Disjunctive Probbilistic Modl Logic is Enough for Bisimilrity on Rective Probbilistic Systems Mrco Bernrdo 1 nd Mrino Miculn 2 1 Dip. di Scienze Pure e Applicte, Università di Urbino, Itly 2 Dip. di Scienze Mtemtiche, Informtiche e Fisiche, Università di Udine, Itly Abstrct. Lrsen nd Skou chrcterized probbilistic bisimilrity over rective probbilistic systems with logic including true, negtion, conjunction, nd dimond modlity decorted with probbilistic lower bound. Lter on, Deshrnis, Edlt, nd Pnngden showed tht negtion is not necessry to chrcterize the sme equivlence. In this pper, we prove tht the logicl chrcteriztion holds lso when conjunction is replced by disjunction, with negtion still being not necessry. To this end, we introduce rective probbilistic trees, fully bstrct model for rective probbilistic systems tht llows us to demonstrte expressiveness of the disjunctive probbilistic modl logic, s well s of the previously mentioned logics, by mens of compctness rgument. 1 Introduction Since its introduction [12], probbilistic bisimilrity hs been used to compre probbilistic systems. It corresponds to Milner s strong bisimilrity for nondeterministic systems, nd coincides with lumpbility for Mrkov chins. Lrsen nd Skou [12] first proved tht bisimilrity for rective probbilistic systems cn be given logicl chrcteriztion: two processes re bisimilr if nd only if they stisfy the sme set of formuls of propositionl modl logic similr to Hennessy-Milner logic [10]. In ddition to the usul constructs,, nd, this logic fetures dimond modlity p φ, which is stisfied by stte if, fter performing ction, the probbility of being in stte stisfying φ is t lest p. Lter on, Deshrnis, Edlt, nd Pnngden [6] showed tht negtion is not necessry for discrimintion purposes, by working in continuous-stte setting. This result hs no counterprt in the nonprobbilistic setting, where Hennessy-Milner logic without negtion chrcterizes simultion equivlence, which is strictly corser thn bisimilrity [8] (while the two equivlences re known to coincide on rective probbilistic systems [2]). Copyright c by the pper s uthors. Copying permitted for privte nd cdemic purposes. V. Biló, A. Cruso (Eds.): ICTCS 2016, Proceedings of the 17th Itlin Conference on Theoreticl Computer Science, 73100 Lecce, Itly, September 7 9 2016, pp. 203 217 published in CEUR Workshop Proceedins Vol-1720 t http://ceur-ws.org/vol-1720
204 M. Bernrdo, M. Miculn In this pper, we show tht cn be used in plce of without hving to reintroduce negtion: the constructs,, nd p suffice to chrcterize bisimilrity on rective probbilistic systems. The intuition is tht from conjunctive distinguishing formul we cn often derive disjunctive one by suitbly incresing some probbilistic lower bounds. Not even this result hs counterprt in the nonprobbilistic setting, where replcing conjunction with disjunction in the bsence of negtion yields trce equivlence (this equivlence does not coincide with bisimilrity on rective probbilistic processes). The proof of our result relies on simple ctegoricl construction of semntics for rective probbilistic systems, which we cll rective probbilistic trees (Sect. 3). This semntics is fully bstrct, i.e., two sttes re probbilisticlly bisimilr if nd only if they re mpped to the sme rective probbilistic tree. Moreover, the semntics is compct, in the sense tht two (possibly infinite) trees re equl if nd only if ll of their finite pproximtions re equl. Hence, in order to prove tht logic chrcterizes probbilistic bisimilrity, it suffices to prove tht it llows to discriminte finite rective probbilistic trees. Indeed, given two different finite trees, we cn construct formul of the considered logic (by induction on the height of one of the trees) tht tells the two trees prt nd hs depth not exceeding the height of the two trees (Sect. 4). Our technique pplies lso to the logics in [12,6], for which it llows us to provide simpler proofs of dequcy, directly in discrete setting. More generlly, this technique cn be used in ny computtionl model tht hs compct, fully bstrct semntics. 