transition relation needs to e inductively dened according to that structure. This is the case of formalisms like Petri nets [30], where a state is a

Electronic Notes in Theoretical Computer Science 4 (1996) Tiles, Rewriting Rules and CCS 1 Faio Gadducci and Ugo Montanari Dipartimento di Informatica Universita di Pisa Corso Italia 40, 56125 Pisa, Italy (fgadducci,ugog@di.unipi.it) Astract In [12] we introduced the tile model, a framework encompassing a wide class of computational systems, whose ehaviour can e descried y certain rewriting rules. We gathered our inspiration oth from the world of term rewriting and of concurrency theory, and our formalism recollects many properties of these sources. For example, it provides a compositional way to descrie oth the states and the sequences of transitions performed y a given system, stressing their distriuted nature. Moreover, a suitale notion of typed proof allows to take into account also those formalisms relying on the notions of synchronization and side-eects to determine the actual ehaviour of a system. In this work we narrow our scope, presenting a restricted version of our tile model and focussing our attention on its expressive power. To this aim, we recall the asic denitions of the process algeras paradigm [3,24], centering the paper on the recasting of this framework in our formalism. 1 Introduction It is not an overstatement to say that, in recent years, there has een an unprecedented ow of proposals, aiming at methodologies to descrie the semantics of rule-ased computational systems. Widely spread in the eld of concurrency theory, transition systems [16] oered a useful tool for recovering suitale descriptions. They are roughly dened as a set of states, representing e.g. the possile memory contents, and a transition relation over states, where each element hs; ti denotes the evolution from the state s to the state t. Due to its simplicity, however, this view is clearly no more adequate when we need to take into account a compositional structure over states, and the 1 Research supported in part y Progetto Speciale del CNR \Strumenti per la Specica e la Verica di Proprieta Critiche di Sistemi Concorrenti e Distriuiti"; and in part y the Information Technology Promotion Agency, Japan, as a part of the Industrial Science and Technology Frontier Program \New Models for Software Architecture" sponsored y NEDO (New Energy and Industrial Technology Development Organization). c 1996 Elsevier Science B. V.

transition relation needs to e inductively dened according to that structure. This is the case of formalisms like Petri nets [30], where a state is a multiset of asic components, and each of them may evolve simultaneously (i.e., in parallel); or term rewriting systems [17], where states are terms of a given algera, and rewrites are freely otained from a set of deduction rules. Furthermore, we may need to consider formalisms relying on the use of synchronization and side-eects in determining the actual ehaviour of a system. Maye, the most important reakthrough is represented y the so-called sos approach [28]: states are compositionally descried as terms of a suitale algera, whose operators express asic features of a system, and the transition relation is dened y means of inference rules, guided y the structure of the states. Along this line, further extensions, which proved fruitful for our view, are context systems [22], where the transition relation is dened not on states ut on contexts, each of them descriing a partially unspecied component of a system; and structured transition systems [9,6], where, in order to give a faithful account of the spatial distriution of a system, also transitions are equipped with an algeraic structure. In [12] we introduced the tile model, as an attempt to encompass the properties of the already mentioned formalisms. As it happened for rewriting logic [23], the underlying idea of the tile model is to take a logical viewpoint, regarding a rule-ased system R as a logical theory, and any transition step making use of rules in R as a sequent entailed y the theory. The entailment relation is dened inductively y a set of inference rules, expressing asic features of the model, like its compositional and spatial properties. In particular, there are three composition rules. First, they allow dierent components of a system to act simultaneously, explicitly descriing parallelism y a monoidal structure over transitions. Moreover, the compositional structure of states is reected on computations: su-components may synchronize and, according to their action, e contextualized. Finally, they can e sequentially composed, expressing in this way the execution of a sequence of transitions. a A sequent : s t is a tuple where s! t is a rewrite step, is a proof term (representing the structure of the step), a is the trigger of the step, and is its eect. Its intuitive meaning is: the context s is rewritten to the context t, producing an eect, ut the rule can e applied only if the variales of s (representing still unspecied su-components) are rewritten with a cumulative eect a. Moreover, two sequents ; can e composed in parallel (), composed sequentially () or contextualized (), varying accordingly the corresponding source, target, trigger and eect. Proof terms allow us to equip each rewriting step with a suitale encoding of its causes, while the fact that sequents carry information also aout the eect of the associated computation expresses certain restrictions aout the class of sequents a given rule can e applied to. Alternatively, a sequent can e considered as synchronized to its context via its trigger and eect components, and the possiility of expressing restrictions and synchronization will e fundamental when applying our 2

