Axiomatizing GSOS with Predicates

M.A. Reniers, P . Sobocinski (Eds.): W orkshop on Structural Operational Semantics 2011 (SOS 2011) EPTCS 62, 2011, pp. 1–15, doi:10.4204/EPTCS.62.1 c  L. Aceto, G. Caltais, E.-I. Goriac, A. Ingolfsdottir This work is licensed under the Creativ e Commons Attribution License. Axiomati zing GSOS with Pr edicates ∗ Luca Aceto Geor giana Caltais Eugen-Ioan Goriac Anna Ingolfsdot tir [luca,gcal tais10,egoria c10,annai]@ru.is ICE-TCS, School of Computer Science, Reykja vik Univ ersity , Iceland In this paper, we in troduce an extension o f th e GSOS rule for mat with predicates such as termin ation, conv ergence and di vergence. For this format we generalize the technique proposed by Aceto, Bloo m and V aandrag er for the automatic gene ration of ground- complete axiomatizations of bisimilarity over GSOS s ystems. Our procedure is implemented in a tool that recei ves SOS specifications as in put and derives th e correspo nding axiomatization s auto matically . This paves the way to check ing stron g bisimilarity over process terms by means of theorem-proving techniques. 1 Introd uction One of the g reatest chall enges in co mputer scien ce is the de velop ment of rigor ous methods for the spe ci- fication and ve rification of reac tiv e systems, i.e. , sys tems that compute by intera cting w ith their e n viron- ment. T ypical examples include embedded systems, control programs and distrib uted communica tion protoc ols. Over the last three decad es, proces s algebr as, such as A CP [4], CCS [16] and CSP [14], ha ve been successfully used as common language s for the descr iption of both actual systems and their specifica tions. In this conte xt, verifying w hether the implementatio n of a reacti ve system complies to its specification reduces to pro ving that the correspo nding process terms are related by some notion of beha vioural equi valenc e or preorde r [13]. One approach to proving equi valenc e between two terms is to explo it the equ ational sty le of re ason- ing supported by process algebras. In this approach, one obtains a (ground -)complete axiomati zation of the beha vioural relation of interes t and uses it to prov e the equi va lence between the terms desc ribing the spec ification and the implementati on by means of e quation al rea soning, possibly in co njunctio n with proof rules to handle recursi vely-de fined process specification s. Finding a “finitely specified”, (ground-) complete axiomatizati on of a beha viour al equi v alence ov er a process alg ebra is often a highly no n-tri vial task. Howe ver , as sho wn in [2] in the se tting of bisimilar - ity [16, 17], this proc ess can be automated for pro cess lang uages with an operati onal semantic s gi ven in terms of rules in the GSOS format of Bloom, Istrail and Meyer [8]. In that reference, Aceto , Bloo m and V a andrage r pro vided an algo rithm that, giv en a GSOS language as inpu t, produ ces as output a “conser - v ati ve extensi on” of the origina l language with auxilia ry operators togeth er with a finite axiom system that is sound and ground- complete with respec t to bisimilarity (see, e.g. , [1, 12, 15, 18] for further re- sults in this line of research). As the ope rational specificatio n of se veral opera tors often requires a clear distin ction bet ween succes sful termination and dea dlock, an ex tension of the a bov e-mention ed appro ach to the setting of GSOS with a predicat e for termina tion wa s propose d in [6]. ∗ The authors hav e been been partially supported by the projects “New De velopments in Operational Semantics” (nr . 0800390 21), “Meta-theory of Algebraic Process Theories” (nr . 100014021), and “Extending and Axiomatizing St ructural Operational Semantics: Theory and T ools” (nr . 110294-0 061) of the Icelan dic Research F und. The work on the paper was partly carried out while Luca Aceto and Anna Ingolfsdottir held an Abel Extraordinary Chair at Univ ersidad Complutense de Madrid, Spain, supported by the NILS Mobility Project. 2 Axiomatizi ng GSOS with Predica tes In this paper we contrib ute to the line of the work in [2 ] and [6]. Inspir ed by [6], we introd uce the pr e g rule format, a natura l extensi on of the GSO S format with an arbitra ry collection of predicat es such as termination, con ver gence and di ver gence. W e further adapt the theory in [2] to this setting and giv e a procedure for obtai ning ground- complete axiomatizati ons for bisimilarity o ver pr e g systems. More specifically , we dev elop a general procedure that, giv en a pr e g languag e as input, automatical ly synthe sizes a conserv ati ve exte nsion of that language and a fi nite axiom system that, in conju nction with an i nfinitary proof rule, yi elds a soun d and ground- complete axio matization of bisimilarit y o ver th e ext ended language. The wo rk we pres ent in this paper is based on the one repor ted in [2, 6]. Howe ver , handli ng more general predica tes than immediate terminati on requires the introductio n of some nov el techni cal ideas. In par ticular , the problem of axi omatizing bisimil arity o ver a pr e g lang uage is reduce d to that of axio matizing that relat ion over finite tree s whose nodes may be labelled with predic ates. In order to do so, one need s to tak e special ca re in axi omatizing negati ve premises in rules that may hav e po siti ve and neg ati ve premises in vol ving predic ates and tran sitions. The results of the curren t paper hav e been used for the implementation of a Maude [10] tool [3] that enables the user to specify pre g systems in a unifo rm fa shion, and that automati cally deri ves the associ ated axiomatiza tions. The tool is av ailable at http://goriac.info/tools/preg- axiomatize r/ . This pa ves the way to checki ng bisimilar ity ov er pro cess terms by means of the orem-pro ving techn iques for a lar ge class of systems that can be express ed using pr e g language specification s. Paper structur e. In Section 2 we introduce the pr e g rule format. In S ection 3 w e introduce an appro - priate “core” language for exp ressing finite trees w ith predica tes. W e also provid e a ground-comp lete axiomati zation for bisimilarity ov er th is type of trees, as ou r aim is to pro ve the completene ss of our fi- nal axiomatizati on by head nor malizing gen eral pr e g terms, and therefore by reduc ing the completen ess proble m for arbitra ry lan guages to th at for trees. Head normalizing gener al pr e g terms is not a straightforw ard proces