2 Processes, Bisimilrity, nd Logics 2.1 Rective Probbilistic Processes nd Strong Bisimilrity Probbilistic processes cn be represented s lbeled trnsitions systems with probbilistic informtion used to determine which ction is executed or which stte is reched. Following the terminology of [9], we focus on rective probbilistic processes, where every stte hs for ech ction t most one outgoing distribution over sttes; the choice mong these rbitrrily mny, differently lbeled distributions is nondeterministic. For countble (i.e., finite or countbly infinite) set X, the set of finitely supported (.k.. simple) probbility distributions over X is given by D(X) = { : X R [0,1] supp( ) < ω, x X (x) = 1}, where the support of distribution is defined s supp( ) {x X (x) > 0}. A rective probbilistic lbeled trnsition system, RPLTS for short, is triple (S, A, ) where S is countble set of sttes, A is countble set of ctions, nd S A D(S) is trnsition reltion such tht, whenever (s,, 1 ), (s,, 2 ), then 1 = 2. An RPLTS cn be seen s directed grph whose edges re lbeled by pirs (, p) A R (0,1]. For every s S nd A, if there re -lbeled edges outgoing from s, then these re finitely mny (imge finiteness), becuse the considered distributions re finitely supported, nd the numbers on them dd up to 1. As usul, we denote (s,, ) s s, where the set of
Disjunctive Probbilistic Modl Logic 205 rechble sttes coincides with supp( ). We lso define cumultive rechbility s (S ) = s S (s ) for ll S S. Probbilistic bisimilrity for the clss of rective probbilistic processes ws introduced by Lrsen nd Skou [12]. Let (S, A, ) be n RPLTS. An equivlence reltion B over S is probbilistic bisimultion iff, whenever (s 1, s 2 ) B, then for ll ctions A it holds tht, if s 1 1, then s 2 2 nd 1 (C) = 2 (C) for ll equivlence clsses C S/B. We sy tht s 1, s 2 S re probbilisticlly bisimilr, written s 1 PB s 2, iff there exists probbilistic bisimultion including the pir (s 1, s 2 ). 2.2 Probbilistic Modl Logics In our setting, probbilistic modl logic is pir formed by set L of formuls nd n RPLTS-indexed fmily of stisfction reltions = S L. The logicl equivlence induced by L over S is defined by letting s 1 =L s 2, where s 1, s 2 S, iff s 1 = φ s 2 = φ for ll φ L. We sy tht L chrcterizes binry reltion R over S when R = = L. We re especilly interested in probbilistic modl logics chrcterizing PB. The logics considered in this pper re similr to Hennessy-Milner logic [10], but the dimond modlity is decorted with probbilistic lower bound s follows: PML : φ ::= φ φ φ p φ PML : φ ::= φ φ p φ PML : φ ::= φ φ φ p φ PML : φ ::= φ φ p φ where p R [0,1] ; triling s will be omitted for ske of redbility. Their semntics with respect to n RPLTS stte s is defined s usul, in prticulr: s = p φ s nd ({s S s = φ}) p Lrsen nd Skou [12] proved tht PML (nd hence PML ) chrcterizes PB. Deshrnis, Edlt, nd Pnngden [6] then proved in mesure-theoretic setting tht PML chrcterizes PB too, nd hence negtion is not necessry. This ws subsequently redemonstrted by Jcobs nd Sokolov [11] in the dul djunction frmework nd by Deng nd Wu [5] for fuzzy extension of RPLTS. The min im of this pper is to show tht PML suffices s well. 3 Compct Chrcteriztion of Probbilistic Bisimilrity 3.1 Colgebrs for Probbilistic Systems It is well known tht the function D defined in Sect. 2.1 extends to functor D : Set Set whose ction on morphisms is, for f : X Y, D(f)( ) = λy. (f 1 (y)). Then, every RPLTS corresponds to colgebr of the functor B RP : Set Set, B RP (X) (D(X) + 1) A. Indeed, for S = (S, A, ), the corresponding colgebr (S, σ : S B RP (S)) is σ(s) λ.(if s then else ). A homomorphism h : (S, σ) (T, τ) is function h : S T tht respects the colgebric structures, i.e., τ h = (B RP h) σ. We denote by Colg(B RP ) the ctegory of B RP -colgebrs nd their homomorphisms. Aczel nd Mendler [1] introduced generl notion of bisimultion for colgebrs, which in our setting instntites s follows:
206 M. Bernrdo, M. Miculn Definition 1. Let (S 1, σ 1 ), (S 2, σ 2 ) be B RP -colgebrs. A reltion R S 1 S 2 is B RP -bisimultion iff there exists colgebr structure ρ : R B RP R such tht the projections π 1 : R S 1, π 2 : R S 2 re homomorphisms (i.e., σ i π i = B RP π i ρ for i = 1, 2). We sy tht s 1 S 1, s 2 S 2 re B RP -bisimilr, written s 1 s 2, iff there exists B RP -bisimultion including (s 1, s 2 ). Proposition 1. The probbilistic bisimilrity over n RPLTS (S, A, ) coincides with the B RP -bisimilrity over the corresponding colgebr (S, σ). B RP is finitry (becuse we restrict to finitely supported distributions) nd hence dmits finl colgebr (cf. [3,15] nd specificlly [14, Thm. 4.6]). The finl colgebr is unique up-to isomorphism, nd cn be seen s the RPLTS whose elements re cnonicl representtives of ll possible behviors of ny RPLTS: Proposition 2. Let (Z, ζ) be finl B RP -colgebr. For ll z 1, z 2 Z: z 1 z 2 iff z 1 = z 2. 