paradigm to the operational description of distriuted systems. The tile model also admits a sosticated characterization y means of doule-categories [19], structures that may e roughly descried as the superposition of a vertical and a horizontal category. In [12] it is shown that, starting from a rewriting system, with a free contruction (y means of a suitale adjunction) a doule-category can e otained whose \arrows" are in a one-to-one correspondence with the sequents entailed y that system. This result generalizes the analogous property for term rewriting [29,5], and it underlines the wide applicaility of our model [11]. Along this line, in this paper we decided to narrow our attention: instead of descriing in full details our formalism, for which we refer the interested reader to [12], we aim at analyzing its expressive power. The focus of the paper, then, is the recasting of the process algeras paradigm [3,14,24] in our model, which can e considered as a real enchmark for any general framework (se e.g. [26]). In particular, we deal with a suitale case study, the Calculus of Communicating Systems (also ccs, [24]), considered as the standard representative of the paradigm. ccs oers a constructive way to descrie concurrent systems, considered as structured entities (the agents) interacting y means of some synchronization mechanism. Each system is then dened as a term of an algera over a set of process constructors: new systems are uilt from existing ones, on the assumption that algeraic operators represent asic features of a concurrent system. The structure over agents allows for an immediate denition of the operational semantics of the language y means of the sos approach: the dynamic ahaviour of an agent is then descried y a suitale laelled transition system, where each transition step is a triple hs; ; ti, with the oservation associated to the transition itself. Finally, a further astraction is otained with the associated notion of isimulation: an equivalence over agents equating those with the same oservale ehaviour. The paper has the following structure. In Section 2.1 we introduce a formalization of term algeras, providing a concrete description which underlines the assumptions implicitly made in the ordinary notion. In Section 2.2 we introduce our rewriting systems, equipping them with a logic that descries the classes of derivations entailed y a system using (possily astract) sequents. In Section 3 we recall the asic denitions of ccs, its operational semantics and the associated strong isimulation equivalence, along with its nite axiomatization. Finally, in Section 4 we show how the process algeras paradigm can e recovered in our framework. In particular, in Section 4.1 we descrie a rewriting system which faithfully recovers the ordinary sos semantics of ccs; in Section 4.2 we introduce the notion of tile isimulation, in order to recast a suitale notion of oservational equivalence in our formalism: this enales us to recover also ccs isimilarity; nally, in Section 4.3 we turn the nite axiomatization of isimilarity in a conuent rewriting system, providing each class of isimilar agents with a canonical representative. 3

2 A Summary of the Tile Model In this section we descrie the asic features of the tile model, within a presentation iased towards the process algeras framework we deal with in Sections 3 and 4. For a comprehensive introduction we refer the reader to [12]. 2.1 Building States We open this section recalling some denitions from graph theory, that will e used to introduce algeraic theories [21,18]. Developed in the early Sixties, these theories received a lot of attention during the Seventies from computer scientists as a suitale characterization of the ordinary notion of term algera. Denition 2.1 (graphs). A graph G is a 4-tuple ho G ; A G ; 0 ; 1 i: O G, A G are sets whose elements are called respectively ojects and arrows (ranged over y a; ; : : : and f; g; : : :), and 0 ; 1 : A G! O G are functions, called respectively source and target. A graph G is reexive when equipped with an identity function id : O G! A G such that 0 (id(a)) = 1 (id(a)) = a for all a 2 O G ; it is with pairing if its class O G of ojects forms a monoid; it is monoidal if it is reexive with pairing and also its class of arrows forms a monoid, such that, if 0 is the neutral element of O G, then id(0) is the neutral element of A G. 2 We can think of a signature as a graph, whose nodes are (underlined) natural numers, and its arcs are univocally laeled y an operator, such that f : n! 1 i f 2 n. The usual notion of term can e formalized along this intuition, which allows to recover also alternative structures. Denition 2.2 (graph theories). Given a signature, the associated graph theory G() is the monoidal graph with ojects the elements of the commutative monoid (IlN; ; 0) of underlined natural numers (where 0 is the neutral oject and the sum is dened as n m = n + m); and arrows those generated y the following inference rules: f : n! 1 2 s : n! m; t : n 0! m 0 (generators) (sum) f : n! 1 2 G() s t : n n 0! m m 0 (identities) n 2 IlN id n : n! n satisfying the monoidality axiom id nm = id n id m for all n; m 2 IlN. 2 Identities could e given just for 0; 1, using the monoidality axiom to dene inductively the operator for all the ojects, so otaining a nitary presentation of the theories. The solution we chose is equivalent, yet easier to descrie, and it is used for all the auxiliary operators introduced in the next denitions. Denition 2.3 (monoidal theories). Given a signature, the associated monoidal theory M() is the monoidal graph with ojects the elements of the commutative monoid (IlN; ; 0) of underlined natural numers and arrows 4

those generated y the following inference rules: f : n! 1 2 s : n! m; t : n 0! m 0 (generators) (sum) f : n! 1 2 M() s t : n n 0! m m 0 n 2 IlN (identities) (composition) s : n! m; t : m! k id n : n! n s; t : n! k Moreover, the composition operator ; is associative, and the monoid of arrows satises the functoriality axiom (s t); (s 0 t 0 ) = (s; s 0 ) (t; t 0 ) (whenever oth sides are dened); the identity axiom id n ; s = s = s; id m for all s : n! m; and the monoidality axiom id nm = id n id m for all n; m 2 IlN. 2 Further enriching the auxiliary structure, we are nally ale to present the more expressive kind of theories we deal with in our paper, algeraic theories. Denition 2.4 (algeraic theories). Given a signature, the associated algeraic theory A() is the monoidal graph with ojects the elements of the commutative monoid (IlN; ; 0) of underlined natural numers and arrows those generated y the following inference rules: f : n! 1 2 s : n! m; t : n 0! m 0 (generators) (sum) f : n! 1 2 S() s t : n n 0! m m 0 (identities) (duplicators) n 2 IlN id n : n! n n 2 IlN r n : n! n n (permutation) (composition) s : n! m; t : m! k s; t : n! k (dischargers) n; m 2 IlN n;m : n m! m n n 2 IlN! n : n! 0 Moreover, the composition operator ; is associative, and the monoid of arrows satises the functoriality axiom (s t); (s 0 t 0 ) = (s; s 0 ) (t; t 0 ) (whenever oth sides are dened); the identity axiom id n ; s = s = s; id m for all s : n! m; the monoidality axioms id nm = id n id m nm;p = (id n m;p ); ( n;p id m )! nm =! n! m r nm = (r n r m ); (id n n;m id m )! 0 = r 0 = 0;0 = id 0 0;n = n;0 = id n for all n; m; p 2 IlN; the coherence axioms r n ; (id n r n ) = r n ; (r n id n ) r n ; (id n! n ) = id n for all n; m 2 IlN; and the naturality axioms (s t); m;q = n;p ; (t s) r n ; n;n = r n n;m ; m;n = id n id m s;! m =! n s; r m = r n ; (s s) for all s : n! m; t : p! q. 2 5

