Uniform Labeled Transition Systems for Nondeterministic, Probabilistic, and Stochastic Process Calculi

L. Aceto and M.R. Mousa vi (Eds.): P A CO 201 1 EPTCS 60, 2011, pp. 66–75, doi:10.4204 /EPTCS .60.5 c  M. Bernardo, R. De Nicola & M. Loreti This work is licensed under the Creativ e Commons Attribution License. Unif orm Labeled T ransition Systems f or Nondeterminist ic, Pr obabilistic , and Stochastic Process Calculi Marco Bernardo Dipartimento di Scienze di Base e Fondamen ti – Universit ` a di Urbino – Italy Rocco De Nicola IMT – Institute for Adv anced Studies Lucca – Italy Dipartimento di Sistemi e Informatica – Uni versit ` a di Firenze – Italy Michele Loreti Dipartimento di Sistemi e Informatica – Uni versit ` a di Firenze – Italy Labeled transition sy stems are typ ically used to represent the beh avior of nondeterm inistic processes, with lab eled tr ansitions d efining a o ne-step state-to -state reachability re lation. This mo del h as been recently made more general by modifying th e transition relation in such a way that it associates with any source s tate and transition label a reachability distribution, i.e. , a function mapp ing each possible target state to a value of some d omain th at e xpresses t he degree of one-step re achability o f that tar get state. In th is extended abstra ct, we show how th e resulting m odel, called U LT R A S from Uniform Labeled T R A nsition System, can b e naturally used to g iv e seman tics to a fu lly n ondeterm inistic, a fully probabilistic, and a fully stochastic variant o f a CSP-lik e proc ess language. 1 Introd uction Process algebr as are one of the most success ful formalis ms f or modeling concurre nt syste ms and proving their properties such as correctness , liv eness or safety . After their initial s uccess in this res pect, th ey ha ve also been exte nded to deal with propertie s related to performance and quality of service. Thus, proce ss algebr as hav e been enrich ed with quant itati v e no tions of time and probab ilities and inte grated theories ha ve been cons idered; for a compr ehensi v e descripti on of this approach, the read er is referred to [1]. Moreo ver , due to the gro wing interest in the analysis of shared-r esourc e systems, stoc hastic va riants of proces s algebras ha v e als o been proposed . T he main aim be ing the integr ation of q ualitat iv e descri ptions with those relativ e to perfo rmance in a single mathematical frame work by build ing on th ecombina tion of labeled transiti on systems (L TS) and cont inuous -time Marko v chains (CTMC). In [9], two of the authors of the presen t paper , tog ether with D. Latella and M. Massink , proposed a v ar iant of L TS, na mely r ate tr ansiti on systems (R TS), as a tool for pro viding semanti cs to some of the most represent ati ve stochastic process langua ges. W ith in L TS, the transi tion relation describes the e v olutio n of a sy stem from o ne st ate to another as determined by the ex ecu tion of sp ecific actio ns, thus it is a set of tr iples ( s t a t e , ac t ion , s t a t e ) . In con trast, within R TS the transi tion relation ֌ as sociate s w ith a giv en state P and a gi v en transition label (actio n) a a function, say P , mapping each term into a non- neg ati ve real number . The transit ion P a ֌ P has the follo wing meaning : if P ( Q ) = v with v 6 = 0, then Q is reac hable from P by exe cuting a , the duration of such an execu tion being exponen tially distrib ut ed with rate v ; if P ( Q ) = 0, then Q is not reach able from P via a . R T Ss ha ve been us ed for providin g a uniform semanti c f rame work for modeling many of the d if ferent stocha stic process langua ges, facili tating reaso ning about them, and thro wing light on their similarities M. Bernardo, R. De Nicola & M. Loreti 67 as well as on their dif fer ences. In [8], we considere d a limited, but representat i ve, number of stochasti c proces s calculi and prov ided the R TS semantics for (fully) stochastic proc ess languag es both based on the CSP-like, multipart interacti on parad igm and on th e CCS-lik e, two-way s i nteract ion parad igm. T hen, in [10], R TS s were exte nded by requiri ng that the domain of P b e a generi c semiring and other v arian ts of stoch astic process algebr as ar e studied , in particular it is shown that also language s, lik e IML [13], that mix stochast icity and nondetermini sm can be easily modeled. In [4], we performed a further step in the direc tion of providing a un iform characte rizatio n o f the semantic s of diffe rent process calcu li and introduced a more gener al frame wor k than R TS , which could be instantiate d to model not only stochastic proc ess algebras but also classical process algebras, usual ly modelled via L TS, and other quant itati v e v aria nts of process algebr as that would consider t ime, proba bili- ties, resource s, etc.; we thus introduce d U L T R A S ( Un iform Labeled T R A nsition S ystems ). The transitio n relatio n of U L T R A S associates with a state and a giv en transi tion label a functio n map ping each state into an element of a generic domain D . An U L T R A S transitio n ( s , a , D ) is written s a − → D , with D ( s ′ ) being a D -v alue quantifyi ng the de gree of reachabi lity of s ′ from s via the ex ecuti on of a and D ( s ′ ) = ⊥ meaning that s ′ is not rea chable from s via a . By appropria tely changing the domain D , dif ferent models of concurr ent systems can be captured. For e xample, if D is the set B con sisting of the two Boolean val - ues t rue and f al se we can capture classical L TSs, whil e if D is the set R [ 0 , 1 ] we do capture probabilis tic models, and when D is the set R ≥ 0 we do captu re stochastica lly timed models. Of course , mod eling state transiti ons a nd their annotations i s one of the ke y ingred ients; ho we ve r , one has also to combin e s ingle transitions to obtain computation s and find out ways for determining when two states giv e rise to “equ iv alent” comput ation trees. T o this aim, in [4] we introd uced the notions of trace equi v ale nce and bisimulation equi v alence ov er U L T R A S . An important component of the equi v ale nces definitio n is a measur e function M M ( s , α , S ′ ) tha t computes the d egre e of multi-step reacha bility of a set of targ et sta tes S ′ from a source state s when pe rforming computatio ns labeled with trace α . For instance, to captu re classical equi v al ences over nondetermini stic sys tems, th e measure yields ⊤ if there exists a computa tion from s to S ′ labele d with α and ⊥ othe rwise. As another exa mple, to capture probabilist ic equi v ale nces, the measure yields a valu e in R [ 0 , 1 ] that repre sents the proba bility o f the set of computa tions labele d w ith α to reach a state in S ′ from s . In this note , we put U L T R A S at work and use them to pro vide a unifo rm semantica l descrip tion for a few (quali tati ve and quantitati v e) v ariants of a very simple process algebra. For the sak e of simplicity , we limit our attention to a purely nonde terministi c, a fully probabili stic, and a fully stochastic calcul us, without allowin g any interplay between nondetermin ism and quantitati v e aspec ts. In our view , the three (ve ry compact) resulting sets of operationa l rules giv e e vide nce of the expres si ve power o f our approach and help in appreciating similarities and dif ference s among the three va riants of the considere d process algebr a. The rest of the paper is org anized as follo ws. In Sect. 2 , we recap the basic notions of U L T R A S introd uced in [4] and define three d if ferent types of beha v ioral equiv a lences ove r them. T o the defini tion of trace an d b isimulati on equi v alen ces already p resent in [4], w e a dd the d efinition o f t esting equi v alenc e togeth er with the se t up of the necessary testing frame w ork that we ha v e intro duced in [5]. In Sect. 3, we sho w ho w U L T R A S can be used to provide the operati onal semantics of classical C SP [6] and of two of its probab ilistic [17, 2] and stochastic [14] varian ts. Finally , Sect. 4 reports on some future work. 68 U L T R A S for Nonde terministi c, Probabilis tic, and Stocha stic Process Calculi 2 Unif orm Labeled T rans ition Systems The beha vio r of seque ntial, concu rrent, and distrib ute d processes can be described by means of the so called labeled transition system (L TS) model [16]. It con sists of a set of state s, a set of tran sition labels , and a transiti on relation. States correspo nd to the operat ional modes that processes can pass throug h. L abels describ e the activi ties that process es can perform intern ally or use to inter act with the en vironment. The transi tion relatio n defines pr ocess e v ol ution as d etermined by the e xec ution of specific acti viti es and