3.2 Rective Probbilistic Trees We now introduce rective probbilistic trees, representtion of the finl B RP - colgebr tht cn be seen s the nturl extension to the probbilistic setting of strongly extensionl trees used to represent the finl P f -colgebr [15]. Definition 2 (RPT). An (A-lbeled) rective probbilistic tree is pir (X, succ) where X Set nd succ : X A P f (X R (0,1] ) re such tht the reltion over X, defined by the rules x x nd x y z succ(y,) x z, is prtil order with lest element, clled root, nd for ll x X nd A: 1. the set {y X y x} is finite nd well-ordered; 2. for ll (x 1, p 1 ), (x 2, p 2 ) succ(x, ): if x 1 = x 2 then p 1 = p 2 ; if the subtrees rooted t x 1 nd x 2 re isomorphic then x 1 = x 2 ; 3. if succ(x, ) then (y,p) succ(x,) p = 1. Rective probbilistic trees re unordered trees where ech node for ech ction hs either no successors or finite set of successors, which re lbeled with positive rel numbers tht dd up to 1; moreover, subtrees rooted t these successors re ll different. See the forthcoming Fig. 1 for some exmples. In prticulr, the trivil tree is nil ({ }, λx,. ). We denote by RPT, rnged over by t, t 1, t 2, the set of rective probbilistic trees (possibly of infinite height), up-to isomorphism. For t = (X, succ), we denote its root by t, its -successors by t() succ( t, ), nd the subtree rooted t x X by t[x] ({y X x y}, λy,.succ(y, )); thus, t[x] = x. We define height : RPT N {ω} s height(t) sup{1 + height(t ) (t, p) t(), A} with sup = 0; hence, height(nil) = 0. We denote by RPT f {t RPT height(t) < ω} the set of rective probbilistic trees of finite height. A (possibly infinite) tree cn be pruned t ny height n, yielding finite tree where the removed subtrees re replced by nil. The pruning function
Disjunctive Probbilistic Modl Logic 207 ( ) n : RPT RPT f, prmetric in n, cn be defined by first truncting the tree t t height n, nd then collpsing isomorphic subtrees dding their weights. We hve now to show tht RPT is (the crrier of) the finl B RP -colgebr (up-to isomorphism). To this end, we reformulte B RP in slightly more reltionl formt. We define functor D : Set Set s follows: D X { } {U P f (X R (0,1] ) (x,p) U p = 1 nd (x, p), (x, q) U p = q} D f λu D X.{(f(x), (x,p) U p) x π 1(U)} for ny f : X Y. Proposition 3. D A = B RP, nd Colg(D A ) = Colg(B RP ); hence the (supports of the) finl D A -colgebr nd the finl B RP -colgebr re isomorphic. RPT is the crrier of the finl B RP -colgebr (up-to isomorphism). In fct, RPT cn be endowed with D A -colgebr structure ρ : RPT (D (RPT)) A defined, for t = (X, succ), s ρ(t)() {(t[x], p) (x, p) succ( t, )}. Theorem 1. (RPT, ρ) is finl B RP -colgebr. By virtue of Thm. 1, given n RPLTS S = (S, A, ) there exists unique colgebr homomorphism : S RPT, clled the (finl) semntics of S, which ssocites ech stte in S with its behvior. This semntics is fully bstrct. Another key property of rective probbilistic trees is tht they re compct: two different trees cn be distinguished by looking t their finite subtrees only. Theorem 2 (Full bstrction). Let (S, A, ) be n RPLTS. For ll s 1, s 2 S: s 1 PB s 2 iff s 1 = s 2. Theorem 3 (Compctness). For ll t 1, t 2 RPT: t 1 = t 2 iff for ll n N : t 1 n = t 2 n. Corollry 1. Let (S, A, ) be n RPLTS. For ll s 1, s 2 S: s 1 PB s 2 iff for ll n N : s 1 n = s 2 n. 4 The Discriminting Power of PML By virtue of the ctegoricl construction leding to Cor. 1, in order to prove tht modl logic chrcterizes PB over rective probbilistic processes, it is enough to show tht it cn discriminte ll rective probbilistic trees of finite height. A specific condition on the depth of distinguishing formuls hs lso to be stisfied, where depth(φ) is defined s usul: depth( ) = 0 depth( φ ) = depth(φ ) depth( p φ ) = 1 + depth(φ ) depth(φ 1 φ 2 ) = depth(φ 1 φ 2 ) = mx(depth(φ 1 ), depth(φ 2 )) Proposition 4. Let L be one of the probbilistic modl logics in Sect. 2.2. If L chrcterizes = over RPT f nd for ny two nodes t 1 nd t 2 of n rbitrry RPT f model such tht t 1 t 2 there exists φ L distinguishing t 1 from t 2 such tht depth(φ) mx(height(t 1 ), height(t 2 )), then L chrcterizes PB over the set of RPLTS models.
208 M. Bernrdo, M. Miculn In this section, we show the min result of the pper: the logicl equivlence induced by PML hs the sme discriminting power s PB. This result is ccomplished in three steps. Firstly, we redemonstrte Lrsen nd Skou s result for PML in the RPT f setting, which yields proof tht, with respect to the one in [12], is simpler nd does not require the miniml devition ssumption (i.e., tht the probbility ssocited with ny stte in the support of the trget distribution of trnsition be multiple of some vlue). This provides proof scheme for the subsequent steps. Secondly, we demonstrte tht PML chrcterizes PB by dpting the proof scheme to cope with the replcement of with. Thirdly, we demonstrte tht PML chrcterizes PB by further dpting the proof scheme to cope with the bsence of. Moreover, we redemonstrte Deshrnis, Edlt, nd Pnngden s result for PML through yet nother dpttion of the proof scheme tht, unlike the proof in [6], works directly on discrete stte spces without mking use of mesuretheoretic rguments. Avoiding the resort to mesure theory ws shown to be possible for the first time by Worrell in n unpublished note cited in [13]. 