s. Therefore, follo wing [2], in Section 4 we introduce the notion of smooth and distincti ve operatio n, adapte d to the current setting. These operation s are des igned to “ca pture the beha viour of general pr e g operat ions”, and are de fined by rules satisfying a series of syntactic constrain ts with the purpose of enabling the constru ction of head normaliz ing axiomat izations. S uch ax iomatizatio ns are based on a collec tion of eq uations that des cribe the interplay between smooth and distincti ve operat ions, and the operations in the signature for finite trees. T he existenc e of a sound and ground -complete axiomati zation charact erizing the bisimilar ity of pr e g processes is finally prove n in Section 5. A technical discu ssion on w hy it is important to handle predic ates as first class notio ns, instead of enco ding them by means of transitio n relation s, is presente d in Section 6. In Section 7 we dra w some conclusio ns and prov ide pointers to fu ture work. 2 GSOS with pr edicates In this section we present the pr e g systems w hich are a genera lization of GSOS [8] syst ems. Consider a countabl y infinite set V of pr ocess variables (usually denoted by x , y , z ) and a signature Σ con sisting of a se t of oper ations (den oted by f , g ). The set o f pr ocess terms T (Σ) is inducti ve ly d efined as foll ows: each vari able x ∈ V is a term; if f ∈ Σ is an ope ration of arity l , and if S 1 , . . . , S l are ter ms, then f ( S 1 , . . . , S l ) is a term. W e write T (Σ) in order to represent the set of closed pr ocess terms ( i.e. , terms that do not contain va riables), ran ged o ver by t, s . A s ubstitut ion σ is a function o f type V → T (Σ) . If the range of a substitutio n is includ ed in T (Σ) , we say that it is a closed substitutio n . Moreov er , w e write [ x 7→ t ] to represent a substitu tion that maps the va riable x to the term t . Let ~ x = x 1 , . . . , x n be L. Aceto, G. Caltais, E.-I. Goriac, A. Ingolfs dottir 3 a sequence of pairwise distinct v ariables . A Σ - conte xt C [ ~ x ] is a term in w hich at most the va riables ~ x appear . For instance , f ( x, f ( x, c )) is a Σ -conte xt, if the binary operatio n f and the constant c are in Σ . Let A be a finite, nonempty set of actions (denoted by a , b , c ). A positi ve trans ition formula is a triple ( S, a, S ′ ) written S a − → S ′ , w ith the intended m eaning : process S perfo rms action a and becomes proces s S ′ . A ne gative tra nsition formula ( S, a ) written S a 9 , states that process S cann ot perf orm actio n a . N ote that S, S ′ may conta in v ariable s. T he “intende d meaning ” app lies to clo sed process terms. W e no w define pr e g – pr edicat e e xten sion of th e G SO S rule for mat. Let P be a fi nite set of pre dicates (denot ed by P , Q ). A positi ve pre dicate formula is a pair ( P, S ) , written P S , sayin g that process S satisfies predicate P . Dually , a ne gative pr edicate formula ¬ P S state s that process S does not satisfy predic ate P . Definition 1 ( pr e g rule format) . C onside r A , a set of ac tions, and P , a set of pr edicat es. 1. A pr e g tran sition rule for an l -ary oper ation f is a dedu ction rule of the for m: { x i a ij − − → y ij | i ∈ I + , j ∈ I + i } { P ij x i | i ∈ J + , j ∈ J + i } { x i b 9 | i ∈ I − , b ∈ B i } {¬ Qx i | i ∈ J − , Q ∈ Q i } f ( x 1 , . . . , x l ) c − → C [ ~ x, ~ y ] wher e (a) x 1 , . . . , x l and y ij ( i ∈ I + , j ∈ J + ) ar e pair wise dis tinct variables ; (b) I + , J + , I − , J − ⊆ L = { 1 , . . . , l } and each I + i and J + i is finite ; (c) a ij , b and c are act ions in A ( B i ⊆ A ); and (d) P ij and Q ar e pr edicates in P ( Q i ⊆ P ). 2. A pr e g predica te rule for an l -ary oper ation f is a deduct ion rule similar to the one abo ve, with the only dif fer ence that its concl usion has the fo rm P ( f ( x 1 , . . . , x l )) for some P ∈ P . Let ρ be a pr e g (transit ion or predic ate) rule for f . The symbol f is the principal oper ation of ρ . All the formulas abov e the line are anteceden ts and the formula belo w is the conseque nt . W e say that a positi on i for ρ is tested posit ively if i ∈ I + ∪ J + and I + i ∪ J + i 6 = ∅ . Similarly , i is tested ne gatively if i ∈ I − ∪ J − and B i ∪ Q i 6 = ∅ . Whenev er ρ is a transition rule for f , we say that f ( ~ x ) is the sour ce , C [ ~ x, ~ y ] is the tar get , and c is the actio n of ρ . Whene ver ρ is a predica te rule for f , we call f ( ~ x ) the test of ρ . In order to a vo id confusio n, if in a certain con text we use more than one rule, e.g. ρ, ρ ′ , we para me- terize the correspo nding sets of indic es with the na me of the r ule, e.g. , I + ρ , J − ρ ′ . Definition 2 ( pr e g system) . A preg system is a pair G = (Σ G , R G ) , wher e Σ G is a finite signatur e and R G = R A G ∪ R P G is a finite set of pre g rules over Σ G ( R A G and R P G r epr esent the trans ition and, r espec tively , the pr edicate rules of G ). Consider a pr e g system G . Formally , the operationa l semantics of the closed process terms in G is fully charact erized by the relations → G ⊆ T (Σ G ) × A × T (Σ G ) and ⋉ G ⊆ P × T (Σ G ) , called the (uniqu e) sound and supported transitio n and, respecti vely , predicat e relatio ns. Intuiti vely , sound ness guaran tees tha t → G and ⋉ G are clos ed with respect to the ap plicatio n of the rul es in R G on T (Σ G ) , i.e . , → G (resp. ⋉ G ) contai ns the set o f all possible transiti ons (res p. predicates) process terms in T (Σ G ) can perfor m (resp. satisfy ) according to R G . The requir ement that → G and ⋉ G be supported mean s tha t all the transitions perfor med (resp. all the predicate s satisfied) by a certain proce ss term can be “deriv ed” from the deducti ve system described by R G . As a notatio nal con ve ntion, we w rite S a − → G S ′ and P G S whene ve r ( S, a, S ′ ) ∈ → G and ( P , S ) ∈ ⋉ G . W e omit the subscri pt G when it is clear from the conte xt. 4 Axiomatizi ng GSOS with Predica tes Lemma 1. Let G be a pre g system. Then, for each t ∈ T (Σ G ) the set { ( a, t ′ ) | t a − → t ′ , a ∈ A} is finite . Next we introduce the notion of bisimilari ty – the equi valen ce ov er proces ses we consid er in this paper . Definition 3 (Bisimulation ) . Consider a preg system G = (Σ G , R G ) . A symmetric r elation R ⊆ T (Σ G ) × T (Σ G ) is a bisimulatio n if f: 1. for all s, t, s ′ ∈ T (Σ G ) , whenever ( s, t ) ∈ R and s a − → s ′ for some a ∈ A , then ther e is some t ′ ∈ T (Σ G ) such tha t t a − → t ′ and ( s ′ , t ′ ) ∈ R ; 2. w hene ver ( s, t ) ∈ R an d P s ( P ∈ P ) then P t . T wo closed ter ms s and t ar e bi similar (writt en s ∼ t ) iff ther e is a bisimulati on r elation R such tha t ( s, t ) ∈ R . Pro position 1. Let G be a preg system. T hen ∼ is an equivalen ce r elation and a congrue nce for all oper ations f of G . Definition 4 (Disjoint extens ion) . A preg system G ′ is a disjoint e xtensio n of a preg system G , written G ⊑ G ′ , if the sig natur e and the rules of G ′ includ e those of G , and G ′ does not intr oduce ne w ru les for oper ations in G . It is well kno wn that if G ⊑ G ′ then two terms in T (Σ G ) are bisimilar in G if and only if they are bisimilar in G ′ . From thi s poi nt onwa rd, our focus is to fi nd a sound and gr ound-c omplete axiomati zation of bisimi- larity on closed terms for an arbitrary pr e g syste m G , i.e. , to identif y a (finite) axiom sy stem E G so that E G ⊢ s = t iff s ∼ t for all s , t ∈ T (Σ G ) . The method we ap ply is an adapt ation of the techniqu e in [2] to the p r e g setting. The s trateg y is to incrementall y build a finite, hea d-normaliz ing axio matization for ge n- eral pr e g terms, i.e. , an axioma tization that, when app lied recursi vely , red uces the comple teness problem for arbitr ary terms to that for synchroniza tion trees. This way , the proo f of groun d-complet eness for G reduce s to sho w ing the equalit y of close d tree terms . 3 Pr eliminary steps towards the axiomatization In this se ction we start by id entifying an approp riate lan guage for expr essing finite trees with p redicates . W e continu e in the style of [2], by extend ing the languag e with a kind of restrict ion operator used for exp ressing th e inability o f a p rocess to perform a certain action or t o satisfy a giv en predi cate. (This oper - ator is used in the axiomatizatio n of ne gati ve premise s.) W e pro vide the struct ural ope rational semantics of the resulting languag e, togeth er with a sound and gro und-compl ete axiomatizat ion of bi similarity on finite trees with predicat es. 3.1 Finite tr ees with predicates The langu age for tre es we use in this paper is an exten sion with predicate s of the langu age BCCSP [13]. The syntax of BCCS P cons ists of close d ter ms b uilt from a constan t δ ( deadloc k ), the b inary operato r + ( nond eterministic choi ce ), and the unary operators a. ( action pre fix ), where a ranges ov er the actions in a set A . Let P be a set of predicates. For each P ∈ P we conside r a process constant κ P , w hich “witness es” the assoc iated predi cate in the definition of a process. Intu itiv ely , κ P stands for a process that only satisfies predic ate P and has no transiti on. L. Aceto, G. Caltais, E.-I. Goriac, A. Ingolfs dottir 5 A finite tree term t is bui lt according to the followin g grammar : t ::= δ | κ P ( ∀ P ∈ P ) | a.t ( ∀ a ∈ A ) | t + t. (1) Intuiti vely , δ represen ts a proces s that does not exhib it any beha viour , s + t is the nondeterminis tic choice between the beh av iours of s an d t , while a.t is a proc ess that first performs action a and beha ves like t afterw ards. The operation al semantic s that captur es th is intuition is gi ven by the rules of BCCSP: a.x a − → x ( r l 1 ) x a − → x ′ x + y a − → x ′ ( r l 2 ) y a − → y ′ x + y a − → y ′ ( r l 3 ) Figure 1: T he semanti cs of BCCSP As ou r goal is to extend BCC SP , th e nex t step is to find an appropria te semantics for predi cates. As can be seen in Fig. 1 , action performance is determine d by the shape of the terms. Consequent ly , we choos e to define predic ates in a similar f ashion. Consider a predicate P and the term t = κ P . As previo usly mentioned, the purpose of κ P is to witness the satisfiabili ty of P . Therefore, it is natural to conside r that κ P satisfies P . T ake for example the immediate terminati on predicat e ↓ . As a term s + s ′ exh ibits the beha viour of both s and s ′ , it is reasonable to state that ( s + s ′ ) ↓ if s ↓ or s ′ ↓ . Note that for a term t = a.t ′ the statemen t t ↓ is in contra diction with the meaning of immediate termination , since t can initially only ex ecute action a . Predicate s of this kind are calle d e xplici t pr edicates in what follo ws. Consider now the eventu al terminatio n predicate  . In this situation, it is prope r to consider that ( s + t )  if s  or t  and, moreo ver , that a.s  if s  . W e refer to predicate s such as  as implicit pre dicates (that range o ver a set P I includ ed in P ), sinc e their satis fiability propagates through the s tructure of tre e terms in an implicit fashion . W e denote by A P (inclu ded in A ) the set consisting of the actions a for which this beha viour is permitted when reasoni ng on the sati sfiability of pre dicate P . The rules expr essing the semantics of pre dicates are: P κ P ( r l 4 ) P x P ( x + y ) ( r l 5 ) P y P ( x + y ) ( r l 6 ) P x P ( a.x ) , ∀ P ∈ P I ∀ a ∈ A P ( r l 7 ) Figure 2: T he semanti cs of pred icates The operati onal semantics of trees with predic ates is gi ven by the set of rules ( r l 1 )–( rl 7 ) illus trated in Fig. 1 and Fig. 2. For notati onal consis tency , we make the following con ventio ns. Let A be an action set and P a set of predicates. Σ FTP repres ents the signatur e of finite trees with predicates . T (Σ FTP ) is the set of (close d) tree terms b uilt o ver Σ FTP , and R FTP is the set of ru les ( rl 1 )–( rl 7 ). Moreov er , by FTP we denote the system (Σ FTP , R FTP ) . Discussion on the desig n decisi ons. At first sight, it see ms reaso nable for our frame work to allow for langua ge specificati ons contain ing rules of th e sha pe P ( x + y ) , or ju st on e of ( r l 5 ) and ( r l 6 ). W e decided , ho wev er , to disallo w them, as their presence would in valida te standard algebraic prope rties such as the idempote nce and the commutat iv ity of + . 