is formalized as a state-to-s tate reachab ility relation. In th is sec tion, we re call from [4] a generaliz ation of the L TS model that aims at provi ding a un iform frame work that can be emplo yed for defining and comparing the beha vior of differe nt types of proc ess. In the new model, named U L T R A S from Uniform Labeled T R A nsition S ystem, the transition relatio n associ ates with any source state and transitio n label a functio n mapping each possible tar get state to an element of a domain D . In other words, the state-t o-state reachabil ity relation typical of the L TS model is replaced by a state-to-st ate-dis tribution reachabilit y relation. This is a cons equen ce of the fact that the con cept of next st ate is genera lized via a function that repr esents a one-st ep reachability distrib ution , which exp resses the degree of reachab ility from the current state of ev ery possib le next state. As sho wn in [4 ], by appropr iately changin g the domain D we can capt ure dif ferent process models, in particul ar quantitati v e ones like Marko v chains [18]. For example : • If D is the suppo rt set B = {⊥ , ⊤} of the B oolean algebra with the standar d conj unction ( ∧ ) and disjun ction ( ∨ ) operators, then w e captu re classical L TS models . • If D = R [ 0 , 1 ] , then we captu re fully proba bilisti c models in the form of act ion-lab eled discrete- time Marko v chains (ADTMC). • If D = R ≥ 0 , th en we c apture full y stocha stic models in th e form of action-l abeled continuou s-time Marko v chains (A CTMC). 2.1 Definition of the Uniform Pr ocess Model The de finition of our un iform model is pa rameteriz ed with respe ct to a complet e partia l order ( D , ⊑ ) whose elements expres s the degree of one-step reachabili ty of a state. In the follo wing, we denote by ⊥ the ⊑ -least elemen t of D and by [ S → D ] the set of functi ons fro m a set S to D , which is ranged over by D . Definition 2.1 Let ( D , ⊑ ) be a complete partial o rder . A un iform label ed transi tion system on ( D , ⊑ ) , or D - U LT R A S for short, is a triple U = ( S , A , − → ) where: • S is an at most counta ble set of states. • A is a coun table set of transition -labeli ng actions. • − → ⊆ S × A × [ S → D ] is a tran sition relation. W e say that the D -U L T R A S U is functional iff − → is a function from S × A to [ S → D ] . Every transition ( s , a , D ) is written s a − → D , with D ( s ′ ) being a D -val ue quantify ing the deg ree of reacha bility of s ′ from s via the exec ution of a and D ( s ′ ) = ⊥ meaning that s ′ is not reachable from s via a . When cons idering a functi onal U LT R A S , we will of ten write D s , a ( s ′ ) to deno te the same D -v al ue. M. Bernardo, R. De Nicola & M. Loreti 69 2.2 Beha vioral Equiv alences for the U L T R A S Model L T S-based models come equipp ed with equiv al ences throu gh which it is possible to compare processes on the basis of their beha v ior and reduc e the state space of a process bef ore analyz ing its pro pertie s. These beha v ioral equiv al ences result in a linear -time /branch ing-time spectrum [11, 15, 3, 1] includi ng se ve ral varia nts of three major approach es: bisi mulation [12], trace [6], and testin g [7]. W e now recall ho w bisimulation , trace, and testing equiv al ences can be uniformly defined o ver the U LT R A S model. Their definiti on is parameterize d with respec t to a measure function that exp resses the de gree of multi- step reacha bility of a set of states. Similar to the one-step reacha bility encod ed within an U L T R A S , in which we conside r indivi dual actions, multi-step reacha bility relies on sequences of actio ns commonly called traces, which are the observ ab le effec ts of the computations performed by an U L T R A S . Definition 2.2 Let A be a countable set of transition-la beling actions. A trace α is an element of A ∗ , where α = ε denote s the empty trace. Definition 2.3 Let U = ( S , A , − → ) be a D -U L T R A S and ( M , ⊕ , ⊗ ) be a lattice . An M -measure function for U is a function M M : S × A ∗ × 2 S → M . Note that diff erent m easure functio ns can induce dif ferent varian ts of a behav ioral equi v a lence on the same D - U L T R A S depending on the support set and the operations of ( M , ⊕ , ⊗ ) . Although D and M may be the same support set, this is not nece ssarily the case: while D -va lues are relat ed to one-step reacha bility , M -v alu es – espe cially those of the form M M ( s , α , S ′ ) – are compute d on the basis of D -v alues to quanti fy multi-step reachabil ity . 