4.1 PML Chrcterizes PB : A New Proof To show tht the logicl equivlence induced by PML implies node equlity =, we reson on the contrpositive. Given two nodes t 1 nd t 2 such tht t 1 t 2, we proceed by induction on the height of t 1 to find distinguishing PML formul whose depth is not greter thn the heights of t 1 nd t 2. The ide is to exploit negtion, so to ensure tht certin distinguishing formuls re stisfied by certin derivtive t of t 1 (rther thn the derivtives of t 2 different from t ), then tke the conjunction of those formuls preceded by dimond decorted with the probbility for t 1 of reching t. The only non-trivil cse is the one in which t 1 nd t 2 enble the sme ctions. At lest one of those ctions, sy, is such tht, fter performing it, the two nodes rech two distributions 1, nd 2, such tht 1, 2,. Given node t supp( 1, ) such tht 1, (t ) > 2, (t ), by the induction hypothesis there exists PML formul φ 2,j tht distinguishes t from specific t 2,j supp( 2,) \ {t }. We cn ssume tht t = φ 2,j t 2,j otherwise, thnks to the presence of negtion in PML, it would suffice to consider φ 2,j. As consequence, t 1 = 1,(t ) j φ 2,j t 2 becuse 1, (t ) > 2, (t ) nd 2, (t ) is the mximum probbilistic lower bound for which t 2 stisfies formul of tht form. Notice tht 1, (t ) my not be the mximum probbilistic lower bound for which t 1 stisfies such formul, becuse j φ 2,j might be stisfied by other -derivtives of t 1 in supp( 1, ) \ {t }. Theorem 4. Let (T, A, ) be in RPT f nd t 1, t 2 T. Then t 1 = t 2 iff t 1 = φ t 2 = φ for ll φ PML. Moreover, if t 1 t 2, then there exists φ PML distinguishing t 1 from t 2 such tht depth(φ) mx(height(t 1 ), height(t 2 )). 4.2 PML Chrcterizes PB : Adpting the Proof Since φ 1 φ 2 is logiclly equivlent to ( φ 1 φ 2 ), it is not surprising tht PML chrcterizes PB too. However, the proof of this result will be useful to
Disjunctive Probbilistic Modl Logic 209 set up n outline of the proof of the min result of this pper, i.e., tht PML chrcterizes PB s well. Similr to the proof of Thm. 4, lso for PML we reson on the contrpositive nd proceed by induction. Given t 1 nd t 2 such tht t 1 t 2, we intend to exploit negtion, so to ensure tht certin distinguishing formuls re not stisfied by certin derivtive t of t 1 (rther thn the derivtives of t 2 different from t ), then tke the disjunction of those formuls preceded by dimond decorted with the probbility for t 2 of not reching t. In the only non-trivil cse, for t supp( 1, ) such tht 1, (t ) > 2, (t ), by the induction hypothesis there exists PML formul φ 2,j tht distinguishes t from specific t 2,j supp( 2,)\{t }. We cn ssume tht t = φ 2,j = t 2,j otherwise, thnks to the presence of negtion in PML, it would suffice to consider φ 2,j. Therefore, t 1 = 1 2,(t ) j φ 2,j = t 2 becuse 1 2, (t ) > 1 1, (t ) nd the mximum probbilistic lower bound for which t 1 stisfies formul of tht form cnnot exceed 1 1, (t ). Notice tht 1 2, (t ) is the mximum probbilistic lower bound for which t 2 stisfies such formul, becuse tht vlue is the probbility with which t 2 does not rech t fter performing. Theorem 5. Let (T, A, ) be in RPT f nd t 1, t 2 T. Then t 1 = t 2 iff t 1 = φ t 2 = φ for ll φ PML. Moreover, if t 1 t 2, then there exists φ PML distinguishing t 1 from t 2 such tht depth(φ) mx(height(t 1 ), height(t 2 )). 4.3 Also PML Chrcterizes PB The proof tht PML chrcterizes PB is inspired by the one for PML, thus considers the contrpositive nd proceeds by induction. In the only non-trivil cse, we will rrive t point in which t 1 = 1 ( 2,(t )+p) j J φ 2,j = t 2 for: derivtive t of t 1, such tht 1, (t ) > 2, (t ), not stisfying ny subformul φ 2,j ; suitble probbilistic vlue p such tht 2, (t ) + p < 1; n index set J identifying certin derivtives of t 2 other thn t. The choice of t is crucil, becuse negtion is no longer vilble in PML. Different from the cse of PML, this induces the introduction of p nd the limittion to J in the formt of the distinguishing formul. An importnt observtion is tht, in mny cses, disjunctive distinguishing formul cn be obtined from conjunctive one by suitbly incresing some probbilistic lower bounds. An obvious exception is when the use of conjunction/disjunction is not necessry for telling two different nodes prt. Exmple 1. The nodes t 1 nd t 2 in Fig. 1() cnnot be distinguished by ny formul in which neither conjunction nor disjunction occurs. It holds tht: t 1 = 0.5 ( b 1 c 1 ) t 2 t 1 = 1.0 ( b 1 c 1 ) = t 2 Notice tht, when moving from the conjunctive formul to the disjunctive one, the probbilistic lower bound decorting the -dimond increses from 0.5 to 1 nd the roles of t 1 nd t 2 with respect to = re inverted. The sitution is similr for
