Deriving Syntax and Axioms for Quantitative Regular Behaviours

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1 Deriving Syntx nd Axioms for Quntittive Regulr Behviours Filippo Bonchi 2, Mrcello Bonsngue 1,2, Jn Rutten 2,3, nd Alexndr Silv 2 1 LIACS - Leiden University 2 Centrum voor Wiskunde en Informtic (CWI) 3 Vrije Universiteit Amsterdm (VUA) Abstrct. We present systemtic wy to generte (1) lnguges of (generlised) regulr expressions, nd (2) sound nd complete xiomtiztions thereof, for wide vriety of quntittive systems. Our quntittive systems include weighted versions of utomt nd trnsition systems, in which trnsitions re ssigned vlue in monoid tht represents cost, durtion, probbility, etc. Such systems re represented s colgebrs nd (1) nd (2) bove re derived in modulr fshion from the underlying (functor) type of these colgebrs. In previous work, we pplied similr pproch to clss of systems (without weights) tht generlizes both the results of Kleene (on rtionl lnguges nd DFA s) nd Milner (on regulr behviours nd finite LTS s), nd includes mny other systems such s Mely nd Moore mchines. In the present pper, we extend this frmework to del with quntittive systems. As consequence, our results now include lnguges nd xiomtiztions, both existing nd new ones, for mny different kinds of probbilistic systems. 1 Introduction Kleene s Theorem [22] gives fundmentl correspondence between regulr expressions nd deterministic finite utomt (DFA s): ech regulr expression denotes lnguge tht cn be recognized by DFA nd, conversely, the lnguge ccepted by DFA cn be specified by regulr expression. Lnguges denoted by regulr expressions re clled regulr. Two regulr expressions re (lnguge) equivlent if they denote the sme regulr lnguge. Slom [32] presented sound nd complete xiomtiztion (lter refined by Kozen in [23]) for proving the equivlence of regulr expressions. The bove progrmme ws pplied by Milner in [26] to process behviours nd lbelled trnsition systems (LTS s). Milner introduced set of expressions for finite LTS s nd proved n nlogue of Kleene s Theorem: ech expression denotes the behviour of finite LTS nd, conversely, the behviour of finite LTS cn be specified by n expression. Milner lso provided n xiomtiztion for his expressions, with the property tht two expressions re provbly equivlent if nd only if they re bisimilr. Colgebrs provide generl frmework for the study of dynmicl systems such s DFA s nd LTS s. For functor G: Set Set, G-colgebr or G-system is pir This work ws crried out during the first uthor s tenure of n ERCIM Alin Bensoussn Fellowship Progrmme. The fourth uthor is prtilly supported by the Fundção pr Ciênci e Tecnologi, Portugl, under grnt number SFRH/BD/27482/2006. M. Brvetti nd G. Zvttro (Eds.): CONCUR 2009, LNCS 5710, pp , c Springer-Verlg Berlin Heidelberg 2009

2 Deriving Syntx nd Axioms for Quntittive Regulr Behviours 147 (S, g), consisting of set S of sttes nd function g:s GS defining the trnsitions of the sttes. We cll the functor G the type of the system. For instnce, DFA s cn be redily seen to correspond to colgebrs of the functor G(S) =2 S A nd imge-finite LTS s re obtined by G(S) =P ω(s) A,whereP ω is finite powerset. Under mild conditions, functors G hve finl colgebr (unique up to isomorphism) into which every G-colgebrcn be mppedvi uniqueso-clled G-homomor-phism. The finl colgebr cn be viewed s the universe of ll possible G-behviours: the unique homomorphism into the finl colgebr mps every stte of colgebr to cnonicl representtive of its behviour. This provides generl notion of behviourl equivlence: two sttes re equivlent iff they re mpped to the sme element of the finl colgebr. In the cse of DFA s, two sttes re equivlent when they ccept the sme lnguge; for LTS s, behviourl equivlence coincides with bisimilrity. For colgebrs of lrge but restricted clss of functors, we introduced in [7] lnguge of regulr expressions; corresponding generlistion of Kleene s Theorem; nd sound nd complete xiomtiztion for the ssocited notion of behviourl equivlence. We derived both the lnguge of expressions nd their xiomtiztion, in modulr fshion, from the functor defining the type of the system. In recent yers, much ttention hs been devoted to the nlysis of probbilistic behviours, which occur for instnce in rndomized, fult-tolernt systems. Severl different types of systems were proposed: rective [24, 29], genertive [16], strtified [36, 38], lternting [18, 39], (simple) Segl [34, 35], bundle [12] nd Pnueli-Zuck [28], mong others. For some of these systems, expressions were defined for the specifiction of their behviours, s well s xioms to reson bout their behviourl equivlence. Exmples include[1,2,4,13,14,21,25,27,37]. Our previous results [7] pply to the clss of so-clled Kripke-polynomil functors, which is generl enough to include the exmples of DFA s nd LTS s, s well s mny other systems such s Mely nd Moore mchines. However, probbilistic systems, weighted utomt [15, 33], etc. cnnot be described by Kripke-polynomil functors. It is the im of the present pper to identify clss of functors () tht is generl enough to include these nd more generlly lrge clss of quntittive systems; nd (b) to which the methodology developed in [7] cn be extended. To this end, we give non-trivil extension of the clss of Kripke-polynomil functors by dding functor type tht llows the trnsitions of our systems to tke vlues in monoid structure of quntittive vlues. This new clss, which we shll cll quntittive functors, now includes ll the types of probbilistic systems mentioned bove. We show how to extend our erlier pproch to the new setting. As it turns out, the min technicl chllenge is due to the fct tht the behviour of quntittive systems is inherently non-idempotent. As n exmple consider the expression 1/2 ε 1/2 ε representing probbilistic system tht either behves s ε with probbility 1/2 or behves s ε with the sme probbility. When ε is equivlent to ε, then the system is equivlent to 1 ε rther thn 1/2 ε. This is problemtic becuse idempotency plyed crucil role in our previous results to ensure tht expressions denote finite-stte behviours. We will show how the lck of idempotency in the extended clss of functors cn be circumvented by clever use of the monoid structure. This will llow us to derive for ech functor in our new extended clss everything we were fter: lnguge of regulr expressions;