2.2.1 T race Equiva lence T race equi v al ence is straightfor ward: two states are trace equiv a lent if ev ery trace has the same measure with respect to the entire set of states when startin g from those two states. Definition 2.4 Let U = ( S , A , − → ) be a D -U L T R A S and M M be an M -measure function for U . W e say that s 1 , s 2 ∈ S are M M -trace equi v alen t, written s 1 ∼ Tr , M M s 2 , if f for all traces α ∈ A ∗ : M M ( s 1 , α , S ) = M M ( s 2 , α , S ) 2.2.2 Bisimulation Equivale nce While trace equiv a lence simply compares any two states without taking into acco unt the state s reached at the end of the trace, bisimulati on equi v alence also poses constraint s on the reach ed states. Definition 2.5 Let U = ( S , A , − → ) be a D - U L T R A S and M M be an M -measure function for U . An equi v ale nce relation B over S is an M M -bisimul ation if f, whene v er ( s 1 , s 2 ) ∈ B , then for all traces α ∈ A ∗ and equi v ale nce classes C ∈ S / B : M M ( s 1 , α , C ) = M M ( s 2 , α , C ) W e say th at s 1 , s 2 ∈ S are M M -bisimila r , written s 1 ∼ B , M M s 2 , if f there exists an M M -bisimul ation B ov er S such that ( s 1 , s 2 ) ∈ B . 2.2.3 T esting Equivalence The definition of t esting equi v alenc e require s the formaliz ation of the notion of test and the co nsidera tion of configurat ions rather than simple state s. A test specifies w hich actio ns of a process are permitted at each step and can be e xpress ed as some suitable U L T R A S that includes a success state, which is used to determin e w hich on es are the successful computations. 70 U L T R A S for Nonde terministi c, Probabilis tic, and Stocha stic Process Calculi Definition 2.6 Let ( D , ⊑ ) be a complete partial order . A D -observ at ion system is a D - U L T R A S O = ( O , A , − → ) w here O contains a distin guishe d success state denote d by ω such that, whene v er ω a − → D , then D ( o ) = ⊥ for all o ∈ O . W e say that a computatio n of O is successfu l iff its leng th is finite and its last state is ω . A D - U L T R A S can be tested only throug h a D -obser v ation system by running them in parallel and enforc ing sy nchro nizatio n on any action . The states of the r esultin g D - U L T R A S are called configuratio ns and are pairs each formed by a state of the D - U L T R A S under test and a state of th e D -obser v ation system. A configuration can ev olv e to a ne w configuration only through the synchroniz ation of two transitions – leavin g the two states constituting the configuration – that are labeled with the same action and reach at least one state, i.e., two identically labeled transitions w hose tar get functions are not identically equal to ⊥ . For each such pair of synchro nizing trans itions, the target functio n of the resulting transition is ob- tained from t he two original tar get func tions by means o f s ome D -valu ed function δ , which c omputes th e deg ree of one -step reach ability of e ve ry po ssible tar get con figuration . Since ⊥ represen ts unreachabi lity , the only const raint on δ is that it is ⊥ -preserving , i.e., th at it yields ⊥ iff at least o ne of its ar gument s is ⊥ . As a consequen ce of this constrain t, in the case of nonde terminist ic process es δ boils down to logical conjun ction, whereas sev eral alter nati v e options are a v ailable in the case of probabi listic and stochast ic proces ses. Definition 2.7 Let U = ( S , A , − → U ) be a D -U L T R A S, O = ( O , A , − → O ) be a D -obser v ation system, and δ be a ⊥ -preserving D -valu ed function. The interactio n system of U and O with respect to δ is the D - U LT R A S I δ ( U , O ) = ( S × O , A , − → ) where: • Every element ( s , o ) ∈ S × O is called a configurat ion and is said to be successful if f o = ω . W e de note by S δ ( U , O ) the set of successf ul configuration s of I δ ( U , O ) . • The transition relatio n − → ⊆ ( S × O ) × A × [( S × O ) → D ] is such that ( s , o ) a − → D if f s a − → U D 1 and o a − → O D 2 with D ( s ′ , o ′ ) being obtained fro m D 1 ( s ′ ) and D 2 ( o ′ ) by applying δ . W e say that a computation of I δ ( U , O ) is successful iff its length is finite and its last configura- tion is succ essful. Definition 2.8 Let U = ( S , A , − → U ) be a D -U L T R A S, M