210 M. Bernrdo, M. Miculn () t 1 t 2 (c) t 5 t 6 0.5 0.5 0.5 0.5 t 1 t 1 t 2 t 2 b c b c 0.25 0.25 t 5 t 5 0.5 b t c 0.5 0.5 t 6 t b c (b) t 3 t 4 0.2 0.2 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.3 0.2 0.2 0.2 b c d b c b d c d b c d b c d b c b d c d (d) (e) (f) t 7 t 8 t 9 t 10 t 11 t 12 t 13 t 7 b t 8 b c 0.5 t b c 0.5 t 0.4 0.4 0.1 0.1 t t 10 b c t t 10 b c b 0.7 0.3 b Fig. 1. RPT f models used in the exmples of Sects. 4.3 nd 4.4. the nodes t 3 nd t 4 in Fig. 1(b), where two occurrences of conjunction/disjunction re necessry: t 3 = 0.2 ( b 1 c 1 d 1 ) t 4 t 3 = 0.9 ( b 1 c 1 d 1 ) t 4 but the roles of t 3 nd t 4 with respect to = cnnot be inverted. Exmple 2. For the nodes t 5 nd t 6 in Fig. 1(c), it holds tht: t 5 = 0.5 ( b 1 c 1 ) = t 6 If we replce conjunction with disjunction nd we vry the probbilistic lower bound between 0.5 nd 1, we produce no disjunctive formul cpble of discriminting between t 5 nd t 6. Nevertheless, distinguishing formul belonging to PML exists with no disjunctions t ll: t 5 = 0.5 b 1 = t 6 The exmples bove show tht the increse of some probbilistic lower bounds when moving from conjunctive distinguishing formuls to disjunctive ones tkes plce only in the cse tht the probbilities of reching certin nodes hve to be summed up. Additionlly, we recll tht, in order for two nodes to be relted by PB, they must enble the sme ctions, so focussing on single ction is enough for discriminting when only disjunction is vilble. Bering this in mind, for ny node t of finite height we define the set Φ (t) of PML formuls stisfied by t feturing:
Disjunctive Probbilistic Modl Logic 211 probbilistic lower bounds of dimonds tht re mximl with respect to the stisfibility of formul of tht formt by t (this is consistent with the observtion in the lst sentence before Thm. 5, nd keeps the set Φ (t) finite); dimonds tht rise only from existing trnsitions tht deprt from t (so to void useless dimonds in disjunctions nd hence keep the set Φ (t) finite); disjunctions tht stem only from single trnsitions of different nodes in the support of distribution reched by t (trnsitions deprting from the sme node would result in formuls like h H h ph φ h, with h1 h2 for h 1 h 2, which re useless for discriminting with respect to PB ) nd re preceded by dimond decorted with the sum of the probbilities ssigned to those nodes by the distribution reched by t. Definition 3. The set Φ (t) for node t of finite height is defined by induction on height(t) s follows: If height(t) = 0, then Φ (t) =. If height(t) 1 for t hving trnsitions of the form t i i with supp( i ) = {t i,j j J i} nd i I, then: Φ (t) = { i 1 i I} hplb( { i. i(t i I =J i,j J ) φ i,j,k t i,j supp( i), φ i,j,k Φ (t i,j )}) i j J j J where is vrint of in which identicl opernds re not dmitted (i.e., idempotence is forced) nd hplb keeps only the formul with the highest probbilistic lower bound decorting the initil i -dimond mong the formuls differring only for tht bound. To illustrte the definition given bove, we exhibit some exmples showing the usefulness of Φ -sets for discrimintion purposes. Given two different nodes tht with the sme ction rech two different distributions, good criterion for choosing t ( derivtive of the first node not stisfying certin formuls, to which the first distribution ssigns probbility greter thn the second one) seems to be the minimlity of the Φ -set. Exmple 3. For the nodes t 7 nd t 8 in Fig. 1(d), we hve: Φ (t 7 ) = { 1, 1 b 1 } Φ (t 8 ) = { 1, 1 b 1, 1 c 1 } A formul like 1 ( b 1 c 1 ) is useless for discriminting between t 7 nd t 8, becuse disjunction is between two ctions enbled by the sme node nd hence constituting nondeterministic choice. Indeed, such formul is not prt of Φ (t 8 ). While in the cse of conjunction it is often necessry to concentrte on severl lterntive ctions, in the cse of disjunction it is convenient to focus on single ction per node when iming t producing distinguishing formul. The fct tht 1 c 1 Φ (t 8 ) is distinguishing formul cn be retrieved s follows. Strting from the two identiclly lbeled trnsitions t 7 7, nd t 8 8, where 7, (t 7) = 1 = 8, (t 8) nd 7, (t 8) = 0 = 8, (t 7), we hve: Φ (t 7) = { b 1 } Φ (t 8) = { b 1, c 1 } If we focus on t 7 becuse 7, (t 7) > 8, (t 7) nd its Φ -set is miniml, then t 7 = c 1 = t 8 with c 1 Φ (t 8) \ Φ (t 7). As consequence, t 7 = 1 c 1 = t 8 where the vlue 1 decorting the -dimond stems from 1 8, (t 7).