3 148 F. Bonchi et l. Tble 1. All the expressions re closed nd gurded. The congruence nd the α-equivlence xioms re implicitly ssumed for ll the systems. The symbols 0 nd + denote, in the cse of weighted utomt, the empty element nd the binry opertor of the commuttive monoid S while, for the other systems, denote the ordinry 0 nd sum of rel numbers. With slight buse of nottion, we write i 1 n pi εi for p1 ε1 pn εn. Weighted utomt S (S Id ) A ε:: = ε ε μx.ε x s (s ε) where s S nd A (ε 1 ε 2) ε 3 ε 1 (ε 2 ε 3) ε 1 ε 2 ε 2 ε 1 ε ε (s ε) (s ε) ((s + s ) ε) s s s + s (0 ε) ε[μx.ε/x] μx.ε γ[ε/x] ε μx.γ ε 0 Strtified systems D ω(id)+(b Id)+1 ε:: = μx.ε x b,ε i 1 n pi εi where b B, pi (0, 1] nd i 1...n pi =1 (ε 1 ε 2) ε 3 ε 1 (ε 2 ε 3) ε 1 ε 2 ε 2 ε 1 (p 1 ε) (p 2 ε) (p 1 + p 2) ε ε[μx.ε/x] μx.ε γ[ε/x] ε μx.γ ε Segl systems P ω(d ω(id)) A ε:: = ε ε μx.ε x ({ε }) where A, p i (0, 1] nd i 1...n ε :: = i 1 n p i ε i pi =1 (ε 1 ε 2) ε 3 ε 1 (ε 2 ε 3) ε 1 ε 2 ε 2 ε 1 ε ε ε ε ε (ε 1 ε 2) ε 3 ε 1 (ε 2 ε 3) ε 1 ε 2 ε 2 ε 1 (p 1 ε) (p 2 ε) (p 1 + p 2) ε ε[μx.ε/x] μx.ε γ[ε/x] ε μx.γ ε Pnueli-Zuck systems P ωd ωp ω(id) A ε:: = ε ε μx.ε x {ε } where A, p i (0, 1] nd i 1...n ε :: = i 1 n p i ε i ε :: = ε ε ({ε}) pi =1 (ε 1 ε 2) ε 3 ε 1 (ε 2 ε 3) ε 1 ε 2 ε 2 ε 1 ε ε ε ε ε (ε 1 ε 2) ε 3 ε 1 (ε 2 ε 3) ε 1 ε 2 ε 2 ε 1 (p 1 ε ) (p 2 ε ) (p 1 + p 2) ε (ε 1 ε 2 ) ε 3 ε 1 (ε 2 ε 3 ) ε 1 ε 2 ε 2 ε 1 ε ε ε ε ε ε[μx.ε/x] μx.ε γ[ε/x] ε μx.γ ε corresponding Kleene Theorem; nd sound nd complete xiomtiztion for the corresponding notion of behviourl equivlence. In order to show the effectiveness nd the generlity of our pproch, we pply it to four types of systems: weighted utomt; nd simple Segl, strtified nd Pnueli- Zuck systems. For simple Segl systems, we recover the lnguge nd xiomtiztion presented in [14]. For weighted utomt nd strtified systems, lnguges hve been defined in [9] nd [38] but, to the best of our knowledge, no xiomtiztion ws ever given. Applying our method, we obtin the sme lnguges nd, more interestingly, we obtin novel xiomtiztions. We lso present completely new frmework to reson bout Pnueli-Zuck systems. Tble 1 summrizes our results.

4 2 Bckground Deriving Syntx nd Axioms for Quntittive Regulr Behviours 149 In this section, we present the bsic definitions for polynomil functors nd colgebrs. We recll, from [7], the lnguge of expressions Exp G ssocited with functor G, the nlogue of Kleene s theorem nd sound nd complete xiomtiztion of Exp G. Let Set be the ctegory of sets nd functions. Sets re denoted by cpitl letters X, Y,... nd functions by lower cse f, g,...the collection of functions from set X to set Y is denoted by Y X. We write g f for function composition, when defined. The product of π 1 π 2 two sets X, Y is written s X Y, with projection functions X X Y Y. The set 1 is singleton set written s 1={ }.WedefineX +Y s the set X Y {, }, where is the disjoint union of sets, with injections X κ 1 κ 2 X Y Y.Note tht the set X + Y is different from the clssicl coproduct of X nd Y, becuse of the two extr elements nd. These extr elements re used to represent, respectively, underspecifiction nd inconsistency in the specifiction of systems. Polynomil functors. In our definition of polynomil functors we will use constnt sets equipped with n informtion order. In prticulr, we will use join-semilttices. A (bounded) join-semilttice is set B endowed with binry opertion B nd constnt B B. The opertion B is commuttive, ssocitive nd idempotent. The element B is neutrl w.r.t. B.EverysetS cn be trnsformed into join-semilttice by tking B to be the set of ll finite subsets of S with union s join. We re now redy to define the clss of polynomil functors. They re functors G : Set Set, built inductively from the identity nd constnts, using, + nd ( ) A. Formlly, the clss PF of polynomil functors on Set is inductively defined by putting: PF G:: = Id B G 1 + G 2 G 1 G 2 G A with B finite join-semilttice nd A finite set. For set S, Id(S) =S, B(S) =B, (G 1 G 2)(S) =G 1(S) G 2(S), (G 1 + G 2)(S) =G 1(S) +G 2(S) nd G A (S) ={f f : A G(S)} nd, for function f : S T, Gf : GS GT is defined s usul [31]. Typicl exmples of polynomil functors re D =2 Id A, M = (B Id) A nd St = A Id. These functors represent, respectively, the type of deterministic utomt, Mely mchines,ndinfinite strems. Our definition of polynomil functors slightly differs from the one of [19, 30] in the use of join-semilttice s constnt functor nd in the definition of +. Thissmll vrition plys n importnt technicl role in giving full colgebric tretment of the lnguge of expressions which we shll introduce lter. The intuition behind these extensions becomes cler if one reclls tht the set of clssicl regulr expressions crries join-semilttice structure. Since ordinry polynomil functors cn be nturlly embedded into our polynomil functors bove (becuse every set cn be nturlly embedded in the generted free join semilttice), one cn use the results of Section 5 to obtin regulr expressions (nd xiomtiztion) for ordinry polynomil functors. Next, we give the definition of the ingredient reltion, which reltes polynomil functor G with its ingredients, i.e. the functors used in its inductive construction. We shll use this reltion lter for typing our expressions. Let PF PF be the lest reflexive nd trnsitive reltion, written infix, such tht G 1 G 1 G 2, G 2 G 1 G 2, G 1 G 1 + G 2, G 2 G 1 + G 2, G G A.