6 Axiomatizi ng GSOS with Predica tes W ithout loss of g enerality we a void rul es of the form P ( a.x ) . A s f ar as the us er is con cerned, in order to express that a.x satisfies a pr edicate P , on e can alw ays add the witness κ P as a summan d: a.x + κ P . This decision helped us a vo id some technical problems for the soundness and completeness proofs for the case of the restrictio n opera tor ∂ B , Q , which is present ed in Sectio n 3.3. Due to the afore mentioned res triction, we also had to leav e out uni versa l pred icates w ith rule s of the form P x P y P ( x + y ) . H o wev er , the elimination of uni vers al pre dicates is not a theore tical limita tion to what one can exp ress, since a un iv ersal predicat e can alwa ys be defined as the negati on of an e xisten tial one. As a last approach, we thought of allo wing the user to specify exi stential predicates using rules of the form P 1 x...P n x P ( x + y ) ( ∗ ) and P 1 y . ..P n y P ( x + y ) ( ∗∗ ) (instead of ( r l 5 ) and ( r l 6 ) ). Howe ver , in order to maintain the v alidity of the axiom x + x = x in the pr esence of rul es of these forms, it wou ld hav e to be the case that one of the predicates P i in the premises is P itself. (If that were not the case, then let t be the sum of the consta nts w itnessi ng the P i ’ s for a rule of the f orm ( ∗ ) abov e with a minimal set of set pre mises. W e ha ve that t + t satisfies P by rule ( ∗ ) . O n the ot her hand , P t doe s not hold since no ne of the P i is equal to P and no rule for P with a smaller s et of premises ex ists.) Now , if a rule of the form ( ∗ ) has a premise of the form P x , then it is subsumed by ( r l 5 ) whic h we must ha ve to ensu re the v alidity of laws such as κ P = κ P + κ P . 3.2 Axiomatizing finite tr ees In what follo ws we prov ide a fi nite sound and groun d-complet e axiomatiz ation ( E FTP ) for bisimilar ity ov er finite trees with predica tes. The axiom system E FTP consis ts of the follo wing axio ms: x + y = y + x ( A 1 ) x + x = x ( A 3 ) ( x + y ) + z = x + ( y + z ) ( A 2 ) x + δ = x ( A 4 ) a. ( x + κ P ) = a. ( x + κ P ) + κ P , ∀ P ∈ P I ∀ a ∈ A P ( A 5 ) Figure 3: T he axiom syste m E FTP Axioms ( A 1 ) – ( A 4 ) ar e well-kno wn [16]. Axiom ( A 5 ) descr ibes th e propagati on of witness constant s for the case of implicit predica tes. W e no w introduce the notio n of terms in head normal form . This concep t play s a ke y role in the proofs of completen ess for the axiom systems gene rated by our frame work. Definition 5 (Head Normal Form) . Let Σ be a signatu r e such that Σ FTP ⊆ Σ . A term t in T (Σ ) is in head normal form (for short, h.n.f.) if t = X i ∈ I a i .t i + X j ∈ J κ P j , and th e P j ar e all th e pr edicates satisfied by t. The empty sum ( I = ∅ , J = ∅ ) is de noted by the deadloc k con stant δ . Lemma 2. E FTP is head normali zing for terms in T (Σ FTP ) . T hat is, for all t in T (Σ FTP ) , the r e e xists t ′ in T (Σ FTP ) in h.n.f. suc h that E FTP ⊢ t = t ′ holds. Pr oof. The reasonin g is by induct ion on th e structure of t . Theor em 1. E FTP is sound and gr ound-complete for bisimilarity on T (Σ FTP ) . T hat is, it holds that ( ∀ t, t ′ ∈ T (Σ FTP )) . E FTP ⊢ t = t ′ if f t ∼ t. L. Aceto, G. Caltais, E.-I. Goriac, A. Ingolfs dottir 7 3.3 Axiomatizing negative p remises A crucial step in finding a complete axiomatization for pr e g sys tems is the “axiomatiz ation” of negati ve premises (of the shape x a 9 , ¬ P x ). In the style of [2], we introdu ce the restri ction op erator ∂ B , Q , where B ⊆ A and Q ⊆ P are the sets of initi ally forb idden actions and predicates, resp ecti vely . The semantics of ∂ B , Q is gi ven by the two ty pes of transition rules in Fig. 4. x a − → x ′ ∂ B , Q ( x ) a − → ∂ ∅ , Q∩P I ( x ′ ) if a 6∈ B ( rl 8 ) P x P ( ∂ B , Q ( x )) if P 6∈ Q ( r l 9 ) Figure 4: T he semanti cs of ∂ B , Q Note that ∂ B , Q beha ves like the one step restriction operator in [2] for the action s in B , as the re- stricti on on the actio n set d isappea rs after one trans ition. On the othe r hand, for the ca se of predicate s in Q , the operator ∂ B , Q resemble s the CCS restric tion operato r [16] since, due to the presence of implicit predic ates, not al l the rest rictions related to predi cate satisfact ion nece ssarily disappear after one step , as will become clear in what follo w s. W e write E ∂ FTP for the e xtension of E FTP with the axioms in volv ing ∂ B , Q presen ted in Fig. 5. R ∂ FTP stands for the set of rules ( r l 1 ) − ( rl 9 ) , while FTP ∂ repres ents the syst em (Σ ∂ FTP , R ∂ FTP ) . ∂ B , Q ( δ ) = δ ( A 6 ) ∂ B , Q ( a.x ) = P P 6∈Q ,P ( a.x ) κ P if a ∈ B ( A 9 ) ∂ B , Q ( κ P ) = δ if P ∈ Q ( A 7 ) ∂ B , Q ( a.x ) = ∂ ∅ , Q ( a.x ) if a 6∈ B ( A 10 ) ∂ B , Q ( κ P ) = κ P if P 6∈ Q ( A 8 ) ∂ ∅ , Q ( a.x ) = a.∂ ∅ , Q∩P I ( x ) ( A 11 ) ∂ B , Q ( x + y ) = ∂ B , Q ( x ) + ∂ B , Q ( y ) ( A 12 ) Figure 5: T he axiom syste m E ∂ FTP \ E FTP Axiom ( A 6 ) st ates that it is useless to impo se restric tions on δ , as δ does not exhibi t any beha viour . The intuit ion behind ( A 7 ) is that since a predicate witness κ P does not perform any action, inhibiting the satisfiability of P leads to a process with no behavi our , namely δ . Consequen tly , if the restric ted predic ates do not include P , the resulting process is κ P itself (see ( A 8 ) ). Inhibitin g the only action a proces s a.t can pe rform leads to a ne w proce ss that, in the best case, satisfies some of the predicates in P I satisfied by t (by ( r l 7 ) ) if Q 6 = P I (see ( A 9 ) ). Whenev er the restrict ed actio n set B d oes not contain the only actio n a process a.t can pe rform, then it is safe to gi ve up B (see ( A 10 ) ). As a proces s a.t on ly satisfies the predic ates also satisfied by t , it is straigh tforward to see that ∂ ∅ , Q ( a.t ) is equi v alent to the proces s obtained by propa gating the restric tions on implicit predi cates deeper into the beha viour of t (see ( A 11 ) ). A xiom ( A 12 ) is giv en in con formity with the semantics of + ( s + t encapsulat es both th e beha viours of s and t ). Remark 1. F or the sake of br evity and r eadabil ity , in Fig . 5 we pr esented ( A 9 ) , which is a schema with infinit ely many instances. However , it can be r eplaced by a finite family of axioms. See Appendi x D in the full vers ion of the pap er available at http://www.ru.is/faculty/luca/PAPERS/axgsos.pdf for detail s. Theor em 2. The following statements hold for E ∂ FTP : 1. E ∂ FTP is soun d for bisimil arity on T (Σ ∂ FTP ) . 8 Axiomatizi ng GSOS with Predica tes 2. ∀ t ∈ T (Σ ∂ FTP ) , ∃ t ′ ∈ T (Σ FTP ) s.t. E ∂ FTP ⊢ t = t ′ . As provi ng compl eteness fo r FT P ∂ can be reduced to sho wing completenes s for FT P (alre ady pr ove d in Theorem 1), the follo wing result is an immediate conseque nce of Theorem 2: Cor ollary 1. E ∂ FTP is sound and complete for bisimila rity on T (Σ ∂ FTP ) . 