212 M. Bernrdo, M. Miculn Exmple 4. For the nodes t 1 nd t 2 in Fig. 1(), we hve: Φ (t 1 ) = { 1, 0.5 b 1, 0.5 c 1 } Φ (t 2 ) = { 1, 0.5 b 1, 0.5 c 1, 1 ( b 1 c 1 )} The formuls with two dimonds nd no disjunction re identicl in the two sets, so their disjunction 0.5 b 1 0.5 c 1 is useless for discriminting between t 1 nd t 2. Indeed, such formul is prt of neither Φ (t 1 ) nor Φ (t 2 ). In contrst, their disjunction in which decortions of identicl dimonds re summed up, i.e., 1 ( b 1 c 1 ), is fundmentl. It belongs only to Φ (t 2 ) becuse in the cse of t 1 the b-trnsition nd the c-trnsition deprt from the sme node, hence no probbilities cn be dded. The fct tht 1 ( b 1 c 1 ) Φ (t 2 ) is distinguishing formul cn be retrieved s follows. Strting from the two identiclly lbeled trnsitions t 1 1, nd t 2 2, where 1, (t 1) = 1, (t 1) = 0.5 = 2, (t 2) = 2, (t 2) nd 1, (t 2) = 1, (t 2) = 0 = 2, (t 1) = 2, (t 1), we hve: Φ (t 1) = { b 1, c 1 } Φ (t 1) = Φ (t 2) = { b 1 } Φ (t 2) = { c 1 } If we focus on t 1 becuse 1, (t 1) > 2, (t 1) nd its Φ -set is miniml, then t 1 = b 1 = t 2 with b 1 Φ (t 2) \ Φ (t 1) s well s t 1 = c 1 = t 2 with c 1 Φ (t 2)\Φ (t 1). Thus, t 1 = 1 ( b 1 c 1 ) = t 2 where vlue 1 decorting the -dimond stems from 1 2, (t 1). Exmple 5. For the nodes t 5 nd t 6 in Fig. 1(c), we hve: Φ (t 5 ) = { 1, 0.25 b 1, 0.25 c 1, 0.5 ( b 1 c 1 )} Φ (t 6 ) = { 1, 0.5 b 1, 0.5 c 1 } The formuls with two dimonds nd no disjunction re different in the two sets, so they re enough for discriminting between t 5 nd t 6. In contrst, the only formul with disjunction, occurring in Φ (t 5 ), is useless becuse the probbility decorting its -dimond is equl to the one decorting the -dimond of ech of the two formuls with two dimonds in Φ (t 6 ). The fct tht 0.5 b 1 Φ (t 6 ) is distinguishing formul cn be retrieved s follows. Strting from the two identiclly lbeled trnsitions t 5 5, nd t 6 6, where 5, (t 5) = 5, (t 5 ) = 0.25, 5, (t ) = 0.5 = 6, (t 6) = 6, (t ), nd 5, (t 6) = 0 = 6, (t 5) = 6, (t 5 ), we hve: Φ (t 5) = { b 1 } Φ (t 5 ) = { c 1 } Φ (t 6) = { b 1, c 1 } Φ (t ) = Notice tht t might be useless for discriminting purposes becuse it hs the sme probbility in both distributions, so we exclude it. If we focus on t 5 becuse 5, (t 5 ) > 6, (t 5 ) nd its Φ -set is miniml fter the exclusion of t, then t 5 = b 1 = t 6 with b 1 Φ (t 6) \ Φ (t 5 ), while no distinguishing formul is considered with respect to t s element of supp( 6, ) due to the exclusion of t itself. As consequence, t 5 = 0.5 b 1 = t 6 where the vlue 0.5 decorting the -dimond stems from 1 ( 6, (t 5 ) + p) with p = 6, (t ). The reson for subtrcting the probbility tht t 6 reches t fter performing is tht t = b 1. We conclude by observing tht focussing on t s derivtive with the minimum Φ -set is indeed problemtic, becuse it would result in 0.5 b 1 when considering t s derivtive of t 5, but it would result in 0.5 ( b 1 c 1 ) when considering t s derivtive of t 6, with the ltter formul not distinguishing be-
Disjunctive Probbilistic Modl Logic 213 tween t 5 nd t 6. Moreover, when focussing on t 5, no formul φ could hve been found such tht t 5 = φ = t s Φ (t ) Φ (t 5 ). j J φ 2,j The lst exmple shows tht, in the generl formt 1 ( 2,(t )+p) for the PML distinguishing formul mentioned t the beginning of this subsection, the set J only contins ny derivtive of the second node different from t to which the two distributions ssign two different probbilities. No derivtive of the two originl nodes hving the sme probbility in both distributions is tken into ccount even if its Φ -set is miniml becuse it might be useless for discriminting purposes nor is it included in J becuse there might be no formul stisfied by this node when viewed s derivtive of the second node, which is not stisfied by t. Furthermore, the vlue p is the probbility tht the second node reches the excluded derivtives tht do not stisfy j J φ 2,j ; note tht the first node reches those derivtives with the sme probbility p. We present two dditionl exmples illustrting some techniclities of Def. 3. The former exmple shows the usefulness of the opertor nd of the function hplb for selecting the right t on the bsis of the minimlity of its Φ -set mong the derivtives of the first node to which the first distribution ssigns probbility greter thn the second one. The ltter exmple emphsizes the role plyed, for the sme purpose s before, by formuls occurring in Φ -set whose number of nested dimonds is not mximl. Exmple 6. For the nodes t 9 nd t 10 in Fig. 1(e), we hve: Φ (t 9 ) = { 1, 0.5 b 1, 0.5 c 1 } Φ (t 10 ) = { 1, 0.5 b 1, 0.5 c 1, 0.6 ( b 1 c 1 )} Strting from the two identiclly lbeled trnsitions t 9 9, nd t 10 10, where 9, (t ) = 9, (t ) = 0.5, 10, (t ) = 10, (t ) = 0.4, 10, (t 10) = 10, (t 10 ) = 0.1, nd 9, (t 10) = 9, (t 10 ) = 0, we hve: Φ (t ) = { b 1, c 1 } Φ (t ) = Φ (t 10) = { b 1 } Φ (t 10 ) = { c 1 } If we focus on t becuse 9, (t ) > 10, (t ) nd its Φ -set is miniml, then t = b 1 = t