5 150 F. Bonchi et l. If F G, thenf is sid to be n ingredient of G. For exmple, 2, Id, 2 Id,nd2 Id A re the ingredients of the deterministic utomt functor D. Colgebrs. For n endofunctor G on Set, G-colgebr is pir (S, f ) consisting of set of sttes S together with function f : S GS. The functor G, together with the function f, determines the trnsition structure of the G-colgebr [31]. Exmples of colgebrs include deterministic utomt,mely mchines nd infinite strems, which re,respectively,colgebrsfor the functors D, M nd St given bove. A G-homomorphism from G-colgebr (S, f ) to G-colgebr (T, g) is function h : S T preserving the trnsition structure, i.e., such tht g h = Gh f. A G-colgebr (Ω,ω) is sid to be finl if for ny G-colgebr (S, f ) there exists unique G-homomorphism beh S : S Ω. For every polynomil functor G there exists finlg-colgebr (Ω G,ω G) [31]. The notion of finlity plys key role in defining bisimilrity. For G-colgebrs (S, f ) nd (T, g) nd s S, t T, we sy tht s nd t re (G-)bisimilr, written s t, if nd only if beh S(s) =beh T(t). Given G-colgebr (S, f ) nd subset V of S with inclusion mp i : V S we sy tht V is subcolgebr of S if there exists g : V GV such tht i is homomorphism. Given s S, s S denotes the subcolgebr generted by s [31], i.e. the set consisting of sttes tht re rechble from s. We will write Colg lf (G) for the ctegory of G-colgebrs tht re loclly finite: objects re G-colgebrs (S, f ) such tht for ech stte s S the generted subcolgebr s is finite; mps re the usul homomorphisms of colgebrs. 2.1 A Lnguge of Expressions for Polynomil Colgebrs In order to be ble to formulte the generliztion of our previous work [7], we first hve to recll the min definitions nd results concerning the lnguge of expressions ssocited to polynomil functor G. Note tht in [7] we ctully treted Kripke polynomil functors, s mentioned lso in the present introduction. In order to give more uniform nd concise presenttion, we omit in this section the cse of the finite powerset P ω (thus, we only present polynomil functors), which cn be recovered s specil instnce of the monoidl vlution functor (Section 3). We strt by introducing n untyped lnguge of expressions nd then we single out the well-typed ones vi n pproprite typing system, thereby ssociting expressions to polynomil functors. Then, we present the nlogue of Kleene s theorem. Let A be finite set, B finite join-semilttice nd X set of fixpointvribles.the set of ll expressions is given by the following grmmr (where A, b B): ε :: = ε ε x μx.ε b l(ε) r(ε) l[ε] r[ε] (ε) An expression is closed if it hs no free occurrences of fixpoint vribles x. We denote the set of closed expressions by Exp. Intuitively, expressions denote elements of finl colgebrs. The expressions, ε ε nd μx.εwill ply role similr to, respectively, the empty lnguge, the union of lnguges nd the Kleene str in clssicl regulr expressions for deterministic utomt. The expressions l(ε), r(ε), l[ε], r[ε] nd (ε) denote the left nd right hnd-side of products nd sums nd function ppliction, respectively.

6 Deriving Syntx nd Axioms for Quntittive Regulr Behviours 151 Next, we present typing ssignment system tht will llow us to ssocite with ech functor G the expressions ε tht re vlid specifictions of G-colgebrs. The typing proceeds following the structure of the expressions nd the ingredients of the functors. We type expressions ε using the ingredient reltion, for A, b B nd x X,s follows: ε : G G : F G b : B G x : Id G μx.ε : G G ε 1 : F G ε 2 : F G ε 1 ε 2 : F G ε : G G ε : Id G ε : F G (ε) :F A G ε : F 1 G ε : F 2 G ε : F 1 G ε : F 2 G l(ε) :F 1 F 2 G r(ε) :F 1 F 2 G l[ε] :F 1 + F 2 G r[ε] :F 1 + F 2 G Most of the rules re self-explntory. The rule involving Id G reflects the isomorphism of the finl colgebr: Ω G = G(ΩG). It is interesting to note tht the rule for the vrible x gurntees tht occurrences of vribles in fixpoint expression re gurded: they occur under the scope of expressions l(ε), r(ε), l[ε], r[ε] nd (ε). For further detils we refer to [7]. The set of G-expressions of well-typed expressions ssocited with polynomil functor G is defined by Exp G = Exp G G,where,forF n ingredient of G: Exp F G = {ε Exp ε : F G}. To illustrte this definition we instntite it for the functor D =2 Id A. Exmple 1 (Deterministic expressions). Let A be finite set nd let X be set of fixpoint vribles. The set Exp D of well-typed D-expressions is given by the BNF: ε:: = x l(0) l(1) r((ε)) ε ε μx.ε where A, x X, ε is closed nd occurrences of fixpoint vribles re within the scope of n input ction, s cn be esily checked by structurl induction on the length of the type derivtions. Our derived syntx for this functor differs from clssicl regulr expressions in the use of ction prefixing nd fixpoint insted of sequentil composition nd str, respectively. However, s we will soon see (Theorem 1), the expressions in our syntx correspond to deterministic utomt nd, in tht sense, they re equivlent to clssicl regulr expressions. The lnguge of expressions induces n lgebric description of systems. In [7], we showed tht such lnguge is colgebr. More precisely, we defined function λ F G : Exp F G F (Exp G) nd then set λ G = λ G G, providing Exp G with colgebric structure. The function λ F G is defined by double induction on the mximum number of nested ungurded occurrences of μ-expressions in ε nd on the length of the proofs for typing expressions. For every ingredient F of polynomil functor G nd ε Exp F G,