M be an M -measure funct ion for U , δ be a ⊥ -preserv ing D -v alued function, and O = ( O , A , − → O ) be a D -obser v ation sys tem. The extensio n of M M to I δ ( U , O ) is the functio n M δ , O M : ( S × O ) × A ∗ × 2 S × O → M whose definition is obtained from that of M M by re placing sta tes and tr ansitio ns of U with configura tions and tra nsition s of I δ ( U , O ) . Definition 2.9 Let U = ( S , A , − → U ) be a D - U L T R A S , M M be an M -measure function for U , and δ be a ⊥ -prese rving D -val ued funct ion. W e say that s 1 , s 2 ∈ S are M δ M -testin g e qui v alen t, written s 1 ∼ T , M δ M s 2 , if f for all D -observ ati on systems O = ( O , A , − → O ) with initial state o ∈ O and for all traces α ∈ A ∗ : M δ , O M (( s 1 , o ) , α , S δ ( U , O )) = M δ , O M (( s 2 , o ) , α , S δ ( U , O )) 3 U L T R A S in Use: Three Experiments with CSP In this sectio n, we sho w that the U L T R A S formalis m can be used for providi ng operation al m odels of dif feren t kinds of process algebra. In parti cular , w e w ill see ho w operationa l semantics of the language of Communica ting Sequential Proc esses (CSP) [6] and two of its v arian ts, which resp ecti v ely extend M. Bernardo, R. De Nicola & M. Loreti 71 the calculu s w ith probab ilistic binary operators and expon entiall y timed actions, can be easily described within the U L T R A S model by appropria tely instantiatin g the domain D . First, we introduce the syntax of the nonde terministi c language and its operat ional se mantics in terms of U L T R A S. For the sake of simplicity , we only consider a kernel of CSP and omit some operators, like hiding and renaming, because their treatment would add very little to the message we wish to con ve y . Then, w e focus on the probabilis tic and stochastic var iants of the ke rnel of CSP by exhibiti ng a suitabl e U L T R A S -based operational semantics for each of them. 3.1 B -U L T R A S Semantics f or a Ker nel of CSP In CS P , syst ems are described as interaction s of components that m ay engage in activi ties. Component s reflect the beha vio r of the import ant parts of a system, whi le acti vitie s cap ture t he actions that the compo- nents perfo rm. The ch oice among the acti vities tha t are enab led in eac h system sta te is no ndeter ministic. Let A be a countable set of activi ties. W e denote by P CSP the set of process terms defined according to the follo wing grammar: P :: = a . P | P + P | P k L P | B where a ∈ A , L ⊆ A , and B is a beh a vioral constant defined by an approp riate equation of the form A ∆ = P for some process term P in w hich constants occur only guarded in P , i.e., inside the scope of an actio n prefix. Component a . P models a process that performs acti vit y a and then beha ves like P . Component P 1 + P 2 models a process that may beha ve either as P 1 or as P 2 . The operator P 1 k L P 2 models instead the parallel execu tion of P 1 and P 2 , which synchroniz e (or cooperate ) on ev ery acti vity in L and proceed indepe ndentl y on eve ry acti vity not in L . The beha vior of constant B is the same as that of the process term P on the right- hand side of its defining equation. The semantics for the conside red kern el of CSP ca n be des cribed in terms of th e foll o wing fu nction al B - U LT R A S : ( P CSP , A , − → ) whose transition relation − → is defined in T able 1. Giv en a transition P a − → D , intuiti v ely we ha v e that D ( Q ) = ⊤ means that Q is reachab le from P via an a -trans ition, while D ( Q ) = ⊥ means that it is not possib le to reach Q from P by execu ting a . Rule A C T states that a . P ev olves via a to [ P 7→ ⊤ ] , w ith the latter being the function assoc iating ⊤ with P and ⊥ with all the other pro cess terms. On th e co ntrary , / 0- A C T establis hes that no sta te is reacha ble from a . P by performing any action b 6 = a . This is formaliz ed by lettin g a . P ev olv e via b 6 = a to [ ] , the function associat ing ⊥ with each process term. Rule S U M describes nondeter ministic choic e: the states reach able from P 1 + P 2 via a are all those that can be reach ed either by P 1 or by P 2 . Indeed, D 1 ∨ D 2 denote s the function D such that D ( Q ) = D 1 ( Q ) ∨ D 2 ( Q ) for all process terms Q . Rules C O O P and I N T go vern parallel compositio n. Rule C O O P is used for computing the next-stat e functi on when a synchroniza tion between P 1 and P 2 occurs . W hene v er P 1 a − → D 1 and P 2 a − → D 2 with a ∈ L , then P 1 k L P 2 e v olv es via a to D 1 k L D 2 , where ( D 1 k L D 2 )( Q ) is D 1 ( Q 1 ) ∧ D 2 ( Q 2 ) if Q = Q 1 k L Q 2 and ⊥ otherwise. Rule I N T deals with a / ∈ L . In that case, if P 1 a − → D 1 and P 2 a − → D 2 , then P 1 k L P 2 e v olv es via a to ( D 1 k L P 2 ) ∨ ( P 1 k L D 2 ) , where D 1 k L P 2 (resp. P 1 k L D 2 ) den otes the function D such that D ( Q ) is D 1 ( P ′ 1 ) (resp. D 2 ( P ′ 2 ) ) if Q = P ′ 1 k L P 2 (resp. Q = P 1 k L P ′ 2 ) and ⊥ otherwise. 