As for identities, also permutation and the other auxiliary operators could e inductively extended to all n 2 IlN starting from the asic cases, interpreting in a constructive way the monoidality axioms. Let us consider the signature = S 2 i=0 i, where 0 = fa; g, 1 = ff; gg and 2 = fhg (that same signature is also used in the following sections). Some of the elements in A( ) are a ; f : 0! 1, f ; g : 1! 1, a ; r 1 ; (f id 1 ); h : 0! 1, intuitively corresponding to the terms f(a); g(f(x)) and h(f(a); a), respectively, for a given variale x. In fact, a classical result we already anticipated proves that algeraic theories are equivalent to the ordinary construction (as it can e found e.g. in [2]) for term algeras. Proposition 2.5 (algeraic theories and term algeras). Let e a signature. Then for all n; m 2 IlN there exists a one-to-one correspondence etween the set of arrows from n to m of A() and the m-tuples of elements of the term algera {over a set of n variales{ associated to. 2 The previous result states that each arrow t : n! 1 identies an element t of the term algera over the set fx 1 ; : : : ; x n g: an arrow n! m is an m-tuple of such elements, and arrow composition is term sustitution. Note that this correspondence requires that r and! are natural: if this were not the case, we get s-monoidal theories [10,12]. In these more concrete structures, such elements as a ; r 1 ; h and (a a ); h, that intuitively represent the same term h(a; a), are dierent. In fact, in the PhD thesis [10] of the rst author it is shown that a fundamental property of correspondence holds etween s- monoidal theories and term graphs (as dened e.g. in the introductory chapter of [8]): each arrow t : n! m identies a term graph t over with a specied m-tuple of roots and a specied n-tuple of variales nodes, and arrow composition is graph replacement. The incremental description of algeraic theories has received little attention in the literature (see [15,20]), despite the relevant fact that, dierently from the usual categorical construction, all the elements of the class A() are inductively dened, making a much handier tool to deal with. In fact, the relevant point for our discussion is that, although their denitions are more involved than the classical, set-theoretical ones, algeraic theories allow for a characterization of terms which is far more general, and at the same time more concrete, than the one allowed y the usual formalization of the elements of a term algera, separating in a etter way the \-structure" from the additional algeraic structure that the meta-operators used in the ordinary description (like sustitution) implicitly enjoy. In this view,! and r represent respectively garage collection and sharing (as discussed in [7,5]). As an example, let us consider the constant a: as a generator, the corresponding arrow is a : 0! 1, while, when considered as an element of the term algera over fx 1 ; x 2 g, the associated arrow is! 2 ; a : 2! 1, where! 2 intuitively corresponds to the garaging of the two variales. Also the dierence etween a ; r 1 ; h and (a a ); h has a similar justication: in the rst element, the a is shared; 6

in the latter, it is not. For our purposes, the dierence etween shared and unshared { discharged and undischarged { suterms does not play a relevant part, while instead s-monoidal theories hold a fundamental role in [12], when dealing with truly concurrent semantics in the setting of process algeras. 2.2 Descriing Systems In this section we recall the asic formulation of our framework, inspired oth from the rewriting logic approach y Meseguer [23] and the sos approach y Plotkin [28]. Intuitively, an algeraic rewriting system is just a set of rules, each of them carrying information (i.e., expressing some conditions) on the possile ehaviours of the terms to which they can e applied. Denition 2.6 (algeraic rewriting systems). An algeraic rewriting system (ars) R is a tuple h ; ; N; Ri, where ; are signatures, N is a set of rule names and R is a function R : N! A( )G( )G( )A( ) such that for all d 2 N, if R(d) = hl; a; ; ri, then l : n! m; r : p! q i a a : n! p; : m! q. We usually write d : l r. 2 A context system [22] is just a very simple ars, where R : N! G( ) G( ), with the further restriction that a : 1! 1 for all a 2 (hence, for all d 2 N, if R(d) = hl; a; ; ri then l; r have the same source and target). Term rewriting systems [17], instead, are given y a pair h; Ri where is an ordinary signature, and R is a set of rules, i.e., of pairs hl; ri for l; r elements of the term algera over. Hence, thanks to Proposition 2.5, they are just a very particular case of algeraic rewriting systems, where is empty: in the following, we will refer to these systems as horizontal rewriting systems (also hrs's), and a rule will e simply denoted as d : l?! r. In fact, an hrs is what is called an unconditional rewriting theory in [23]. It is actually less general, since in this paper we decided to consider rewriting systems uilt over signatures instead that over equational theories (; E), even if the extension of ars's to deal with them is quite straightforward. Let us consider the signatures (already introduced) and = 1, where 1 = fu; v; wg. Our running example will e the algeraic rewriting system R e = h ; ; N e ; R e i, where the function R e is descried y R e = fd : a u u ; d 1 : f v g; d 2 : f w v uv f; d 3 : h w! 1 gg. (where is a shorthand for the identitiy arrow id 0 2 M( )). The intuitive meaning of the rules is: for d, the element a can e rewritten to, producing an eect u; for d 1, f can e rewritten to g, producing an eect v, whenever there exists a suitale rewriting with an eect u. Or, in the ordinary term rewriting view: the term f(x) is rewritten to g(x), producing an eect v, ut the rule can e applied only if the suterm associated to x is rewritten with an eect u; and so on for the other rules. 7