7 152 F. Bonchi et l. Tble 2. The function Plus F G Empty F G F (Exp G) Empty Id G Empty B G Empty F1 +F 2 G Empty F1 F 2 G Empty F A G : F (Exp G) F (Exp G) F (Exp G) nd the constnt = = B = = Empty F1 G, Empty F2 G = λ.empty F G Plus Id G (ε 1,ε 2 ) = ε 1 ε 2 Plus B G (b 1, b 2 ) = b 1 B b 2 Plus F1 +F 2 G(x, ) = Plus F1 +F 2 G(, x) = Plus F1 +F 2 G(x, ) = Plus F1 +F 2 G(, x) =x Plus F1 +F 2 G(κ i(ε 1),κ i(ε 2)) = κ i(plus Fi G(ε 1,ε 2)), i {1, 2} Plus F1 +F 2 G(κ i (ε 1 ),κ j (ε 2 )) = for i, j {1, 2} nd i j Plus F1 F 2 G( ε 1,ε 2, ε 3,ε 4 ) = Plus F1 G(ε 1,ε 3), Plus F2 G(ε 2,ε 4) Plus F A G(f, g) = λ. Plus F G (f (), g()) the mpping λ F G(ε) is given by : λ F G( ) = Empty F G λ F G(ε 1 ε 2) = Plus F G(λ F G(ε 1),λ F G(ε 2)) λ G G(μx.ε) = λ G G(ε[μx.ε/x]) λ Id G (ε) = ε for G Id λ B G(b) = b λ F1 F 2 G(l(ε)) = λ F1 G(ε), Empty F2 G λ F1 F 2 G(r(ε)) = Empty F1 G,λ F2 G(ε) λ F1 +F 2 G(l[ε]) = κ 1(λ F1 G(ε)) λ F1 +F 2 G(r[ε]) = κ 2(λ { F2 G(ε)) λf G(ε) λ F A G((ε)) = λ =. Empty F G otherwise Here, ε[μx.ε/x] denotes syntctic substitution, replcing every free occurrence of x in ε by μx.ε. The uxiliry constructs Empty nd Plus re defined in Tble 2. Note tht we use λ in the right hnd side of the eqution for λ F A G ((ε)) to denote lmbd bstrction. This overlp of symbols is sfe since when we use it in λ F G it is lwys ccompnied by the type subscript. It is interesting to remrk tht λ G is the generliztion of the well-known notion of Brzozowski derivtive [8] for regulr expressions nd, moreover, it provides n opertionl semntics for expressions. We now present the generliztion of Kleene s theorem. Theorem 1 ([7, Theorem 4]). Let G be polynomil functor. 1. For every loclly finite G-colgebr (S, g) nd for ny s S there exists n expression ε s Exp G such tht ε s s. 2. For every ε Exp G, we cn construct colgebr (S, g) such tht S is finite nd there exists s S with ε s. Note tht ε s s mens tht the expression ε s nd the (system with initil) stte s hve the sme behviour. For instnce, for DFA s, this would men tht they denote nd ccept the sme regulr lnguge. Similrly for ε nd s in item 2.. In [7], we presented sound nd complete xiomtiztion wrt bisimilrity for Exp G. We will not recll it here becuse this xiomtiztion cn be recovered s n instnce of the one presented in Section 4.

8 Deriving Syntx nd Axioms for Quntittive Regulr Behviours Monoidl Vlution Functor In the previous section we introduced polynomil functors nd lnguge of expressions for specifying colgebrs. Colgebrs for polynomil functors cover mny interesting types of systems, such s deterministic nd Mely utomt, but not quntittive systems. For this reson, we recll the definition of the monoidl vlution functor [17], which will llow us to define colgebrs representing quntittive systems. In the next section, we will provide expressions nd n xiomtiztion for these. A monoid M is n lgebric structure consisting of set with n ssocitive binry opertion + nd neutrl element 0 for tht opertion. A commuttive monoid is monoid where + is lso commuttive. Exmples of commuttive monoids include 2, the two-element {0, 1} boolen lgebr with logicl or, nd the set R of rel numbers with ddition. A property tht will ply crucil role in the rest of the pper is idempotency: monoid is idempotent, if the ssocited binry opertion + is idempotent. For exmple, the monoid 2 is idempotent, while R is not. Notice tht n idempotent commuttive monoid is join-semilttice. Given function ϕ from set S to monoid M, wedefinesupport of ϕ s the set {s S ϕ(s) 0}. Definition 1 (Monoidl vlution Functor). Let M be commuttive monoid. The monoidl vlution functor M ω :Set Set is defined s follows. For ech set S, M S ω is the set of functions from S to M with finite support. For ech function h : S T, M h ω:m S ω M T ω is the function mpping ech ϕ M S ω into ϕ h M T ω defined, for every t T,s ϕ h (t) = ϕ(s ) s h 1 (t) Proposition 1. The functor M ω hs finl colgebr. Note tht the (finite) powerset functor P ω( ) coincides with 2 ω.thisisoftenused to represent non-deterministic systems. For exmple, (imge-finite) LTS s (with lbels over A) re colgebrs for the functor P ω( ) A. In the following, to simplify the nottion we will lwys write M insted of M ω. By combining the monoidl vlution functor with the polynomil functors, we cn model quntittive systems s colgebrs. As n exmple, we mention weighted utomt. Weighted Automt. A semiring S is tuple S, +,, 0, 1 where S, +, 0 is commuttive monoid nd S,, 1 is monoid stisfying certin distributive lws. Weighted utomt [15, 33] re trnsition systems lbelled over set A nd with weights in semiring S. Moreover, ech stte is equipped with n output vlue 1 in S. From colgebric perspective weighted utomt re colgebrs for the functor S (S Id ) A, where we use S to denote, the commuttive monoid of the semiring S.More concretely, colgebr for S (S Id ) A is pir (Q, o, t ), whereq is set of sttes, 1 In the originl formultion lso n input vlue is considered. To simplify the presenttion nd following [10] we omit it.