with b 1 Φ (t ) \ Φ (t ), t = b 1 = t 10 with b 1 Φ (t 10) \ Φ (t ), nd t = c 1 = t 10 with c 1 Φ (t 10 )\Φ (t ). Thus, t 9 = 0.6 ( b 1 c 1 ) = t 10 where the formul belongs to Φ (t 10 ) nd the vlue 0.6 decorting the -dimond stems from 1 10, (t ). If were used in plce of, then in Φ (t 10 ) we would lso hve formuls like 0.5 ( b 1 b 1 ) nd 0.5 ( c 1 c 1 ). These re useless in tht logiclly equivlent to other formuls lredy in Φ (t 10 ) in which disjunction does not occur nd, most importntly, would pprently ugment the size of Φ (t 10 ), n inpproprite fct in the cse tht t 10 were derivtive of some other node insted of being the root of tree. If hplb were not used, then in Φ (t 10 ) we would lso hve formuls like 0.1 b 1, 0.4 b 1, 0.1 c 1, nd 0.4 c 1, in which the probbilistic lower bounds of the -dimonds re not mximl with respect to the stisfibility of formuls of tht form by t 10 ; those with mximl probbilistic lower bounds ssocited with -dimonds re 0.5 b 1 nd 0.5 c 1, which lredy belong to Φ (t 10 ). In the cse tht t 9 nd t 10 were derivtives of two nodes under
214 M. Bernrdo, M. Miculn comprison insted of being the roots of two trees, the presence of those dditionl formuls in Φ (t 10 ) my led to focus on t 10 insted of t 9 for resons tht will be cler in Ex. 8 thereby producing no distinguishing formul. Exmple 7. For the nodes t 11, t 12, t 13 in Fig. 1(f), we hve: Φ (t 11 ) = { 1 } Φ (t 12 ) = { 1, 1 b 1 } Φ (t 13 ) = { 1, 0.7 b 1 } Let us view them s derivtives of other nodes, rther thn roots of trees. The presence of formul 1 in Φ (t 12 ) nd Φ (t 13 ) lthough it hs not the mximum number of nested dimonds in those two sets ensures the minimlity of Φ (t 11 ) nd hence tht t 11 is selected for building distinguishing formul. If 1 were not in Φ (t 12 ) nd Φ (t 13 ), then t 12 nd t 13 could be selected, but no distinguishing formul stisfied by t 11 could be obtined. The criterion for selecting the right t bsed on the minimlity of its Φ -set hs to tke into ccount further spect relted to formuls without disjunctions. If two derivtives with different probbilities in the two distributions hve the sme formuls without disjunctions in their Φ -sets, then distinguishing formul for the two nodes will hve disjunctions in it (see Exs. 4 nd 6). If the formuls without disjunctions re different between the two Φ -sets, then one of them will tell the two derivtives prt (see Ex. 3). A prticulr instnce of the second cse is the one in which for ech formul without disjunctions in one of the two Φ -sets there is vrint in the other Φ -set i.e., formul without disjunctions tht hs the sme formt but my differ for the vlues of some probbilistic lower bounds nd vice vers. In this event, regrdless of the minimlity of the Φ -sets, it hs to be selected the derivtive such tht (i) for ech formul without disjunctions in its Φ -set there exists vrint in the Φ -set of the other derivtive such tht the probbilistic lower bounds in the former formul re thn the corresponding bounds in the ltter formul nd (ii) t lest one probbilistic lower bound in formul without disjunctions in the Φ -set of the selected derivtive is < thn the corresponding bound in the corresponding vrint in the Φ -set of the other derivtive. We sy tht the Φ -set of the selected derivtive is (, <)-vrint of the Φ -set of the other derivtive. Exmple 8. Let us view the nodes t 5 nd t 6 in Fig. 1(c) s derivtives of other nodes, rther thn roots of trees. Bsed on their Φ -sets shown in Ex. 5, we should focus on t 6 becuse Φ (t 6 ) contins fewer formuls. However, by so doing, we would be unble to find distinguishing formul in Φ (t 5 ) tht is not stisfied by t 6. Indeed, if we look crefully t the formuls without disjunctions in Φ (t 5 ) nd Φ (t 6 ), we note tht they differ only for their probbilistic lower bounds: 1 Φ (t 6 ) is vrint of 1 Φ (t 5 ), 0.5 b 1 Φ (t 6 ) is vrint of 0.25 b 1 Φ (t 5 ), nd 0.5 c 1 Φ (t 6 ) is vrint of 0.25 c 1 Φ (t 5 ). Therefore, we must focus on t 5 becuse Φ (t 5 ) contins formuls without disjunctions such s 0.25 b 1 nd 0.25 c 1 hving smller bounds: Φ (t 5 ) is (, <)-vrint of Φ (t 6 ). Consider now the nodes t 9 nd t 10 in Fig. 1(e), whose Φ -sets re shown in Ex. 6. If function hplb were not used nd hence Φ (t 10 ) lso contined 0.1 b 1,