4 Smooth and distincti ve operations Recall that our goal is to pro vide a soun d an d groun d-complete a xiomatizat ion for bi similarity on sys tems specified in the pr e g format. As the pr e g format is too permissi ve for achie ving this result directly , our nex t task is to find a class of operati ons for which we can b uild such an axiomatizatio n by “easily” reduci ng it to the comple teness resu lt for FTP , presented in Theore m 1. In th e lit erature, the se oper ations are kno wn as smooth and distinc tive [2]. As we will see, th ese opera tions are incrementa lly identified by imposing suitabl e restrictio ns on pr e g rul es. The standard procedu re is t o first find the smoot h op erations , based on which one determines the distinc tive ones . Definition 6 (Smooth operati on) . 1. A preg tran sition rule is smooth if it is of the following format: { x i a i − → y i | i ∈ I + } { P i x i | i ∈ J + } { x i b 9 | i ∈ I − , b ∈ B i } {¬ Qx i | i ∈ J − , Q ∈ Q i } f ( x 1 , . . . , x l ) c − → C [ ~ x, ~ y ] wher e (a) I + , J + , I − , J − disjoi ntly co ver the set L = { 1 , . . . , l } , (b) in the tar get C [ ~ x, ~ y ] w e allow only: y i ( i ∈ I + ) , x i ( i ∈ I − ∪ J − ) . 2. A preg pr edicate rule is smooth if it has the form above , its pr emises sat isfy cond ition (1a) and its conclu sion is P ( f ( x 1 , . . . , x l )) for some P ∈ P . 3. A n ope ratio n f of a preg system is smooth if all its (tr ansition an d pr edicate) rules ar e smooth. By Definition 6, a rule ρ is smooth if it satisfies the follo w ing properties : • a position i cannot be tested both positi vel y and neg ati vely at the same time, • positions tested positi vely are ei ther from I + or J + and the y are not tested for the performanc e of multiple t ransition s (res pecti vel y , for the satisfiabilit y of multiple pred icates) within th e same rule , and • if ρ is a transition rul e, then the occurre nce of variab les at positions i ∈ I + ∪ J + is not allowed in the tar get of the conseq uent of ρ . Remark 2. Note that we can always consider a posit ion i that does not occur as a pre mise in a rule for f as being ne gative , with the empty set of constrai nts (i.e. either i ∈ I − and B i = ∅ , or i ∈ J − and Q i = ∅ ). Definition 7 (Distincti ve operation) . An operatio n f of a preg syst em is dist incti ve if it is smooth and: • for each ar gument i , either al l rules for f test i positiv ely , or none of them does, and • for any two distinct rules for f ther e exis ts a positio n i tested positi vely , such that one of the followin g hold s: L. Aceto, G. Caltais, E.-I. Goriac, A. Ingolfs dottir 9 - both rules have actions that ar e diff er ent in t he pr emise at position i , - both rules have pr edicates that ar e dif fer ent in the pr emise at position i , - one rule has an action pr emise at position i , and the other rule has a pr edicate test at the same posi tion i . Accordin g to the first requiremen t in Definition 7, we st ate that fo r a smooth and di stincti ve oper ation f , a position i is positi ve (resp ecti vely , ne gative ) fo r f if there is a rule f or f such that i is tes ted positi vely (respe ctiv ely , negati vely) for that rule. The existence of a fa mily of smooth and distincti ve opera tions “describ ing the beha viour” of a gen eral pr e g op eration is formalized by the follo wing lemma: Lemma 3. Consider a preg syst em G . Then th er e e xist a pre g sys tem G ′ , whic h is a disjoint e xtension of G and FTP, and a finite axiom system E such that 1. E is sound for bisimilarity over any disjoin t e xtension G ′′ of G ′ , and 2. for each term t in T (Σ G ) ther e is some term t ′ in T (Σ G ′ ) such that t ′ is built solely using smoot h and distin ctive ope ratio ns and E pr oves t = t ′ . 4.1 Axiomatizing smooth and distinctive preg op erations T o start with, consider , for the good fl o w of the prese ntation, that we only handle expli cit predi cates ( i.e. , we take P I = ∅ ). T ow ards the end of the section we discuss ho w to exten d the presen ted theory to implicit predic ates. W e proceed in a similar fashi on to [2] by defining a set of laws used in the constr uction of a complete axiomatiza tion for bisimila rity on terms buil t over smooth and distin cti ve operat ions. T he streng th of these laws lies in their capab ility of reduci ng terms to their head normal form, thus reducing co mpleteness for general pr e g systems to complet eness of E FTP (which has already been prov ed in Section 3.2). Definition 8 . Let f be a smooth a nd dist inctive l -ary o perat ion of a pre g syste m G , suc h tha t FTP ∂ ⊑ G . 1. F or a pos itive pos ition i ∈ L = { 1 , . . . , l } , the distrib uti vity law fo r i w .r . t. f is given as follo ws: f ( X 1 , . . . , X ′ i + X ′′ i , . . . , X l ) = f ( X 1 , . . . , X ′ i , . . . , X l ) + f ( X 1 , . . . , X ′′ i , . . . , X l ) . 2. F or a rule ρ ∈ R for f the trigger law is, depend ing on w hether ρ is a transi tion or a pr edicate rule: f ( ~ X ) =  c.C [ ~ X , ~ y ] , ρ ∈ R A (action law) κ P , ρ ∈ R P (predi cate la w) wher e X i ≡    a i .y i , i ∈ I + κ P i , i ∈ J + ∂ B i , Q i ( x i ) , i ∈ I − ∪ J − . 3. Suppose that for i ∈ L , te rm X i is in o ne of the for ms δ, z i , κ P i , a.z i , a.z i + z ′ i or κ P i + z i . Suppose furthe r that for each rule for f ther e ex ists X j ∈ ~ X ( j ∈ { 1 , . . . , l } ) s.t. one of the following hold s: • j ∈ I + and ( X j ≡ δ or X j ≡ b.z j ( b 6 = a j ) or X j ≡ κ Q , for some Q ), 10 Axiomatizi ng GSOS with Predica tes • j ∈ J + and ( X j ≡ δ or X j ≡ κ Q ( Q 6 = P j ) or X j ≡ b.z j , for some b ), • j ∈ I − and X j ≡ b.z j + z ′ j , wher e b ∈ B j , • j ∈ J − and X j ≡ κ Q + z j , wher e Q ∈ Q j . Then the deadl ock la w is as follows: f ( ~ X ) = δ. Example 1. Consider the right-b iased sequent ial composition oper ation ; r , whose semantic s is give n by the rules x ↓ y a − → y ′ x ; r y a − → y ′ , x ↓ y ↓ ( x ; r y ) ↓ , and x ↓ y ↑ ( x ; r y ) ↑ , wher e ↓ and ↑ ar e, r espectively , the immediat e termi- nation and immediate di ver gence pr edicates. ; r is one of the auxiliary operati ons ge nerat ed by the algori thm for deriving smoot h and distin ctive opera tions when axiomatizin g the sequential compo sition in the pr esence of the two mention ed pr edicates. The laws derived accor ding to Definit ion 8 for this system ar e: ( x + y ) ; r z = x ; r z + y ; r z δ ; r y = δ x ; r ( y + z ) = x ; r y + z ; r z k ↑ ; r y = δ k ↓ ; r a.y = a.y a.x ; r y = δ k ↓ ; r k ↓ = k ↓ x ; r δ = δ k ↓ ; r k ↑ = k ↑ . . . Theor em 3. Consider G a preg system such that FTP ∂ ⊑ G . L et Σ ⊆ Σ G \ Σ ∂ FTP be a collection of smooth and distinctive oper ations of G . Let E G be the finite axiom system that ex tends E ∂ FTP with the followin g axioms for eac h f ∈ Σ : • for each posi tive ar gument i of f , a distrib utivity law (Definition 8.1), • for each tr ansition rule for f , an action law (Definitio n 8.2), • for each pr edicate rule for f , a pr edica te law (Definiti on 8.2), and • all deadloc k laws for f (Definitio n 8.3). The followin g statemen ts hol d for E G , for any G ′ suc h that G ⊑ G ′ : 1. E G is soun d for bisimil arity on T (Σ G ′ ) . 