility mesures, IEEE Trns. Fuzzy Syst., vol. 23, no. 6, pp , Dec Hengyng Wu received the B.S. degree from Xuzhou Norml University, in 1996, nd M.Sc. nd Ph.D. degrees from Shnghi Norml University, in 2004 nd 2007, respectively, ll in mthemtics. From 2008 to 2011, he ws Postdoctorl Resercher with Est Chin Norml University. From 2011 to July 2016, he ws n Associte Professor of computer science with Hngzhou Dinzi University. Currently, he is n Associte Reserch Fellow with Est Chin Norml University. His current reserch interests include forml methods nd domin theory. Yuxin Deng received the B.Eng. nd M.Sc. degrees from Shnghi Jio Tong University in 1999 nd 2002, respectively, nd the Ph.D. in computer science from Ecole des Mines de Pris in He is Professor in Est Chin Norml University. His reserch interests include concurrency theory, especilly bout process clculi, nd forml semntics of progrmming lnguges, s well s forml verifiction. He uthored the book titled Semntics of Probbilistic Processes: An Opertionl Approch (Springer, 2015).