An ars R can e considered as a logical theory, and any rewriting {using rules in R{ as a sequent entailed y the theory. An algeraic sequent is then a 5-tuple h; s; a; ; ti, where s! t is a rewrite step, is a proof term (encoding of the causes of the step), a and are respectively the input and output conditions, the actions associated to the rewrite. We say that s rewrites to t via (using a trigger a and producing an eect ) if we otain the sequent a : s t y nitely many applications of a set of inference rules. Denition 2.7 (algeraic tile logic). Let R = h ; ; N; Ri e an ars. We say that R entails the class R of the algeraic sequents : s a t otained y nitely many applications of the following inference rules: asic rules (generators) d : s a t 2 R d : s a t 2 R (h-re) s : n! m 2 A( ) id n id s a : n! m 2 M( ) (v-re) id : s s id m 2 R a a : id n a id m 2 R ; composition rules (p-comp) : s a t; 0 : s 0 a 0 t 0 2 R 0 0 : s s 0 aa 0 t t 0 2 R 0 (h-comp) : s a c t; 0 : s 0 c t 0 2 R 0 : s; s 0 a t; t 0 2 R (v-comp) : s a u; 0 a : u 0 t 2 R 0 ; 0 a;a : s 0 t 2 R ; 0 auxiliary rules (perm) a : n! m; : n0! m 0 2 M( ) a; : a n;n 0 a m;m 0 2 R a : n! m 2 M( ) (dupl) (disch) a : n! m 2 M( ) a r a : r n aa r m 2 R! a :! a n id 0! m 2 R : 2 The dierent sets of rules are self-explaining. Basic rules provide the generators of the sequents, together with suitale identity arrows, whose intuitive meaning is that an element of A( ) or M( ) can e rewritten to itself (showing no eectusing no trigger, so to say). Composition rules provide all the possile ways in which sequents can e composed, while auxiliary rules are the counterpart of the auxiliary operators for algeraic theories. 8

Gadducci and Montanari Let us consider the ars R e we previously dened. It entails the sequent d : a u u d 1 : f v g (h-comp:) d d 1 : a; f v ; g d 2 : f w v f (h-comp:) (d d 1 ) d 2 : a; f; f w ; g; f where is a shorthand for id 0 and the entailment is descried in a natural deduction style. It also entails the sequent d : a u d 1 u : f v g d : a u d d 1 : a; f v ; g d (d d 1 ) : a (a; f) uv (; g) d 3 : h uv w! 1 g (d (d d 1 )) d 3 : (a (a; f)); h w ; g; g where the entailment is still descried in a natural deduction style, ut without using the rule names. The class R is too concrete, in the sense that many sequents that intuitively should represent the same rewrite have a dierent representation. An equivalence over sequents can then e considered as a way to astract away from implementation details, identifying computationally equivalent derivations. Denition 2.8 (astract algeraic sequents). Let R = h ; ; N; Ri e an ars. We say that it entails the class R E of astract algeraic sequents: equivalence classes of algeraic sequents entailed y R modulo the set E of axioms, which are intended to apply to the corresponding proof terms. The set E contains three associativity axioms, stating that all the composition operators are associative; the functoriality axioms ( ) ( ) = ( ) ( ) ( ) ( ) = ( ) ( ) ( ) ( ) = ( ) ( ) (satised whenever oth sides are dened); the identity axioms id s = = a id t and id = = id a for all : s t; the monoidality axioms id st = id s id t id a = id a id id s;t = id s id t id a; = id a id id id0 = = id id0 a;c = (id a ;c ) ( a;c id )! a =! a! r a = (r a r ) (id a a; id )! id0 = r id0 = id0 ;id 0 = id id0 id0 ;a = a;id0 = a for all 2 R, s; t 2 A( ) and a; ; c 2 M( ); the coherence axioms r a (id a r a ) = r a (r a id a ) r a a;a = r a r a (id a! a ) = id a a; ;a = id a id for all a; 2 M( ); and the naturality axioms ( 0 ) ; 0 = a;a 0 ( 0 )! =! a r = r a ( ) 9