2. E G is head normaliz ing for T (Σ ∪ Σ ∂ FTP ) . Obtainin g the sound ness of the action law (Definition 8.2) requires some care when allowing for specifica tions with implicit predicates ( P I 6 = ∅ ). Conside r a scenario in which a trans ition rule for a smooth and distincti ve operat ion f is of the form H f ( ~ X ) c − → C [ ~ X ,~ y ] . Assume the closed insta ntiation ~ X = ~ s , ~ y = ~ t and assume that P ( c.C [ ~ s, ~ t ]) holds for some pr edicate P in P I . This means that P ( C [ ~ s, ~ t ]) holds. In order to preserv e the soundness of the action law , P ( f ( ~ s )) shoul d also hold, bu t this is impossib le since f is distincti ve. One possibl e way of ensuring the soundness of the action law in the pres ence of implicit pred icates is to stipula te some s yntactic consist ency requirements on the la nguage specificati on. One suf ficient requirement would be that if predicate rule H ′ P ( C [ ~ z ,~ y ]) is deriv able, then the system shoul d contai n a predicate rule H ′′ P ( f [ ~ z ]) with H ′′ ⊆ H ′ . This is eno ugh to guarante e that if the righ t-hand side of the action law satis fies P then so does the left-hand side. L. Aceto, G. Caltais, E.-I. Goriac, A. Ingolfs dottir 11 5 Soundn ess and completeness Let us su mmarize our resu lts so far . By Theorem 3, it follo ws that, for an y pr e g system G ⊒ FTP ∂ , there is an axiomatizatio n that is head no rmalizing for T (Σ ∪ Σ ∂ FTP ) , where Σ ⊆ Σ G \ Σ ∂ FTP is a collection of smooth and distincti ve operatio ns of G . Also, as hinted in S ection 4 (Lemma 3), there exist s a sound algori thm for trans forming gene ral pr e g op erations to smooth an d distincti ve one s. So, for any pr e g system G , we can bui ld a pr e g system G ′ ⊒ G and an axiomatiz ation E G ′ that is head normalizin g for T (Σ G ′ ) . T his stat ement is forma lized as follows: Theor em 4. Let G be a preg system. Then ther e exist G ′ ⊒ G and a finite axiom system E G ′ suc h that 1. E G ′ is sound for bisimila rity on T (Σ G ′ ) , 2. E G ′ is head normaliz ing for T (Σ G ′ ) , and mor eover , G ′ and E G ′ can be ef fectively construct ed fr om G . Pr oof. The result follo ws immediately by Theorem 3 and by the exist ence of an algorithm used for transfo rming genera l pr e g to smooth and dist incti ve operations . Remark 3. Theor em 4 gu aran tees gr ound-complete ness of the ge nerat ed axiomatiza tion fo r well-foun ded pre g specificatio ns, that is, preg specifi cations in whic h each pr ocess can only e xhibit finite behavio ur . Let us furthe r recall an example gi ven in [2]. Conside r the const ant ω , specified by the rule ω a − → ω . Obviously , the correspond ing actio n law ω = a.ω will apply for an infinite number of times in the normaliz ation proces s. So the last step in obtaining a complete axiomatizat ion is to handle infinite beha viour . Let t and t ′ be two proces ses with infinite behav iour (remark that the infinite beha viour is a conse- quenc e of performing a ctions for an infinite number o f times, so the extens ion to predic ates is not a caus e for this issue). Since we are dealing with finitely branchi ng proces ses, it is well kno w n that if two proc ess terms are bisimil ar at each finite depth, then they a re bisimilar . One way o f formalizing this requir ement is to use the well-kno w n Appr oximation Indu ction Princ iple (AIP) [5, 7]. Let us first consid er the operatio ns π n ( · ) , n ∈ N , known as pr ojection opera tions . The purpose of these op erations is to stop th e e v olution of processes after a certain n umber of s teps. The AIP is giv en by the follo w ing condition al equation : x = y if π n ( x ) = π n ( y ) ( ∀ n ∈ N ) . W e furthe r adapt the idea i n [2] to o ur conte xt, and model th e infinite f amily of proje ction operation s π n ( · ) , n ∈ N , by a bina ry op eration · / · defined as follo ws: x a − → x ′ h c − → h ′ x/h a − → x ′ /h ′ ( r l 10 ) P x P ( x/h ) ( r l 11 ) where c is an arbitrary action. Note that · / · is a smooth and distinc tiv e ope ration. The role of v ariable h is to “control” the e vo lution of a pr ocess, i.e. , to stop t he proc ess in performing action s, after a gi ven number of ste ps. V ar iable h (th e “h our glass” in [2]) will al ways be instantiat ed with terms of the shape c n , inducti vely de fined as: c 0 = δ , c n +1 = c.c n . Let G = (Σ G , R G ) be a pr e g system. W e use the notation G / to refer to the pr e g system (Σ G ∪ {· / ·} , R G ∪ { ( r l 10 ) , ( r l 11 ) } ) – the extens ion of G with · / · . Moreov er , we use the notation E AIP to refer to the axioms for the smooth and distinc tiv e ope ration · / · , deriv ed as in Section 4.1 – Definition 8. W e reformulate AIP accordin g to the ne w operatio n · / · : x = y if x/c n = y /c n ( ∀ n ∈ N ) 12 Axiomatizi ng GSOS with Predica tes Lemma 4. AIP is sound for bisimilar ity on T (Σ FTP / ) . In what follo ws we provid e the final ingredien ts for pro ving the exist ence of a ground-co mplete axiomati zation for bisimilarit y on pr e g systems. A s pre vious ly stated, this is achie ved by reducing com- pleten ess to pro ving equal ity in FTP . So, based on AIP , it would suf fice to sho w that for any close d proces s term t and natural number n , there exis ts an F TP term equ iv alent to t at moment n in time: Lemma 5. Consider G a preg system. Then ther e exist G ′ ⊒ G / and E G ′ with the pr operty : ∀ t ∈ T (Σ G ′ ) , ∀ n ∈ N , ∃ t ′ ∈ T (Σ FTP ) s.t. E G ′ ⊢ t/c n = t ′ . At this po int we can prov e the ex istence of a sound and gro und-compl ete axiomatizatio n for bisimi- larity on general pr e g systems: Theor em 5 (Soundn ess and Completeness) . Consider G a preg syste m. Then ther e exist G ′ ⊒ G / and E G ′ a finite axiom system, suc h that E G ′ ∪ E AIP is sound and complete for bisimila rity on T (Σ G ′ ) . 