9 154 F. Bonchi et l. o : Q S is the function tht ssocites n output weight to ech stte q Q nd t : Q (S Q ) A is the trnsition reltion tht ssocites weight to ech trnsition: q,s q t(q)()(q )=s. Bisimilrity for weighted utomt hs been studied in [9] nd it coincides with the colgebric notion of bisimilrity. (For proof, see [6].) Proposition 2. Bisimilrity for S (S Id ) A coincides with the weighted utomt bisimilrity defined in [9]. 4 A Non-idempotent Algebr for Quntittive Regulr Behviours In this section, we will extend the frmework presented in Section 2 in order to del with quntittive systems, s described in the previous section. We will strt by defining n pproprite clss of functors H, proceed with presenting the lnguge Exp H of expressions ssocited with H together with Kleene like theorem nd finlly we introduce sound nd complete xiomtiztion of Exp H. Formlly, the set QF of quntittive functors on Set is defined inductively by putting: QF H :: = G M H (M H ) A M H 1 1 M H 2 2 M H M H 2 2 where G is polynomil functor, M is commuttive monoid nd A is finite set. Note tht we do not llow mixed functors, such s G + M H or G M H. The reson for this restriction will become cler lter in this section when we discuss the proof of Kleene s theorem. In Section 5, we will show how to del with such mixed functors. Every definition we presented in Section 2 needs now to be extended to quntittive functors. We strt by observing tht tking the current definitions nd replcing the subscript F G with F H does most of the work. In fct, hving tht, we just need to extend ll the definitions for the cse M F H. We strt by introducing new expression m ε, with m M, extending the set of untyped expressions, which is now given by: ε :: = ε ε x μx.ε b l(ε) r(ε) l[ε] r[ε] (ε) m ε The intuition behind the new expression is tht there is trnsition between the current stte nd the stte specified by ε with weight m. The ingredient reltion is extended with the rule H M H, the type system nd λ M F H with the following rules: ε : F H m ε : M F H Empty M F H = λε.0 Plus M F H (f, g) =λε {.f (ε )+g(ε ) m if λ M F H (m ε) =λε λf H (ε) =ε. 0 otherwise where 0 nd + re the neutrl element nd the binry opertion of M. Recll tht the function λ H = λ H H provides n opertionl semntics for the expressions. We will soon illustrte this for the cse of expressions for weighted utomt (Exmple 2). Finlly, we introduce n equtionl system for expressions of type F H. We will use the symbol Exp F H Exp F H, omitting the subscript F H, for the lest reltion stisfying the following:

10 Deriving Syntx nd Axioms for Quntittive Regulr Behviours 155 (Idempotency) ε ε ε, ε Exp F G (Commuttivity) ε 1 ε 2 ε 2 ε 1 (Associtivity) ε 1 (ε 2 ε 3) (ε 1 ε 2) ε 3 (Empty) ε ε (FP) γ[μx.γ/x] μx.γ (Unique) γ[ε/x] ε μx.γ ε (B ) B (B ) b 1 b 2 b 1 B b 2 ( L) l( ) ( L) l(ε 1 ε 2) l(ε 1) l(ε 2) ( R) r( ) ( R) r(ε 1 ε 2) r(ε 1) r(ε 2) ( A ) ( ) ( A ) (ε 1 ε 2) (ε 1) (ε 2) (M ) (0 ε) (M ) (m ε) (m ε) (m + m ) ε (+ L) l[ε 1 ε 2] l[ε 1] l[ε 2] (+ R) r[ε 1 ε 2] r[ε 1] r[ε 2] (α equiv ) μx.γ μy.γ[y/x] (+ ) l[ε 1] r[ε 2] l[ ] r[ ] if y not free in γ (Cong) If ε ε then ε ε 1 ε ε 1, μx.ε μx.ε, l(ε) l(ε ), r(ε) r(ε ), l[ε] l[ε ], r[ε] r[ε ], (ε) (ε ),ndm ε m ε. We shll write Exp/ for the set of expressions modulo. Note tht (Idempotency) only holds for ε Exp F G. The reson why it cnnot hold for the remining functors comes from the fct tht monoid is, in generl, not idempotent. Thus, (Idempotency) would conflict with the xiom (M ), which llows us to derive, for instnce, (2 ) (2 ) 4. In the cse of n idempotent commuttive monoid M, (Idempotency) follows from the xiom (M ). Lemm 1. Let M be n idempotent commuttive monoid. For every expression ε Exp M F H, one hs ε ε ε. Exmple 2 (Expressions for weighted utomt). The syntx utomticlly derived from our typing system for the functor W = S (S Id ) A is the following. ε :: = ε ε x μx.ε l(s) r(ε ) ε :: = ε ε (ε ) ε :: = ε ε s ε where s S, A nd ll the occurrences of x re gurded. The semntics of these expressions is given by the function λ W W (herefter denoted by λ W ) which is n instnce of the generl λ F H defined bove. It is given by: λ W ( ) = 0,λ.λε.0 λ W (ε 1 ε 2)= s 1 + s 2,λ.λε.(f ()(ε)+g()(ε)) where s 1, f = λ W (ε 1) nd s 2, g = λ W (ε 2) λ W (μx.ε) = λ W (ε[μx.ε/x]) λ W (l(s)) = s,λ.λε.0 λ W (r(ε )) = 0,λ (S Id ) A W (ε ) λ (S Id ) A W ( ) = λ.λε.0 λ (S Id ) A W (ε 1 ε 2)=λ.λε.(f 1()(ε)+f 2(s)(ε)) where f i = λ (S Id ) A W { (ε i), i {1, 2} λ (S Id ) A W ((ε )) = λ λs. Id W (ε ) = λε.0 oth. λ (S Id ) W ( ) = λε.0 λ (S Id ) W (ε 1 ε 2)=λε.(f 1(ε)+f 2(ε)) where f i = λ (S Id ) W (ε { i), i {1, 2} s ε = ε λ (S Id ) W (s ε) = λε. 0 oth.

