To this purpose, ordinary transition systems have een extended in the literature in several ways. From the operational point of view, certain commutin

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1 Electronic Notes in Theoretical Computer Science 7 (1997) URL: 26 pages Tiles for Concurrent and Located Calculi? GianLuigi Ferrari Dipartimento di Informatica, Universit di Pisa, giangi@di.unipi.it Ugo Montanari Computer Science Laoratory, SRI International, Menlo Park, ugo@csl.sri.com Astract When concurrency is a primitive notion, models of process calculi usually include commuting diamonds and oservations of causality links or of astract locations. However it is still deatale if the existing approaches are natural, or rather if they are an ad hoc addition to the more asic interleaving semantics. In the paper a treatment of concurrent process calculi is proposed where the same operational and astract concurrent semantics descried in the literature now descend from general, uniform notions. More precisely we introduce a tile-ased semantics for located CCS and we show it consistent with the ordinary concurrent (via permutation of transitions) and isimilarity ased location semantics. Tiles are rewrite rules with side eects, reminiscent of oth Plotkin SOS and Meseguer rewriting logic rules. We argue that the tile model is particularly well suited for dening directly operational and astract semantics of concurrent process calculi in a compositional style. 1 Introduction Process calculi are usually equipped with notions of operational semantics ased on transition systems and of astract semantics ased on oserved actions and isimilarity. Sometimes it is convenient to consider concurrency as a primitive notion, rather than to reduce it to nondeterminism via interleaving.? Research supported y Oce of Naval Research Contracts N C-0225 and N C-0114 and y the Information Technology Promotion Agency, Japan, as part of the Industrial Science and Technology Frontier Program New Models for Software Architechture sponsored y NEDO (New Energy and Industrial Technology Development Organization). Also research supported in part y CNR Integrated Project Metodi e Strumenti per la Progettazione e la Verica di Sistemi Eterogenei Connessi mediante Reti di Comunicazione and Esprit Working Groups CONFER2 and COORDINA. The second author is on leave from Dipartimento di Informatica, Pisa, Italy. c1997 Pulished y Elsevier Science B. V. Open access under CC BY-NC-ND license.

2 To this purpose, ordinary transition systems have een extended in the literature in several ways. From the operational point of view, certain commuting diamonds are introduced (see e.g. [16,5]), whose role is to dene as concurrent those pairs of events which can occur in any order. Concurrent astract semantics is dened instead y decorating actions with causality links or with astract locations and possily y introducing specialized versions of isimulation [13,12,4,22,10,30,28]. However, while concurrent semantics of process calculi has een given a remarkale attention in the past several years, it is still deatale if the existing approaches are natural, or rather if they are an ad hoc addition to the more asic interleaving semantics. We elieve a more natural treatment of concurrency should e possile, as we feel has een achieved (at least from an operational point of view) for other models of computations, like Petri nets [14] and term [26,23], graph [7], and term graph [9] rewriting, where axioms generating commuting diamonds are automatically imposed y the framework of denition. The aim of this paper is to propose a treatment of concurrent process calculi where the same operational and astract concurrent semantics descried in the literature now descend from general, uniform notions. Our approach is ased on the tile model [1820]. The tile model relies on certain rewrite rules with side eects, called tiles, reminiscent of oth SOS rules [31] and rewriting logic rules [26]. Also related models 1 are SOS contexts [24] and structured transition systems [11]. Tiles have een used for coordination formalisms equipped with exile synchronization primitives [29,6] and for calculi for moile processes, like the asynchronous -calculus [17]. The main advantage of the tile model for handling concurrent process calculi is to integrate a distriuted representation of agents and a partial order representation of oservations within an SOS-like compositional framework. In particular, with respect to the location approach of [4] the tile version has the advantage of employing only local names and of avoiding innite ranching. Tiles are naturally equipped with a isimulationased equivalence relation, which yields the correct notion of process isimilarity. We now riey introduce the tile model. A tile has the form: s a! s 0 and states that the initial conguration s of the system evolves to the - nal conguration s producing an 0 eect. However s is in general open (not closed) and the rewrite step is actually possile only if the sucomponents of s also evolve producing the trigger a. Both trigger and eect are called 1 While tiles can e considered as a generalization of SOS inference rules, their algeraic structure is new. Larsen and Xinxin contexts [24] are analogous, ut their algeraic structure is limited to ordinary terms and not axiomatized. Structured transition systems and rewriting logic have similar aims and similar algeraic structure, ut do not account for side eects and synchronization. 2