72 U L T R A S for Nonde terministi c, Probabilis tic, and Stocha stic Process Calculi a . P a − → [ P 7→ ⊤ ] A C T b 6 = a a . P b − → [ ] / 0- A C T B ∆ = P P a − → D B a − → D C A L L P 1 a − → D 1 P 2 a − → D 2 P 1 + P 2 a − → D 1 ∨ D 2 S U M P 1 a − → D 1 P 2 a − → D 2 a ∈ L P 1 k L P 2 a − → D 1 k L D 2 C O O P P 1 a − → D 1 P 2 a − → D 2 a / ∈ L P 1 k L P 2 a − → ( D 1 k L P 2 ) ∨ ( P 1 k L D 2 ) I N T T able 1: U LT R A S -based operational semantic rules for CSP 3.2 R [ 0 , 1 ] -U L T R A S Semantics for PCSP W e no w consid er a probabilis tic varian t of CSP that we call PCSP . While in CSP the next action to ex ecute is se lected nondetermini stically , in PCSP it is selected ac cording to some discre te prob ability distrib u tion that can be dif fe rent fro m state to state. T aking inspira tion from [17, 2], the probabili stic calcul us PCSP is obt ained from CSP by decor ating the alternati v e and parallel composition ope rators with a probabi lity valu e p ∈ R [ 0 , 1 ] . W e deno te by P PCSP the set of proces s terms defined according to the follo wing grammar: P :: = a . P | P + p P | P k L p P | B Component P 1 + p P 2 models a pr ocess that, aft er performing an ac tion, beha v es as the co ntinua tion of P 1 with probability p or the conti nuatio n of P 2 with probability 1 − p . Similarly , in P 1 k L p P 2 the valu e p is used to regu late the interlea vin g of P 1 and P 2 . The semantic s for PCS P can be desc ribed in terms of the follo wing functio nal R [ 0 , 1 ] -U L T R A S: ( P PCSP , A , − → ) whose transition relation − → is defined in T able 2. Giv en a transition P a − → D , intuiti v ely we ha v e that D ( Q ) > 0 means that Q is reachable from P via an a -transition w ith prob ability D ( Q ) , while D ( Q ) = 0 means that it is not possibl e to reach Q from P by exec uting a . Note that ∑ Q D ( Q ) ∈ { 0 , 1 } . The first three rules are identical to the fi rst three rules of T able 1, with the dif ference that [ P 7→ 1 ] denote s the functi on associa ting 1 with P and 0 with all the other process terms, w hile [ ] den otes th e functi on associating 0 with each process term. Rule S U M relies on the follo wing notatio n: • D 1 + D 2 denote s the function D such that D ( Q ) = D 1 ( Q ) + D 2 ( Q ) for all process terms Q . • ⊕ D = ∑ Q D ( Q ) . • x y · D denotes the function D ′ such that D ′ ( Q ) = x y · D ′ ( Q ) if y 6 = 0 and 0 otherwise. This rule asser ts that the states reachab le from P 1 + p P 2 via a are obta ined by aggre gatin g according to p the probabili ty distrib ution s associa ted with P 1 and P 2 after a . When both P 1 and P 2 can perform a , i.e., P 1 a − → D 1 and P 2 a − → D 2 with D 1 and D 2 both differ ent from [ ] , then ⊕ D 1 = ⊕ D 2 = 1 and hence the M. Bernardo, R. De Nicola & M. Loreti 73 a . P a − → [ P 7→ 1 ] A C T b 6 = a a . P b − → [ ] / 0- A C T B ∆ = P P a − → D B a − → D C A L L P 1 a − → D 1 P 2 a − → D 2 P 1 + p P 2 a − → p ·⊕ D 1 p ·⊕ D 1 +( 1 − p ) ·⊕ D 2 · D 1 + ( 1 − p ) ·⊕ D 2 p ·⊕ D 1 +( 1 − p ) ·⊕ D 2 · D 2 S U M P 1 a − → D 1 P 2 a − → D 2 a ∈ L P 1 k L p P 2 a − → D 1 k L D 2 C O O P P 1 a − → D 1 P 2 a − → D 2 a 6∈ L P 1 k L p P 2 a − → p ·⊕ D 1 p ·⊕ D 1 +( 1 − p ) ·⊕ D 2 · ( D 1 k L P 2 ) + ( 1 − p ) ·⊕ D 2 p ⊕ D 1 +( 1 − p ) ·⊕ D 2 · ( P 1 k L D 2 ) I N T T able 2: U LT R A S -based operational semantic rules for PCSP aggre gate pro babilit y distrib ut ion reduces to p · D 1 + ( 1 − p ) · D 2 . In contrast, w hen D 1 (resp. D 2 ) is equal to [ ] , then ⊕ D 1 = 0 (resp. ⊕ D 2 = 0) and henc e the aggre gate probability distrib ut ion reduces to D 2 (resp. D 1 ). Rules C O O P and I N T gov ern parallel compositio n. They are similar to the two correspo nding rules of T able 1, with the differ ences that (i) in the synchro nizatio n case ( D 1 k L D 2 )( Q ) is D 1 ( Q 1 ) · D 2 ( Q 2 ) if Q = Q 1 k L p Q 2 and 0 otherwise, while (ii) in the interlea vin g case a S U M -like aggre gation based on p of the probab ility distrib ution s associa ted w ith P 1 and P 2 after a comes into play . 