a for all : s t; 0 : s 0 0 Gadducci and Montanari a 0 t 0 2 R. 2 This axiomatization properly extends the one given for unconditional rewriting logic in [23]. Note also that, as already happened for the theories of Section 2.1, even in this case we could have inductively dened identities, permutations and the other auxiliary operators starting from the asic cases, interpreting in a constructive way the monoidality axioms. As an example, if we consider the ars R e, from identity and monoidality axioms we have that hence the entailed proof terms d (d d 1 ) = (d d) (id u d 1 ) (d (d d 1 )) d 3 ((d d) (id u d 1 )) d 3 are equivalent, even if the latter has a derivation unrelated to the one already shown for (d (d d 1 )) d 3. 3 Operational Semantics for CCS It is quite common in concurrency theory to deal with formalisms relying on the notion of side-eects and synchronization in determining the actual ehaviour of a system, features wich are quite dicult to recast in frameworks like (classical) term rewriting. Process (Description) Algeras [3,14,24] oer a constructive way to descrie concurrent systems, considered as structured entities (the agents) interacting y means of some synchronization mechanism. They dene each system as a term of an algera over a set of process constructors, uilding new systems from existing ones, on the assumption that algeraic operators represent asic features of a concurrent system. We present here one of the etter known example of process algera, the Calculus of Communicating Systems (ccs), introduced y Milner in the early Eighties [24], restricting ourselves, for the sake of exposition, to the case of nite ccs. Denition 3.1 (the Calculus of Communicating Systems). Let Act e a set of atomic actions, ranged over y, with a distinguished symol and equipped with an involutive function such that =. Moreover, let ; ; : : : range over Actnfg. A ccs process (also agent) is a term generated y the following syntax P ::= nil; :P; P n ; P []; P 1 + P 2 ; P 1 jjp 2 where : Act! Act is a relaeling function, preserving involution and. Usually, we let P; Q; R; : : : range over the set Proc of processes. 2 In the following, we indicate as ccs the signature associated with ccs processes (for example, nil is a constant, a unary operator for each element in Act, and so on...). Given a process P, its dynamic ehaviour can e descried y a suitale transition system, along the lines of the sos approach, where the transition relation is freely generated from a set of inference rules. 10

Denition 3.2 (operational semantics of CCS. The ccs transition system is the relation T ccs P roc Act P roc inductively generated from the following set of axioms and inference rules?! Q where P :P?! for P 2 Act P P?! Q P jjr?! QjjR P []?! () Q[] P?! Q for 62 f; g P n?! Qn P?! Q P + R?! Q P P?! Q R + P?! Q?! Q; P 0?! Q 0 P P jjp 0?! QjjQ 0 RjjP for relaeling?! Q?! RjjQ?! Q means that hp; ; Qi 2 T ccs. 2 A process P can execute an action and ecome Q if we can inductively construct a sequence of rule applications, such that the transition hp; ; Qi 2 T ccs. As an example, to infer that from P = (:nil + :nil)jj:nil we can deduct P?! Q = niljj:nil, three dierent rules must e applied. Moreover, a process P can e rewritten into Q if there exists a computation from P to Q, i.e., a chain P = 1 n P 0?! P1 : : : P n?1?! Pn = Q of one-step reductions. The operational semantics we just dened is however too intensional, and more astract semantics have een introduced y dening suitale ehavioural equivalences, which identify processes exhiiting the same oservational ehaviour. Most of them are dened on the asic notion of isimulation [27]: intuitively, two processes P; Q are isimilar if, whenever P performs an action evolving to a state P 0, then also Q may execute that same action, evolving to a state Q 0 which is still isimilar to P 0. Denition 3.3 (isimulation equivalence). A symmetric equivalence relation P roc P roc is a isimulation if, whenever P Q for generic P; Q processes, then for any transition P?! P 0 there exists a corresponding transition Q?! Q 0 with Q Q 0. The maximal isimulation equivalence is called strong isimulation, and denoted y. 2 It is well-known that strong isimilarity for ccs is also a congruence, and that it can e descried y an equational theory over ccs [24]. In [3], the authors dened a nitary equational theory for an oservational equivalence over their Algera of Communicating Processes, introducing auxiliary operators. An ovious extension of their formalism can e adapted to get a nite description of strong isimilarity for ccs, introducing three auxiliary operators, which intuitively split the parallel operator into three distinct cases, corresponding to left, right and synchronous composition of the su-agents. On the other hand, Moller [25] has proved that isimilarity cannot e nitely axiomatized without resorting to auxiliary operators. So, let eccs e the signature otained extending ccs with the operators f; c; j : 2! 1g. 11