11 156 F. Bonchi et l. The function λ W ssigns to ech expression ε pir s, t, consisting of n output weight s S nd function t : A S Exp W. For concrete exmple, let S = R nd consider ε = μx.r((2 x 3 )) l(1) l(2). The semntics of this expression, obtined by λ W is described by the weighted utomton below.,2,3 ε 3 0 In Tble 1 more concise syntx for expressions for weighted utomt is presented. To derive tht syntx from the one utomticlly generted, we first write ε s ε :: = ε ε (s ε) using the xioms ( ) nd (ε 1 ε 2 ) (ε 1 ) (ε 2 ). Similrly, using r( ) = nd r(ε 1 ε 2) r(ε 1) r(ε 2), we cn write ε s follows. ε :: = ε ε x μx.ε l(s) r((s ε)) In Tble 1, we bbrevite l(s) to s nd r((s ε)) to (s ε), without ny risk of confusion. Note tht the xioms presented in Tble 1 lso reflect the chnges in the syntx of the expressions. We re now redy to formulte the nlogue of Kleene s theorem for quntittive systems. Theorem 2 (Kleene s theorem for quntittive functors). Let H be quntittive functor. 1. For every loclly finite H -colgebr (S, h) nd for every s S there exists n expression ε s Exp H such tht s ε s. 2. For every ε Exp H, there exists finite H -colgebr (S, h) with s S such tht s ε. The proof of the theorem cn be found in [6], but let us explin wht re the technicl difficulties tht rise when compred with Theorem 1, where only polynomil functors re considered. In the proof of item 2. in Theorem 1, we strt by constructing the subcolgebr generted by ε, using the fct tht the set Exp G hs colgebr structure given by λ G. Then, we observe tht such subcolgebr might not be finite nd, following similr result for clssicl regulr expressions, we show tht finiteness cn be obtined by tking the subcolgebr generted modulo (Associtivity), (Commuttivity) nd (Idempotency) (ACI). Consider for instnce the expression ε = μx.r((x x)) of type D =2 Id A.The subcolgebrs generted with nd without pplying ACI re the following: ε ε ε ε (ε ε) (ε ε)...

12 Deriving Syntx nd Axioms for Quntittive Regulr Behviours 157 We cnnot pply ACI in the quntittive setting, since the idempotency xiom does not hold nymore. However, surprisingly enough, in the cse of the functor M H,we re ble to prove finiteness of the subcolgebr ε just by using (Commuttivity) nd (Associtivity). The key observtion is tht the monoid structure will be ble to void the infinite scenrio described bove. In fct, for the functor M H one cn prove tht, if ε ε is one of the successors of ε then the successors of ε ε will ll be contined in the set of direct successors of ε, which we know is finite. Wht hppens is concisely cptured by the following exmple. Tke the expression ε = μx.2 (x x) for the functor R Id. Then, the subcolgebr generted by ε is depicted in the following picture: ε 2 ε ε 4 In this mnner, we re ble to del with the bse cses G (polynomil functor) nd M H of the inductive definition of the set of quntittive functors. Moreover, the functors M H M H nd M H +M H inherit the bove property from M H nd do not pose ny problem in the proof of Kleene s theorem. The syntctic restriction tht excludes mixed functors is needed becuse of the following problem. Tke s n exmple the functor M Id Id A. A well-typed expression for this functor would be ε = μx.r((x x l(2 x) l(2 x))). It is cler tht we cnnot pply idempotency in the subexpression x x l(2 x) l(2 x) nd hence the subcolgebr generted by ε will be infinite: ε ε ε ε (ε ε ) (ε ε ) with ε = ε ε l(2 ε) l(2 ε). We will show in the next section how to overcome this problem. Let us summrize wht we hve chieved so fr: we hve presented frmework tht llows, for ech quntittive functor H QF, the derivtion of lnguge Exp H. Moreover, Theorem 2 gurntees tht for ech expression ε Exp H, there exists finite H -colgebr (S, h) tht contins stte s S bisimilr to ε nd, conversely, for ech loclly finite H -colgebr (S, h) nd for every stte in s there is n expression ε s Exp H bisimilr to s. The proof of Theorem 2, which cn be found in [6], shows how to compute the H -colgebr (S, h) corresponding to n expression ε nd vice-vers. The xiomtiztion presented bove is sound nd complete: Theorem 3 (Soundness nd Completeness). Let H be quntittive functor nd let ε 1,ε 2 Exp H. Then, ε 1 ε 2 ε 1 ε 2. The proof of this theorem follows similr strtegy s in [7, 20] nd cn be found in [6]. 5 Extending the Clss of Functors In the previous section, we introduced regulr expressions for the clss of quntittive functors. In this section, by employing stndrd results from the theory of colgebrs,