3.3 R ≥ 0 -U L T R A S Semantics f or PEP A Building on [9 , 8], we finally cons ider a stochastica lly timed va riant of C SP called Performan ce Eval u- ation Process Algebra (PEP A) [14]. In this calcul us, e very action is equip ped with a rate λ ∈ R > 0 that uniqu ely characte rizes the exponen tially distrib ut ed random variab le quant ifying the duration of the ac- tion itself (the expected duration is 1 / λ ). The choic e among the actions that are enabled in each state is gov erne d by the race policy: the action to execu te is the one that samples the least duratio n. Therefore , (i) the sojourn time in each state is exp onenti ally distrib u ted with rate gi v en by the sum of the rates of the transit ions departi ng from that state, (ii) the ex ecution probabil ity of each transition is proport ional to its rate, and (iii) the alterna ti ve and parallel composition operators are implicitly probabil istic. W e deno te by P PEP A the set of proces s terms defined accordin g to the follo wing grammar: P :: = ( a , λ ) . P | P + P | P k L P | B Component ( a , λ ) . P models a process that can perform action a at rate λ and then beha v es like P . The semantic s for PEP A can be describ ed in terms of the follo wing functiona l R ≥ 0 -U L T R A S: ( P PEP A , A , − → ) whose transition relation − → is defined in T able 3. Giv en a transition P a − → D , intuiti v ely we ha v e that D ( Q ) > 0 m eans that Q is reachable from P via an a -tra nsition at rate D ( Q ) , while D ( Q ) = 0 means that it is not possibl e to reach Q from P by exec uting a . The rules of T able 3 are similar to those of T able 2, with the diffe rences that (i) [ P 7→ λ ] denote s the functi on associ ating λ with P and 0 with all the other proce ss terms, (ii) no normalizati on is need ed in 74 U L T R A S for Nonde terministi c, Probabilis tic, and Stocha stic Process Calculi ( a , λ ) . P a − → [ P 7→ λ ] A C T a 6 = b ( a , λ ) . P b − → [ ] / 0- A C T B ∆ = P P a − → D B a − → D C A L L P 1 a − → D 1 P 2 a − → D 2 P 1 + P 2 a − → D 1 + D 2 S U M P 1 a − → D 1 P 2 a − → D 2 a ∈ L P 1 k L P 2 a − → min {⊕ D 1 , ⊕ D 2 } ⊕ D 1 ·⊕ D 2 · ( D 1 k L D 2 ) C O O P P 1 a − → D 1 P 2 a − → D 2 a / ∈ L P 1 k L P 2 a − → ( D 1 k L P 2 ) + ( P 1 k L D 2 ) I N T T able 3: U LT R A S -based operational semantic rules for PEP A rules S U M and I N T because tra nsition rate s simpl y sum up du e to the r ace po licy , and (iii) the multip lica- ti ve f actor in rule C O O P is spec ific to the PEP A c oopera tion discipline based on the slo w est compon ent. 4 Conclusions and Futur e W ork After recall ing the U L T R A S model from [4, 5], in this pape r we ha ve extend ed the scope of the work done in [9, 8, 10] by exhi biting the U L T R A S-based operational semantic rules for CSP and two of its probabili stic and stochastical ly timed va riants. These three exp eriments seem to indica te that the U L T R A S model naturally lends itself to be used as a compact and uniform semantic frame work for dif feren t classes of process calculi. W ith respect to future work, we plan to continue our expe riments by using the U LT R A S model for descri bing the operatio nal semantics of othe r process desc ription languag es of nondete rministic , prob- abilist ic, or stochastic nature, as well as proc ess calculi combini ng nondeterminis m and probabilit y or stocha sticity . This should help to assess the relativ e expressi v enes s of their operators and establ ish gen- eral pr opertie s for the vari ous langu ages. Moreov er , the unif orm characte rizatio n of th e equi v ale nces might help in ev alua ting and discerning among the m any relatio ns proposed in the literature. It would be, indeed, interesting to determine which of the existin g relations can be obtaine d as instan ces of the genera l framewo rk. This stud y may also lead to the definition of a uniform proces s calculus with an U L T R A S -based operat ional semanti cs and th e de vel opment of uniform axiomatizati ons of bisimulat ion, trace, and testing equi v ale nces. From this calcu lus, it shoul d be possible to retriev e the origina lly proposed calculi by v arying the targ et domain and the beha vioral operators . W e shall also consider further options