Denition 3.4 (B-K axioms). Let P; Q e eccs processes. The Bergstra- Klop (also B-K) axiomatization is given y the following axioms for the parallel, relaelling and restriction operators (:P )j( 0 :Q) = P jjq = ((P Q) + (P cq)) + (P jq); (:P )Q = P c(:q) = :(P jjq); 8 >< (:P )n = :(P jjq) if 0 = and 6=, >: nil 8 >< >: nil otherwise; :(P n ) if 62 f; g, otherwise; (:P )[] = ():(P []); nilp = P cnil = niljp = P jnil = niln = nil[] = nil; extended with the Hennessy-Milner (also H-M) axioms for the choice operator P + P = P + nil = P P + Q = Q + P (P + Q) + R = P + (Q + R): We usually write P BK Q if P and Q are in the same equivalence class with respect to the B-K axioms. 2 The H-M axioms simply state the associativity, commutativity, identity and idempotency of the non-deterministic operator (see [13]). The importance of the B-K axioms is given y their soundness and completness with respect to the isimulation equivalence, as stated in the following result. >From our point of view, however, equally relevant is the fact that these axioms can e easily turned into rewriting rules, otaining a conuent rewriting system, that identies isimilar agents: more on this in the next section. Proposition 3.5 (B-K axioms and strong isimulation). Let P; Q e ccs processes. Then P Q i P BK Q. 2 4 Operational Semantics from Rewriting Systems In this section we show how the ccs operational semantics can e recovered y suitale rewriting systems. In particular, in Section 4.1 we dene an algeraic rewriting system R ccs which faithfully corresponds to the ccs transition system T ccs. Then, in Section 4.2 we dene the notion of tile isimulation, roughly identifying sequents with the same eect: when applied to the sequents entailed y R ccs, it provides a recasting of strong isimilarity for ccs processes. Finally, in Section 4.3 we descrie a horizontal rewriting system R BK, that derives, for each element of a class of isimilar ccs processes, a canonical representative of the class itself. 4.1 Using Tiles for CCS As shown in the previous section, from an operational point of view a process algera can e faithfully descried y a triple h; A; Ri, where is the 12

signature of the algera of agents, A is the set of actions and R is the set of deduction rules. Note that these rules are conditional: you need information on the action performed y the transitions in the premise, efore applying a rule. Moreover, the rewriting steps are always performed on top: the order in which the rewrites are actually executed is important since, as an example, the correct operational ehaviour of the agent P = ::nil is expressed saying that it executes rst and then. If we let A ccs e the signature containing all the atomic actions of Act (i.e., A ccs = f : 1! 1 j 2 Actg), then oth those features are easily descried in the framework of tile logic. Denition 4.1 (the CCS rewriting system). The ars R ccs associated to ccs is the tuple h ccs ; A ccs ; N; Ri, with the following set of rules: id act : 1 id1 rel : () res : n n for 62 f; g id 1 id id 1! 1 +i : + 1! 1 id 1 h+ : + id l : jj 1 jj r : jj id 1 jj : jj jj s (where we omitted the suscripts for the sake of readiility). 2 Note that there is exactly one asic rule for each operational rule of ccs; some of them (such as act and rel ) are parametric with respect to the set of actions or to the set of relaeling functions, since the corresponding rules are so. The eect indicates that the process is actually \running", outputting the action. For example, the rule act prexes an idle process with the action, and then starts the execution, consuming that same action. There are also three rules dealing with the parallel operator: s synchronizes two running processes, while l and r perform an asynchronous move, taking a running and an idle process. As an example of sequent construction, let us consider again the process P = ::nil, executing sequentially rst the action, then the action. The computation is represented y the sequent (id nil; act ) (id nil act ) : nil; ; ; nil whose two-steps entailment is the following id nil; : nil; nil; act : id 1 id nil; act : nil; ; nil; id nil : nil nil act : id 1 id nil act : nil; nil (where is a shorthand for oth id 0 and id 1, since no confusion can arise), showing the importance of eects in expressing the ordering constraints: P can execute only if the underlying process P 0 = :nil is actually idle. 13

For the agent P = ((:nil)n )n, instead, the execution of the action is represented y the sequent ((id nil act ) res ) res, whose entailment is id 1 id nil act : nil; nil res : n n (id nil act ) res : nil; ; n nil; n res : n ((id nil act ) res ) res : nil; ; n ; n nil; n ; n where the asic sequent act has een provided with a suitale context. Note that the axioms impose an equivalence relation over sequents (i.e., over computations), and then oer a description that, even if more concrete than the one given y the set{theoretical relation entailed y a transition system, is still somewhat \astract": there are many derivations that are identied, corresponding to \essentially" equivalent ccs computations. There is however an ovious adequacy result, stated y the following theorem. id nil : nil nil act : Proposition 4.2 (computational correspondence). Let P; Q e ccs agents, and P R ; Q R the associated elements of A( ccs ). Then the transition P?! Q is entailed y the ccs transition system T ccs i an astract algeraic sequent id : 0 P R QR is entailed y R ccs. 2 The correspondence is instead one-to-one if we consider the restriction R p of R ccs over A( ccs ) G(A ccs ) G(A ccs ) A( ccs ): i.e., the relation otained y dropping the proof terms from sequents. Or, equivalently, if we take into account the class of astract sequents modulo the set of axioms E 0, otained adding to E the axiom t : = Proposition 4.3 (interleaving correspondence). Let P; Q e ccs agents, and P R ; Q R the associated elements of A( ccs ). Then the transition P?! Q is entailed y the ccs transition system T ccs (i.e., hp; ; Qi 2 T ccs ) i the astract algeraic sequent (modulo the set of axioms E 0 id ) : 0 P R QR is a : s a t; : s entailed y R ccs (i.e., hp R ; id 0 ; ; Q R i 2 R p ). 2 4.2 Recovering Bisimulation for Tiles It seems quite reasonale that the notion of isimulation could e extended to deal with our framework. In this section we introduce tile isimulation, showing its (intuitive) correspondence with strong isimilarity for ccs processes. Denition 4.4 (tile isimulation). Let R = h ; ; N; Ri e an ars. A symmetric equivalence relation A( ) A( ) is a tile isimulation for R if, whenever s t for generic s; t elements of A( ), then for any astract sequent : s a s 0 entailed y R there exists a corresponding one a : t t 0 with s 0 t 0. The maximal tile isimulation equivalence is called strong tile isimulation, and denoted y st. 2 14 n