6 Motivation for handling predicates as first-class notions In the liter ature on the th eory of rule for mats for Structura l Operational Semant ics (espe cially , the work de v oted to cong ruence formats fo r variou s no tions of bisi milarity), pr edicates are often neglected at first and are only added to the considerati ons at a later stage . T he reason is that one can encode predic ates quite ea sily by means of trans ition relations. One can find a n umber of such en codings in the liter ature— see, for instance, [11, 19]. In each of these encodi ngs, a predica te P is represen ted as a transition relatio n P − → (assuming that P is a fresh action label) with a fi xed co nstant symbol as tar get. Using this transla tion, one can axiomat ize bisimilarity ov er pr e g langua ge specifications by first con verting them into “equi val ent” standard GSOS systems, and then apply ing the algorithm from [2] to obtain a finite axiomati zation of bisimila rity o ver the resu lting GSOS sy stem. In ligh t of thi s approa ch, it is natural to wo nder wheth er it is worthwhi le to de velop an algorithm to axiomati ze pr e g language speci fications directly . One possib le answer , w hich has been p resented se vera l times in the literature [19], is th at of ten one does not wan t to en code a language sp ecification with pred- icates using one with transition s only . Sometimes, specifications using predica tes are the most natural ones to write, an d one should not for ce a lan guage desig ner to cod e predicate s using transition s. (H o w- e ver , one can write a tool to perform the transla tion of predica tes into transiti ons, which can therefore be carried out transpa rently to the user/langu age design er .) Also, dev eloping an algorithm to axiomatize GSOS langu age specificati ons with predic ates dir ectly yie lds insight into the d ifficu lties tha t result from the first-clas s us e of, and the interpla y amo ng, v arious types of predic ates, as f ar as axiomat izability pro b- lems are conc erned. These issues would be hidde n by encoding pre dicates as transitio ns. Moreove r , the algori thm resulting from the encodi ng would generate axioms in vo lving predicate-p refixing opera tors, which are some what unintuiti ve. Naturaln ess is, ho w e ver , oft en in the eye of th e beholder . Therefore, we now pr ovide a more tec hnical reason wh y it may be wort hwhile to de velop t echnique s tha t apply to GSOS languag e specifications with predic ates as first-class notions, su ch as the pr e g one s. Indeed, we no w sho w ho w , using predicates , one can con vert an y standard GSOS lang uage spe cification G into an equi valen t posi tive one with predica tes G + . Giv en a GS OS lan guage G , the syste m G + will ha ve the same sign ature and the same set of actions as G , b ut uses predicates cannot ( a ) for each action a . The idea is simply tha t “ x cannot ( a ) ” is the p redicate formula that express es that “ x does not a ffo rd an a -lab elled trans ition”. The t ranslatio n works as follo ws. L. Aceto, G. Caltais, E.-I. Goriac, A. Ingolfs dottir 13 1. E ach rule in G is also a rule in G + , but one replaces each negati ve premise in each rule with its corres ponding po siti ve predicate premise. This means that x a 9 become s x cannot ( a ) . 2. O ne adds to G + rules defining the predicates cannot ( a ) , for each action a . This is done in such a way that p cannot ( a ) hol ds in G + exa ctly when p a 9 in G , for each close d term p and action a . More prec isely , we proceed as follo w s. (a) For each constan t symbol f and action a , add the rule f cannot ( a ) whene ve r there is no tran sition rule in G with f as pr incipal operatio n and with an a -labelled transit ion as its con sequent. (b) For each operation f with arity at least one and action a , let R ( f , a ) be the set of rules in G that hav e f as principa l operati on and an a -labelled transiti on as consequent. W e want to add rules for the predicate canno t ( a ) to G + that allo w us to prov e the predicate formula f ( p 1 , . . . , p l ) cannot ( a ) ex actly when f ( p 1 , . . . , p l ) does not afford an a -labelled transition in G . This occurs if, for each rule in R ( f , a ) , there is some premise that is not satisfied when the argu ments of f are p 1 , . . . , p l . T o formalize this idea, let H ( R ( f , a )) be the collection of premises of rules in R ( f , a ) . W e say that a choice function is a functi on φ : R ( f , a ) → H ( R ( f , a )) that maps each rule in R ( f , a ) to one of its premises. L et neg ( x a − → x ′ ) = x cannot ( a ) and neg ( x a 9 ) = x a − → x ′ , for some x ′ . Then, for each choic e func tion φ , we ad d to G + a predic ate rule of the form { ne g ( φ ( ξ )) | ξ ∈ R ( f , a ) } f ( x 1 , . . . , x l ) cannot ( a ) , where the tar gets of the positi ve transitio n formulae in the premises are chosen to be all dif ferent. The abo ve constr uction ensures the v alidity of the follo w ing lemma. Lemma 6. F or each closed term p and action a , 1. p a − → p ′ in G if, and only if , p a − → p ′ in G + ; 2. p cannot ( a ) in G + if, and onl y if, p a 9 in G + (and ther efor e in G ). This means that two closed terms are bisimilar in G if, and only if, they are bisimilar in G + . M ore- ov er , two closed terms are bisimilar in G + if f they are bisimilar w hen we only consi der the transitions (and not the predic ates cannot ( a ) ). The language G + modulo bisimilarity can be axiomatize d using our algo rithm without the need for the expon entially man y restrictio n operators. T he con versi on to positi ve GSOS with predicates d iscussed abo ve d oes inc ur in an e xponen tial blo w-up in the n umber of rules, b ut it giv es an altern ati ve way of gen- erating ground- complete axiomatizati ons for standard GSOS languag es to the one propose d in [2]. In genera l, it is usefu l to ha ve se veral appro aches in one’ s toolbox, since one may choose the one that is “less exp ensi ve” for the specific task at han d. Moreov er , using positi ve GSOS opera tions, one can also try to extend the methods from the full version of the paper [1] (see Section 7.1 in the techn ical report 14 Axiomatizi ng GSOS with Predica tes a va ilable at http://www.ru.is/ ~ luca/PAPERS/cs0119 94. ps ) to opt imize these axiomat izations . W e are c ur - rently working on applying such methods to positi ve pr e g systems with univ ersal as well as existen tial predic ates, and on ext ending our tool [3] accor dingly . It is worth noting that the predica tes cannot ( a ) are not implicit, therefo re the restrictions presented at the end of Section 4.1 need not to be imposed. 