related to quanti tati v e aspects like including quantities within actions ( inte grated quantity appr oach ) or attaching them to traditi onal operators or providi ng specific operator s for them ( orthog onal quantity appr oa ch ). Finally , it would be interesting to see whether is is possible to bui ld generic tools for supp orting ver ifications th at are based on the unif orm mo del and ha ve only to be instantia ted to deal with the spec ific calcul i. Acknowledgment : This work has been partial ly supported by the EU project ASCE NS 257 414. M. Bernardo, R. De Nicola & M. Loreti 75 Refer ences [1] A. Aldini, M. Bernardo & F . Corradin i (201 0): A Pr ocess Algebraic Appr oach to Software Ar chitectur e Design . Spring er , do i: 10.1007/978- 1- 8 4800- 223- 4 . [2] J.C.M. Baeten, J.A. Bergstra & S.A. Smo lka (19 95): Axio matizing Pr obabilistic Pr ocesses: ACP with Gen- erative Pr ob abilities . Inform ation an d Computation 121, pp. 234– 255, doi: 10.1006/ inco.1995.1135 . [3] C. Baier , J.-P . Katoen, H. Hermanns & V . W olf (2 005): Compa rative Br anching-T ime Seman tics for Markov Chains . Inform ation an d Computation 200, pp. 149– 214, doi: 10.1016/j.ic.2005.03. 001 . [4] M. Berna rdo, R. De Nicola & M. Loreti (2 010): Un iform Lab eled T r ansition S ystems for Nondeterministic, Pr obabilistic, and S tochastic Pr ocesses . In: Proc. of th e 5th I nt. Symp . o n T rustworthy Glob al Compu ting (TGC 2010) , LNCS 6084, Springer, pp. 35– 56, doi: 10.1007/978- 3 - 642- 15640- 3 . [5] M. Bernard o, R. De Nicola & M. Loreti (201 1): A Uniform F r amework for Pr ocess Mod els and Beh av- ioral Eq uivalences o f Nondeterministic, P r obabilistic, Stochastic, or Mixed Natur e . Submitted for journa l publication . [6] S.D. Brookes, C.A.R. Hoare & A.W . Roscoe (1 984): A The ory of Communica ting Sequentia l Pr ocesses . Journal of the A CM 31, pp. 560–5 99, d oi: 10.1145/828.833 . [7] R. De Nicola & M. Hennessy (1 984): T esting Equivalen ces fo r Pr ocesses . Theoretical C ompute r Science 34, pp. 83–1 33, d oi: 10.1016/0304- 3975( 84)90 113- 0 . [8] R. De Nico la, D. Latella, M. Lo reti & M. Massink (2009 ): On a Uniform F ramework for the Defi nition of Stochastic Pr ocess Languages . In: Proc. of the 14th Int. W orksho p on Formal Metho ds for Industrial Critical Systems (FMICS 2009) , LNCS 5825, Springe r , pp . 9–25, doi: 10.1007/978- 3- 6 42- 04570- 7_ 2 . [9] R. De Nicola, D. Latella, M. Loreti & M. Massink (2009) : Rate-Based T r ansition Systems for Stochastic Pr ocess Calculi . In: Proc. o f the 36th In t. Coll. on Au tomata, L anguag es and Prog ramming (ICALP 2 009) , LNCS 5556, Springe r , pp . 435–446, doi: 10.1007/978- 3 - 642- 02930- 1_ 36 . [10] R. De Nicola, D. Latella, M. Loreti & M. Massink (2011): State to Functio n Labelled T ransi- tion Sy stems: A Un iform F r amework fo r Defining S tochastic P r ocess Calculi . T echn ical Report, CNR-ISTI. A v ailable at ht tp://puma.isti.cnr.it/download . php?DocFile=2 011- TR- 012_ 0.pdf& idcode =2011 - TR- 0 12& author ity=c nr. isti&collectio n=cnr . isti . [11] R.J. v an Glabbeek (2001): The Linear T ime – Branching T ime Spectru m I . In: Handbo ok of Process Algebra , Elsevier , pp. 3– 99, doi: 10.1007/BFb00390 66 . [12] M. Hen nessy & R. Milner (1 985): A lgebraic Laws for No ndeterminism an d Concurr ency . Journal of the A CM 32, pp. 137–1 62, d oi: 10.1145/2455.2460 . [13] H. Her manns (200 2): Interactive Ma rkov Chains . Springer, doi: 10 .1007/3- 540 - 45804- 2 . V olu me 2 428 of LNCS. [14] J. Hillston ( 1996) : A Compositional App r oach to P erforman ce Modelling . Cambridge Uni versity Press, doi: 10.1017/CBO97 805115 69951 . [15] C.-C. Jou & S.A. Smolka (19 90): E quivalence s, Congru ences, and Complete Axiomatizations fo r Pr obabilis- tic Pr o cesses . In: Proc. of the 1st Int. Conf. on Concurrency Theo ry (CONC UR 1990) , LNCS 458, Springer, pp. 367– 383, doi: 10.1007/ BFb003 9071 . [16] R.M. Keller (1976 ): F ormal V erificatio n o f P arallel Pr ogr ams . Communica tions o f the ACM 19, pp. 371 – 384, doi: 10.1145/36024 8. 36025 1 . [17] K. Seidel (1 995): Pr obabilistic Commun icating Pr o cesses . Theoretical C ompute r Science 152, pp. 219 –249, doi: 10.1016/0304- 3975 (94)00 286- 0 . [18] W .J. Stew art (1 994): I ntr oduction to th e Numerical Solution of Markov Chains . Princeton Un i versity Press. A vailable at http://press.princeton . edu/titles/5640.html .