This is an ovious generalization of Denition 3.3, due to the more concrete representation of states and the richer structure on eects shown y sequents with respect to ccs transitions. But of course there is a complete coincidence etween isimilarity over ccs processes and tile isimilarity over the corresponding elements of A( ccs ). Proposition 4.5 (isimulation corrispondence). Let P; Q e ccs agents, and P R ; Q R the associated elements of A( ccs ). Then P Q i P R st Q R.2 We need now to develop a concept analogous to congruence. Usually, an equivalence is a congruence whenever it preserves the operators. In our case, this \operator preserving" property can e restated in terms of parallel and horizontal composition. Denition 4.6 (tile functoriality). Let R = h ; ; N; Ri e an ars. A symmetric equivalence relation f A( ) A( ) is functorial for R if, whenever s f t; s 0 f t 0 for generic s; s 0 ; t; t 0 elements of A( ), then s; s 0 f t; t 0 (whenever dened) and s s 0 f t t 0. 2 It is not in general true that a tile isimulation equivalence is also functorial. The following results provide a characterization of such a property in terms of tile decomposition. Denition 4.7 (tile decomposition). Let R e an ars. We say that R veries the (tile) decomposition property if i) whenever it entails an astract sequent : s; t a a u, then it entails also two sequents : s c s 0 and c : t t 0 with u = s 0 ; t 0 ; and ii) whenever it entails an astract sequent : st a a 1 u, then it entails also two sequents : s 1 s 0 a 2 and : t 2 t 0 with u = s 0 t 0, a = a 1 a 2 and = 1 2. 2 A very simple system not verifying the (tile) decomposition property is given y R a = h an ; a ; N a ; R a i, where an = fnil : 0! 1; a : 1! 1g, a = fa 1 : 1! 1; a 2 : 1! 1g and id a R a = fact : nil; a a 1 nil; cons : a 1 a id1 1 g: The asic sequent act cannot e decomposed, while its source oviously can. Proposition 4.8 (decomposition and isimulation). Let R e an ars. If it veries the decomposition property, then the associated strong tile isimulation is functorial. 2 The converse is not true. In fact, the tile isimulation associated to the ars R a is functorial, and it is freely generated from the asic classes fnilg; fid 0 g; fid 1 g; fa; a; a; : : :g = fa n jn 1g, ut the system does not verify the decomposition property. Note also the importance of a 2 2 a, which is responsile for the non-equivalence of id 1 and a: on the contrary, that equivalence would have destroyed functoriality. While it may e dicult to check out if a given rewriting system is \decomposale", the following proposition provides an easy syntactical property that implies decomposition. 15

Gadducci and Montanari Proposition 4.9 (asic source and decomposition). Let R e the ars a h ; ; N; Ri such that, for all d : s t 2 R, s 2 (hence, the source is a asic operator). Then R satises the decomposition property. 2 Proof (sketch). The proof can e carried out in two steps. First, each astract sequent can e decomposed into the vertical composition 1 2 : : : n of \concrete" sequents i such that the operator does not appear in any of them. These kind of sequents are called one-step, and they can e otained without using the v-comp rule. Then, let us suppose that : s a t is one-step. Now, since the source of each rule must e a asic operator, we have that the structure of exactly mirrors the one of its source s. And since also the axioms of algeraic sequents mirror those of algeraic theories, the result holds. 2 In fact, oth R ccs and R e verify this \asic source" property, hence the decomposition one, so that the following corollary holds. Corollary 4.10 (strong isimulation is functorial). The strong tile isimulation st associated to R ccs is functorial. Thanks to Proposition 4.5, this result implies that strong isimilarity for ccs processes is also a congruence. In fact, if an equivalence is functorial it preserves contexts, and a fortiori also operators. As an example, let P; Q e ccs agents, P R ; Q R the associated elements of A( ccs ), and let us assume that P Q. Hence P R st Q R (also Q R st P R y symmetry) and, y functoriality, (P R Q R ); jj st (Q R P R ); jj, so that also P jjq QjjP holds. In general, it should e worthy to identify suitale \formats" for the rules such that, given a rewriting system R, then whenever its rules t a format then R is decomposale. An analogous work has een done on process algeras: see e.g. [1] for more details on the so-called gsos format. For our tile model, some preliminary considerations can e found in [12]. 4.3 B-K Axioms as Rewriting Rules The aim of this section is to show that the axiomatization given in Denition 3.4 can e turned into a horizontal rewriting system, which is adequate for isimilarity, in the sense that, given two ccs processes, they are isimilar i they may evolve to the same element. Denition 4.11 (B-K axioms as rewriting rules). The hrs R BK associated to the B-K axioms is the tuple h eccs ; ;; N BK ; R BK i, with the following set of rules: dec : jj?! r 2 ; ((r 2 ; (c); +) j); + l : ( id 1 );?! jj; r : (id 1 ); c?! jj; : ( ); j?! jj; nl : (nil id 1 ); j?!! 1 ; nil le : (nil id 1 );?!! id1 ; nil re : (id 1 nil); c?!! id1 ; nil e : ( 0 ); j?!! 2 ; nil for 0 6= or = nr : (id 1 nil); j?!! 1 ; nil res : ; n?! n ; for 62 f; g 16