13 158 F. Bonchi et l. we show how to use such regulr expressions to describe the colgebrs of mny more functors, including the mixed functors we mentioned in Section 4. Given F nd G two endofunctors on Set, nturl trnsformtion α:f G is fmily of functions α S:F (S) G(S) (for ll sets S), such tht for ll functions h:t U, α U F (h) =G(h) α T. If ll the α S re injective, then we sy tht α is injective. Proposition 3. An injective nturl trnsformtion α:f G induces functor α ( ) :Colg lf (F ) Colg lf (G) tht preserves nd reflects bisimilrity. This result (proof cn be found in [6]) llows us to extend both regulr expressions nd xiomtiztion to mny functors. Indeed, consider functor F tht is not quntittive, but tht hs n injective nturl trnsformtion α into some quntittive functor H.A (loclly finite) F-colgebr cn be trnslted into (loclly finite) H -colgebr vi the functor α ( ) nd then it cn be chrcterized by using expressions in Exp H (s we will show soon, for the converse some cre is needed). The xiomtiztion for Exp H is still sound nd complete for F-colgebrs, since the functor α ( ) preserves nd reflects bisimilrity. However, notice tht Kleene s theorem does not hold nymore, becuse not ll the expressions in Exp H denote F-regulr behviours or, more precisely, not ll expressions of Exp H re equivlent to H -colgebrs tht re in the imge of α ( ).Thus,inorderto retrieve Kleene s theorem, one hs just to exclude such expressions. In mny situtions, this is fesible by simply imposing some syntctic constrints on Exp H. As n exmple, we recll the definition of the probbility functor tht, in the next section, will llow us to derive regulr expressions for probbilistic systems. Definition 2 (Probbility functor). A probbility distribution over set S is function d : S [0, 1] such tht d(s) =1. The probbility functor Dω:Set Set is defined s S s follows. For ll sets S, D ω(s) is the set of probbility distributions over S with finite support. For ll functions h : S T, D ω(h) mps ech d D ω(s) into d h s defined in Definition 1. Now recll the functor R Id from Section 3. Note tht for ny set S, D ω(s) R S since probbility distributions re lso functions from S to R.Letι be the fmily of inclusions ι S:D ω(s) R S. It is esy to see tht ι is nturl trnsformtion between D ω nd R Id (the two functors re defined in the sme wy on rrows). Thus, in order to specify D ω- colgebrs, we cn use ε Exp R Id which re the closed nd gurded expressions given by ε :: = ε ε x μx.ε r ε,forr R. However, this lnguge llows us to specify R Id -behviours tht re not D ω-behviours, such s for exmple, μx.2 x nd μx.0 x. In order to obtin lnguge tht specifies ll nd only the regulr D ω-behviours, it is enough to restrict the syntx of Exp R Id, s follows: ε :: = x μx.ε p i ε i for p i (0, 1] such tht p i =1 i 1...n i 1...n where, with slight buse of nottion, pi εi denotes p1 ε1 pn εn. i 1...n In the next section, we will use this kind of syntctic restrictions for defining regulr expressions of probbilistic systems. For nother exmple, consider the functors Id nd P ω(id). Letτ be the fmily of functions τ S : S P ω(s) mpping ech s S in the singleton set {s}. It is esy to see

14 Deriving Syntx nd Axioms for Quntittive Regulr Behviours 159 tht τ is n injective nturl trnsformtion. With the bove observtion, we cn lso get regulr expressions for the functor M Id Id A tht, s discussed in Section 4, does not belong to our clss of quntittive functors. Indeed, by extending τ, we cn construct n injective nturl trnsformtion M Id Id A M Id P ω(id) A. In the sme wy, we cn construct n injective nturl trnsformtion from the functor D ω(id)+(a Id)+1 (tht is the type of strtified systems) into R Id +(A P ω(id))+1. Since the ltter is quntittive functor, we cn use its expressions nd xiomtiztion for strtified systems. But since not ll its expressions define strtified behviours, we gin hve to restrict the syntx. The procedure of ppropritely restricting the syntx usully requires some ingenuity. We shll see tht in mny concrete cses, s for instnce D ω bove, it is firly intuitive which restriction to choose. 6 Probbilistic Systems Mny different types of probbilistic systems hve been defined in literture: rective, genertive, strtified, lternting, (simple) Segl, bundle nd Pnueli-Zuck. Ech type corresponds to functor, nd the systems of certin type re colgebrs for the corresponding functor. A systemtic study of ll these systems s colgebrs ws mde in [5]. In prticulr, Fig.1 of [5] provides full correspondence between types of systems nd functors. By employing this correspondence, we cn use our frmework in order to derive regulr expressions nd xiomtiztions for ll these types of probbilistic systems. In order to show the effectiveness of our pproch, we hve derived expressions nd xioms for three different types of probbilistic systems: simple Segl, strtified nd Pnueli-Zuck. Tble 1 shows the expressions nd the xiomtiztions tht we hve obtined, fter some simplifiction of the cnoniclly derived syntx (which is often verbose nd redundnt). b 1/2 1/2 2/3 1/3 1 1/2 1/2 1/3 2/3 b 1/3 2/3 b b (i) (ii) (iii) Fig. 1. (i) A simple Segl system, (ii) strtified system nd (iii) Pnueli-Zuck system 1 Simple Segl systems. Simple Segl systems re colgebrs of type P ω(d ω(id)) A (recll tht P ω is the functor 2 ). These re like lbelled trnsition systems, but ech lbelled trnsition leds to probbility distribution of sttes insted of single stte. An exmple is shown in Fig.1(i).