3 initial input interface h initial conguration s - h initial output interface trigger a eect nal input interface h? -? h s 0 nal output interface nal conguration Fig. 1. A tile. oservations, and model the interaction, during a computation, of the system eing descried with its environment. More precisely, oth system congurations are equipped with an input and an output interface, and the trigger just descries the evolution of the input interface from its initial to its - nal conguration. Similarly for the eect. It is convenient to visualize a tile as a two-dimensional structure (see Fig. 1), where the horizontal dimension corresponds to the extension of the system, while the vertical dimension corresponds to the extension of the computation. Actually, we should also imagine a third dimension (the thickness of the tile), which models parallelism: con- gurations, oservations, interfaces and tiles themselves are all supposed to consist of several components in parallel. To match the SOS style as much as possile and to make more readale the notation, we will more often use the form: s! s 0 Both congurations and oservations are assumed to e equipped with operations of parallel and sequential composition (represented y inx operators and respectively) which allow us to uild more parallel and larger components, extended horizontally for the congurations and vertically for the oservations. Similarly, tilesthemselves possess three operations of composition 2 : parallel ( ), horizontal ( ), and vertical composition. If we consider tiles as logical sequents, it is natural to dene the three operations via inference rules called composition rules (see Denition 3.2). a 2 In general, tiles are also equipped with proof terms which distinguish etween sequents with the same congurations and oservations, ut derived in dierent ways. axioms for normalizing proof terms are also provided [1820]. Suitale 3

4 The operation of parallel composition is self explanatory. Vertical composion models sequential composition of transitions and computations. Horizontal composition corresponds to synchronization: the eect of the rst tile acts as trigger of the second tile, and the resulting tile expresses the synchronized ehavior of oth. Computing in a tile logic consists of starting from a set of asic tiles called rewrite rules (and from a set of auxiliary tiles which depend on the version of the tile model at hand), and of applying the composition rules in all possile ways. A tile logic also can e seen as a doule category [15] and tiles themselves as doule cells. The categorical interpretation [18,19] is useful since it makes the model more general (congurations and oservations can e arrows of any category), allows for universal constructions (e.g. a tile logic is the doule category freely generated y its rewrite rules) and suggests analogies with fruitful concepts of algeraic semantics, like institutions. However, the tile model is presented here in a purely logical form. In this paper, oservations and congurations are term graphs [3] and term cographs respectively. Term graphs are similar to terms, ut two term graphs may explicitly share some of their suterms. Thus in a term graph it is in general not allowed to copy the shared suterms to make the two terms disjoint, since this would yield a dierent term graph. An axiomatization of term graphs y means of gs-monoidal theories has een recently proposed y Corradini and Gadducci [8,9], and it is reported in the Appendix. Term cographs are like term graphs, ut their direction is inverted: while term graphs are oriented from leaves to roots, term cographs are visited from roots to leaves. Term graphs are convenient structures for modeling congurations of distriuted systems and their partial ordering oservations, since they are equipped with an operation of parallel composition (which models indipendent juxtaposition) and with the possiility ofsharing sucomponents. Sharing is used within congurations for modeling the operator j of process algeras, which in this context means sharing the same location. Within oservations, sharing is used to express the fact that two events share the same cause, or, equivalently, that the same location has two dierent sulocations. For instance, the CCS agent a:nil j :nil is represented y the term cograph G = f!(a(e))!((e))g. The shared variale e represents the common location (the only one in this case) of the two components!(a(e)) and!((e)). Component!(a(e)) has one variale ut has no root, since the discharger operator!( ) disposes of the result of the suterm a(e). Thus oth component!(a(e)) and term graph G are arrows from the underlined natural numer 0 (i.e. zero roots) to the natural numer 1 (i.e. one variale). Term cographs initiating from 0 represent closed agents, and in fact the discharger operator represents the agent nil. Notice that while garage collection is automatic in term algera (e.g.!(a(e)) =!(e) since oth memers represent the empty tuple of terms), in the algera of term graphs a term graph with no root may carry nontrivial information. Notice also that variales have only local meaning, i.e. 4