Disjunctive Probbilistic Modl Logic 215 0.4 b 1, 0.1 c 1, nd 0.4 c 1, then the formuls without disjunctions in Φ (t 9 ) would no longer be equl to those in Φ (t 10 ). More precisely, the formuls without disjunctions would be similr between the two sets, with those in Φ (t 10 ) hving smller probbilistic lower bounds, so tht we would erroneously focus on t 10. j J φ 2,j, Summing up, in the PML distinguishing formul 1 ( 2,(t )+p) the steps for choosing the derivtive t, on the bsis of which ech subformul φ 2,j is then generted so tht it is not stisfied by t itself, re the following: 1. Consider only derivtives to which 1, ssigns probbility greter thn the one ssigned by 2,. 2. Within the previous set, eliminte ll the derivtives whose Φ -sets hve (, <)-vrints. 3. Among the remining derivtives, focus on one of those hving miniml Φ -set. Theorem 6. Let (T, A, ) be in RPT f nd t 1, t 2 T. Then t 1 = t 2 iff t 1 = φ t 2 = φ for ll φ PML. Moreover, if t 1 t 2, then there exists φ PML distinguishing t 1 from t 2 such tht depth(φ) mx(height(t 1 ), height(t 2 )). 4.4 PML Chrcterizes PB : A Direct Proof for Discrete Systems By dpting the proof of Thm. 6 consistently with the proof of Thm. 4, we cn lso prove tht PML chrcterizes PB by working directly on discrete stte spces. j J φ 2,j t 2. For ny node t of The ide is to obtin t 1 = 1,(t )+p finite height, we define the set Φ (t) of PML formuls stisfied by t feturing, in ddition to mximl probbilistic lower bounds nd dimonds rising only from trnsitions of t s for Φ (t), conjunctions tht (i) stem only from trnsitions deprting from the sme node in the support of distribution reched by t nd (ii) re preceded by dimond decorted with the sum of the probbilities ssigned by tht distribution to tht node nd other nodes with the sme trnsitions considered for tht node. Given t hving trnsitions of the form t i i with supp( i ) = {t i,j j J i} nd i I, we let: Φ (t) = { i 1 i I} splb({ i i(t i,j ) φ i,j,k K K i,j, t i,j supp( i), φ i,j,k Φ (t i,j ) }) i I k K where { nd } re multiset prentheses, K i,j is the index set for Φ (t i,j ), nd function splb merges ll formuls possibly differring only for the probbilistic lower bound decorting their initil i -dimond by summing up those bounds (such formuls stem from different nodes in supp( i )). A good criterion for choosing t occurring in the PML distinguishing formul t the beginning of this subsection is the mximlity of the Φ -set. Moreover, in tht formul J only contins ny derivtive of the second node different from t to which the two distributions ssign two different probbilities, while p is the probbility of reching derivtives hving the sme probbility in both distributions tht stisfy j J φ 2,j. Finlly, when selecting t, we hve to leve out ll the derivtives whose Φ -sets hve (, <)-vrints.
216 M. Bernrdo, M. Miculn Theorem 7. Let (T, A, ) be in RPT f nd t 1, t 2 T. Then t 1 = t 2 iff t 1 = φ t 2 = φ for ll φ PML. Moreover, if t 1 t 2, then there exists φ PML distinguishing t 1 from t 2 such tht depth(φ) mx(height(t 1 ), height(t 2 )). 5 Conclusions In this pper, we hve studied modl logic chrcteriztions of strong bisimilrity over rective probbilistic processes. Strting from previous work by Lrsen nd Skou [12] (who provided chrcteriztion bsed on probbilistic extension of Hennessy-Milner logic) nd by Deshrnis, Edlt, nd Pnngden [6] (who showed tht negtion is not necessry), we hve proved tht conjunction cn be replced by disjunction without hving to reintroduce negtion. Thus, in the rective probbilistic setting, conjunction nd disjunction re interchngeble to chrcterize (bi)simultion equivlence, while they re both necessry for simultion preorder [7]. As side result, with our proof technique we hve provided lterntive proofs of the expressiveness of PML nd PML. The intuition behind our result is tht from conjunctive distinguishing formul it is often possible to derive disjunctive one by suitbly incresing some probbilistic lower bounds. On the model side, this corresponds to summing up the probbilities of reching certin sttes tht re in the support of trget distribution. In fct, stte of n RPLTS cn be given semntics s rective probbilistic tree, nd hence it is chrcterized by the countble set of formuls (pproximted by the Φ -set) obtined by doing finite visits of the tree. On the ppliction side, the PML -bsed chrcteriztion of bisimilrity over rective probbilistic processes my help to prove conjecture in [4]. This work studies the discriminting power of three different testing equivlences respectively using rective probbilistic tests, fully nondeterministic tests, nd nondeterministic nd probbilistic tests. Numerous exmples led to conjecture tht testing equivlence bsed on nondeterministic nd probbilistic tests my hve the sme discriminting power s bisimilrity. Given two PB -inequivlent rective probbilistic processes, the ide of the tenttive proof is to build distinguishing nondeterministic nd probbilistic test from distinguishing PML formul. One of the min difficulties with crrying out such proof, i.e., the fct tht choices within tests fit well together with disjunction rther thn conjunction, my be overcome by strting from distinguishing PML formul. References 1. P. Aczel nd N. Mendler. A finl colgebr theorem. In Proc. of the 3rd Conf. on Ctegory Theory nd Computer Science (CTCS 1989), volume 389 of LNCS, pges 357 365. Springer, 1989. 2. C. Bier nd M. Kwitkowsk. Domin equtions for probbilistic processes. Mthemticl Structures in Computer Science, 10:665 717, 2000. 3. M. Brr. Terminl colgebrs in well-founded set theory. Theoreticl Computer Science, 114:299 315, 1993.
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