15 160 F. Bonchi et l. Tble 1 shows expressions nd xioms for simple Segl systems. In the following we show how to derive these. As described in Section 5, we cn derive the expressions for P ω(r Id ) A insted of P ω(d ω(id)) A, nd then impose some syntctic constrints on Exp Pω (R Id ) A in order to chrcterize ll nd only the P ω(d ω(id)) A behviours. By simply pplying our typing systems to P ω(r Id ) A, we derive the expressions: ε :: = ε ε x μx.ε (ε ) ε :: = ε ε 1 ε 0 ε ε :: = ε ε p ε where A, p R nd 0 nd 1 re the elements of the boolen monoid 2. Now, observe tht the syntx for ε, due to the xiom 0 ε cn be reduced to ε :: = ε ε 1 ε which, becuse of (ε 1) (ε 2) (ε 1 ε 2) nd ( ) is equivlent to the simplified syntx: ε :: = ε ε x μx.ε ({ε }) Here, nd in wht follows, {ε } bbrevites 1 ε. Note tht the xiomtiztion would hve to include the xiom ({ε }) ({ε }) ({ε }), s consequence of (M ) nd ( A ). However, this xiom is subsumed by the (Idempotency) xiom, which we dd to the xiomtiztion, since it holds for expressions ε Exp Pω (R Id ) A.This, combined with the restrictions to obtin D ω out of R Id, leds to the expressions nd the xiomtiztion in Tble 1 where, in order to void confusion, we use insted of, mking cler distinction between the idempotent nd non-idempotent sums. As n exmple, the expression ({1/2 1/2 }) ({1/3 2/3 }) b({1 }) describes the simple Segl system in Fig.1(i). Strtified systems. Strtified systems re colgebrs of the functor D ω(id)+(b Id)+ 1. Ech stte of these systems either performs unlbelled probbilistic trnsitions or one B-lbelled trnsition or it termintes. To get the intuition for the syntx presented in Tble 1, note tht the strtified system in Fig.1.(ii) would be specified by the expression 1/2 (1/3, 2/3 b, ) 1/2,. Agin, we dded some syntctic sugr to our originl regulr expressions:, denoting termintion, corresponds to our expression r[r[1]], while b,ε corresponds to r[l[l(b) r({ε})]]. The derivtion of the simplified syntx nd xioms follows similr strtegy s in the previous exmple nd thus is omitted here. As described in Section 5, we first derive expressions nd xioms for R Id +(B P ω(id)) + 1 nd then we restrict the syntx to chrcterize only D ω(id) + (B Id)+1-behviours. Pnueli-Zuck systems. These systems re colgebrs of the functor P ωd ω(p ω(id)) A. Intuitively, the ingredient P ω(id) A denotes A-lbelled trnsitions to other sttes. Then, D ω(p ω(id)) A corresponds to probbility distribution of lbelled trnsitions nd then, ech stte of P ωd ω(p ω(id)) A -colgebr performs non deterministic choice mongst probbility distributions of lbelled trnsitions. The expression {1/3 ({ }) ({ }) 2/3 (b({ }) ({ }))} {1 b({ })} specifies the Pnueli-Zuck system in Fig.1 (iii). Notice tht we use the sme symbol for denoting two different kinds of non-deterministic choice. This is sfe, since they stisfy the sme xioms. Agin, the derivtion of the simplified syntx nd xioms is omitted here.

16 7 Conclusions Deriving Syntx nd Axioms for Quntittive Regulr Behviours 161 We presented generl frmework to cnoniclly derive expressions nd xioms for quntittive regulr behviours. To illustrte the effectiveness nd generlity of our pproch we derived expressions nd equtions for weighted utomt, simple Segl, strtified nd Pnueli-Zuck systems. We recovered the syntxes proposed in [10, 14, 38] for the first three models nd the xiomtiztion of [14]. For weighted utomt nd strtified systems we derived new xiomtiztions nd for Pnueli-Zuck systems both novel lnguge of expressions nd xioms. It should be remrked tht [10, 14, 38] considered process clculi tht re lso equipped with the prllel composition opertor nd thus they slightly differ from our lnguges, which re more in the spirit of Kleene nd Milner s expressions. Also [4, 13, 37] study expressions without prllel composition for probbilistic systems. These provide syntx nd xioms for genertive systems, Segl systems nd lternting systems, respectively. For Segl systems our pproch will derive the sme lnguge of [13], while the expressions in [37] differ from the ones resulting from our pproch, since they use probbilistic choice opertor + p. For lternting systems, our pproch would bring some new insights, since [4] considers only expressions without recursion. Acknowledgments. The uthors re grteful to Ctusci Plmidessi for interesting discussions nd pointers to the literture. References 1. Aceto, L., Ésik, Z., Ingólfsdóttir, A.: Equtionl xioms for probbilistic bisimilrity. In: Kirchner, H., Ringeissen, C. (eds.) AMAST LNCS, vol. 2422, pp Springer, Heidelberg (2002) 2. Beten, J., Bergstr, J., Smolk, S.: Axiomiztion probbilistic processes: Acp with genertive probbililties (extended bstrct). In: Clevelnd [11], pp Beten, J., Klop, J. (eds.): CONCUR LNCS, vol Springer, Heidelberg (1990) 4. Bndini, E., Segl, R.: Axiomtiztions for probbilistic bisimultion. In: Orejs, F., Spirkis, P.G., vn Leeuwen, J. (eds.) ICALP LNCS, vol. 2076, pp Springer, Heidelberg (2001) 5. Brtels, F., Sokolov, A., de Vink, E.: A hierrchy of probbilistic system types. TCS 327(1-2), 3 22 (2004) 6. Bonchi, F., Bonsngue, M., Rutten, J., Silv, A.: Deriving syntx nd xioms for quntittive regulr behviours. CWI technicl report (2009) 7. Bonsngue, M., Rutten, J., Silv, A.: An lgebr for Kripke polynomil colgebrs. In: LICS (to pper, 2009) 8. Brzozowski, J.: Derivtives of regulr expressions. Journl of the ACM 11(4), (1964) 9. Buchholz, P.: Bisimultion reltions for weighted utomt. TCS 393(1-3), (2008) 10. Buchholz, P., Kemper, P.: Quntifying the dynmic behvior of process lgebrs. In: de Luc, L., Gilmore, S. (eds.) PROBMIV 2001, PAPM-PROBMIV 2001, nd PAPM LNCS, vol. 2165, pp Springer, Heidelberg (2001) 11. Clevelnd, R. (ed.): CONCUR LNCS, vol Springer, Heidelberg (1992) 12. D Argenio, P., Hermnns, H., Ktoen, J.-P.: On genertive prllel composition. ENTCS 22 (1999) 13. Deng, Y., Plmidessi, C.: Axiomtiztions for Probbilistic Finite-Stte Behviors. In: Sssone, V. (ed.) FOSSACS LNCS, vol. 3441, pp Springer, Heidelberg (2005)

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