7 Conclusions and futur e work In this paper we hav e introduc ed the pr e g rule format , a natur al e xtensio n of GSOS with arbitrary predi- cates. Moreov er , w e ha ve p rovid ed a procedure (simila r to the on e in [2]) for deri ving sound and gro und- complete axiomatiz ations for bisimilari ty of systems that match this format. In the curren t approac h, exp licit predic ates are handled by conside ring constants w itness ing their satisfiabilit y as summands in tree exp ressions . C onseq uently , there is no expli cit pr edicate P satisfied by a term of shape Σ i ∈ I a i .t i . The procedur e introduce d in this pap er ha s also ena bled the implementa tion of a tool [3] tha t can be used to automati cally reason on bisimil arity of systems specified as terms bu ilt over op erations defined by pr e g rules. Sev eral possibl e ex tension s are left as future work. It would be worth in vestig ating the propert ies of positi ve pr e g langua ges. By allo wing only positi ve premise s we elimina te the need of the restri ction operat ors ( ∂ B , Q ) during the axiomat ization proces s. This would enable us to deal w ith more general predic ates over trees, such as those that may be satisfied by terms of the form a.t where a ranges ove r some subset of the collect ion of action s. Another directio n for future research is that of understandi ng the presented wo rk from a coalgebraic perspe cti ve. The exte nsions from [2] to th e prese nt paper , might be tho ught as an extens ion from coal ge- bras for a functo r P ( A × Id ) to a fu nctor P ( P ) × P ( A × Id ) where P is the po w erset functor , A is the set of actions and P is the set of predica tes. A lso th e langua ge FT P coincides , apart from the recu rsion operat or , with the one that would be o btained for the func tor P ( P ) × P ( A × Id ) in the conte xt of Kripk e polyn omial coalg ebras [9]. Finally , we pla n to e xtend our axio matization theory in order to reason on the bis imilarity of g uarded recurs iv ely de fined ter ms, follo wing the lin e presented in [1]. Acknowledgments. The authors are g rateful for the us eful comments an d sugg estions from Alexandra Silv a and three anony mous re vie wers. Refer ences [1] Luc a Aceto (199 4): Deriving Comp lete Inference Systems for a Class of GSOS Langu ages Gener ation Reg- ular Behaviou rs . In Jonsson & Parrow [15], pp. 449–464 , d oi: 10.1007/BFb001 5025 . [2] Luc a Ace to, Bard Bloom & Frits V aa ndrager (1994): T u rning SOS rules into equations . Inf. Comput. 111, pp . 1–52 , doi: 10 .1006/inco.1994.1040 . A vailable at h ttp:/ / porta l.acm.org/citation.cfm? id=184 662.184663 . [3] Luc a Aceto , Geo rgiana Caltais, Eugen-I oan Goriac & Anna Ing olfsdottir (2 011): PR EG Axiomatizer : A gr o und bisimilarity chec ker for GS OS with pr edicates . In: CALCO 2011 , LNCS, Spring er . A vailable at http:/ / gori ac.info/tools/preg- axio matiz er/ . T o app ear . [4] J. C. M. Baeten, T . Basten & M. A. R eniers ( 2010) : Pr ocess A lgebra: Equa tional Theo ries of Commu nicating Pr oc esses . Cambridge T racts in Theoretical Computer Science 50, Cambridge Univ ersity Press, Cambridge. L. Aceto, G. Caltais, E.-I. Goriac, A. Ingolfs dottir 15 [5] J. C. M. Baeten & W . P . W eijland (199 0): Pr ocess Algebra . Cambridg e Un iv ersity Press, New Y ork, NY , USA. [6] Jos C. M. Baeten & E rik P . de V ink (2004 ): Axiomatizing GSOS with termination . J. Log. Alg ebr . Prog ram. 60-61 , pp. 323–35 1, doi: 10.1016/j.jlap.2004.03.001 . [7] J A Bergstra & J W Klop (19 86): V erification of an alternating bit pr otocol by means of pr ocess algebra . In: Proceeding s of the International Sp ring School on M athematical method of specification and s ynth esis of software system s ’85 , Spr inger-V erlag New Y ork, Inc., Ne w Y ork, NY , USA, pp. 9– 23. A vailable at http:// portal . acm.org/cit ation . c fm?id=1666 3. 1666 4 . [8] Bard Bloom, Sorin I strail & Albert R. Meyer (1995): B isimulation can’t be tr aced . J. A CM 42, pp. 2 32–2 68, doi: 10.1145/20083 6. 2008 76 . [9] Marce llo M. Bonsan gue, Jan J. M. M. Rutten & A lexandra Silv a ( 2009) : An Algebra for Kripke P olyn omial Coalgebras . In : LICS , IEEE Computer Society , pp. 49–5 8, doi: 10.1109/ LICS.2009.18 . [10] Man uel Clav el, Francisco Du r ´ an, Stev en Eker, Patrick Linco ln, Nar ciso Mart´ ı-Oliet, Jos ´ e Me seguer & Car - olyn L. T alcott, editors (2007): All About Maude - A High-P erformance Logical F ramew ork, How to Specify , Pr ogram and V erify Systems in Rewr iting Logic . Lecture Notes in Computer Science 4350, Springe r . [11] Sjoer d Cranen, Mohamm adReza Mou savi & Michel A. Reniers (200 8): A Rule F ormat for Associativity . In Franck van Bre ugel & Marsh a Chechik, ed itors: Proceeding s of th e 19th Intern ational Confer ence on Concurre ncy Theor y (CONCUR’08) , Lecture No tes in Comp uter Science 5201, Springer-V erlag, Berlin, Germany , T oronto ,Canada, pp. 447–461 , doi: 10.1007/978- 3- 5 40- 853 61- 9_35 . [12] M. Gazda & W .J. Fokk ink (201 0): T urning GS OS into equations for lin ear time- branching time semantics . 2nd Y oung Researcher s W orksho p on Conc urrency Theory - YR-CONCUR’10, P aris A vailable at ht tp:// www.cs.vu.nl/ ~ wanf/pubs/gsos.pdf . [13] R.J. van Glabbeek (2001): The Linear T ime - Branching T ime Spectrum I. The Semantics of Con cr ete, Se- quential Pr ocesses . In A. Po nse S.A. Smolka J.A. Bergstra, e ditor: Handbo ok of Proc ess Algebr a , Else vier, pp. 3–99 . [14] C.A.R. Ho are (1985): Communicating Sequential Pr ocesses . Pr entice-Hall Internationa l, Englew ood Clif fs. [15] Beng t Jonsson & Joachim Parrow , editors (19 94): CONCUR ’94 , Con curr ency Theory , 5th Interna tional Confer ence, Uppsala, Sweden, August 22- 25, 1994, Pr oceeding s . Lecture Notes in Compu ter Science 836, Springer . [16] R. Miln er (1989 ): Communica tion and C oncu rr e ncy . Prentice-Hall Intern ational, Englew ood Clif fs. [17] D. Park (1981) : Conc urr ency and automata on infinite sequen ces . In P . Deussen, editor: 5th GI Confer ence, Karlsruhe, Germany, Lecture Notes in Compu ter Science 104, Springer-V erlag, p p. 167–1 83, doi: 10.1007/ BFb001 7309 . [18] Ir ek Ulidowski (2000): F inite ax iom systems for testing preor der an d De Simo ne pr o cess lang uages . Theor . Comput. Sci. 239(1 ), pp. 97–139, doi: 10.1016/S0304- 39 75(99 )00214- 5 . [19] Chr is V erhoef (1995 ): A Congruence The or em for Structu r ed Operationa l Semantics with P r edicates an d Ne gative Pr emises . Nordic Journal on Computing 2(2), pp. 274 –302.