res e : ; n?!! 1 ; nil rel : ;?! ; () together with the rules for for the choice operator res n : nil; n?! nil rel n : nil;?! nil idem : r 1 ; +?! id 1 idnil : (id 1 nil); +?! id 1 comm : +?! 1;1 ; + assoc : (id 1 ; +); +?! (+ id 1 ); + (where we omitted the suscripts for the sake of readiility). With R BBK we denote the hrs otained without the rules for the choice operator; with R I the one containing only the rules idem and idnil and nally with R AC the one containing only the rules comm and assoc. 2 The system we dened is convergetnt ut not terminating: it is well-known that the axioms for associativity and commutativity of an operator cannot e in general turned into terminating rules. In fact, it is easy to see that +?! 1;1 ; +?! 1;1 ; 1;1 ; + = +?! 1;1 ; +?! : : : However, let sccs = fnil; ; +g ccs e the signature of sequential ccs (sccs) processes: next result shows that R BK is still adequate with respect to strong isimulation. Proposition 4.12 (isimulation as normal form, I). Let P; Q e eccs processes. Then P BK Q i there exists a sccs process S such that R BK entails two sequents : P?! S and : Q?! S. 2 Notice that the normalization procedure is totally orthogonal to the usual notion of transition in the sos framework. In fact, let us consider the ccs processes P = ::nil and Q = ((:nil)n )n : the associated computations evolving from them have een shown in the previous section. Note instead that, from a normalization point of view, P cannot move. Instead, Q sequentially executes two dierent res operations (one causally dependent from the other), and nally it evolves to :nil, as shown y the following sequents id nil : nil?! nil res : ; n?! n ; id nil res : nil; ; n?! nil; n ; id n : n?! n (id nil res ) id n : nil; ; n ; n?! nil; n ; ; n id nil;n : nil; n?! nil; n res : ; n?! n ; id nil;n res : nil; n ; ; n?! nil; n ; n ; res e : nil; n?! nil id n ; : n ;?! n ; res e id n ; : nil; n ; n ;?! nil; n ; res e : nil; n?! nil id :?! res e id : nil; n ;?! nil; such that, y the axioms of Denition 2.8, we have ((id nil res ) id n ) (res e res ) (res e id ) : nil; ; n ; n?! nil; : 17

The aove denotation of the astract sequent in the example suggests a reduction where two steps are executed in parallel. Proposition 4.12 also has a stronger formulation, which is stated y the following result. Proposition 4.13 (isimulation as normal form, II). Let P; Q e eccs processes. Then P BK Q i there exists a sccs process S such that the sequents : P?! BBK P 0?! AC P 00?! I S and : Q?! BBK Q 0?! AC Q 00?! I S are entailed y R BK (where P?! BBK P 0 means that it is entailed y R BBK, and so on). 2 Since oth R BBK and R I are terminating, then, if we considered an equational extension of our tile model, each class of isimilar processes would e provided with a normal form, modulo associativity and commutativity. References [1] L. Aceto, B. Bloom, F.W. Vaandrager, Turning SOS Rules into Equations, Information and Computation 111 (1), pp. 1-52. [2] J.A. Goguen, J.W. Tatcher, E.G. Wagner, J.R. Wright, Initial Algera Semantics and Continuous Algeras, Journal of the ACM 24 (1), 1977, pp. 68-95. [3] J.A. Bergstra, J.W. Klop, Process Algera for Synchronous Communication, Information and Computation 60, 1984, pp. 109-137. [4] A. Corradini, F. Gadducci, CPO Models for Innite Term Rewriting, in Proc. AMAST'95, LNCS 936, 1995, pp. 368{384. [5] A. Corradini, F. Gadducci, U. Montanari, Relating Two Categorical Models of Concurrency, in Proc. RTA'95, LNCS 914, 1995, pp. 225{240. [6] A. Corradini, U. Montanari, An Algeraic Semantics for Structured Transition Systems and its Application to Logic Programs, Theoretical Computer Science 103, 1992, pp. 51-106. [7] A. Corradini, Term Rewriting, in Parallel, sumitted. [8] M.C.J.D. van Eekelen, M.J. Plasmeijer, M.R. Sleep, Term Graph Rewriting, Theory and Practice, John Wiley & Sons, 1993. [9] G. Ferrari, U. Montanari, Towards the Unication of Models for Concurrency, in Proc. CAAP'90, LNCS 431, 1990, pp. 162-176. [10] F. Gadducci, On the Algeraic Approach to Concurrent Term Rewriting, PhD Thesis, Universita di Pisa, Pisa. Technical Report TD-96-02, Department of Computer Science, University of Pisa, 1996. [11] F. Gadducci, U. Montanari, Enriched Categories as Models of Computations, in Proc. Fifth Italian Conference on Theoretical Computer Science, ICTCS'95, World Scientic, 1996, pp. 1-24. 18

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