5 in G = f!(a(e))!((e))g variale e just represents the only existing variale. In other words also f!(a(e 0 ))!((e 0 ))g would denote the same term graph G. Only the ordering of variales is meaningful. a The computations a:nil j :nil! nil j :nil! nil j nil and a:nil j :nil! a! nil j nil oth correspond to the same tile with empty trigger 3 : a:nil j nil :!(a(e))!((e)) e 0 :=a(e) e 00 :=(e) e!!(e)!(e 0 )!(e 00 ): The eect e 0 := a(e) e 00 := (e) e of is a term graph with one variale e and three roots e, 0 e and e, i.e. it is an arrow from 00 1 to 3. Its meaning in terms of events is as follows. At the eginning there is only an initial event e. After the computation, we still have the same initial event, ut also two new events e, 0 e have happened, laelled y 00 a and respectively, and oth caused y e. In terms of locations, we cansaythattwo sulocations e and 0 e of the initial 00 location e have een created y actions a and. However, there is no left or right location, and new locations are introduced only when something takes place at them. The meaning of the nal conguration!(e)!(e 0 )!(e 00 ) is that all the components in the three locations are inactive. A detailed derivation of the aove tile is shown in Section 5. We now show a rewrite rule of our system: (Prefix ) e 0 e 0 := (e) e 0 :=(e) e! e 0!(e): It represents the ring of a prex and corresponds to the following SOS axiom for located CCS [4]: :p! l l :: p: Notice that Prefix does not descrie the evolution of a closed system, as it is the case for the SOS axiom, since its initial conguration is a partial system e 0 := (e) : 1! 1. However only a trivial (identity) trigger e is 0 required for applying the rule. A rule with this eect is availale in our logic for any agent p and called id [[p]]. Thus the tile relevant for agent :p is the horizontal composition id [[p]] Prefix. This tile descries the creation of a new location, where the nal conguration [p ] is positioned, and states that there is no nontrivial component left at the initial location. Notice that our tile is deterministic while the SOS axiom yields an innite ranching. The paper is organized as follows. Section 2 introduces term graphs, and the operations of parallel and sequential composition on them. Section 3 presents the tile model, in the simplied version needed in the paper, while Section 4 descries located CCS in a strong version (at our knowledge original) and in the weak version. For the operational semantics of the strong version, the commuting diamonds are dened following the axiomatic approach of [16]. 3 In all our tile rewrite systems modeling process algeras, the tiles with a closed agent as initial conguration have empty trigger, i.e. they can e considered as transitions of a laelled transition system. 5

6 Section 5 denes a tile rewrite system for oth the strong and the weak versions, and Section 6 outlines its equivalence with the ordinary semantics of Section 4. In particular, it is shown that the equality of computations speci- ed y the commuting diamonds holds in the tile model, and that its uniform notion of isimilarity yields in the weak case the same equivalence on agents as ordinary location isimilarity. 2 Term Graphs In this section we review term graphs on which it is ased the data stucture we use for modeling congurations and oservations. Term graphs [3,1,8,9], have a nice algeraic structure and can e nitely axiomatized as gs-monoidal theories [18,19,8,9]. We follow a style of presentation similar to [1]. Denition 2.1 (one-sorted term graphs) Let us consider a one-sorted, ranked signature, with f h 2 h. Furthermore let V e a totally ordered (y ) innite set of names, a name eing denoted y n and similar letters. A term graph is a triple G = (S var rt) where: S is a nite set of sentences, which are assignments of the form n := f h (n 1 ::: n h ), or of the form n := n 0. In addition, every name must e assigned at most once and no cycles (with the ovious meaning) must e present var is a list, without repetitions, of the variales of G, variales eing oth all the names which are not assigned in S and possily other names which do not appear in S. Variales are usually denoted y v or similar letters rt is a list, without repetitions, of the roots of G, i.e. the names which appear as left memers of assignments of the form n := n 0. Roots are usually denoted y r or similar letters. The ordering of variales and roots must respect the ordering in V. Furthermore, term graphs are dened up to isomorphic renaming. Given a term graph G, let h (resp. k) e the numer of variales (roots). Then G can e seen as an arrow of type h! k. We write G : h! k and call h and k the source and target of G respectively. 2 For instance, given the signature with 1 = f g and 2 = d, and the names n 1 ::: n 4 v 1 ::: v 3 r 1 r 2, let G : 3! 2 = (S var rt) with S = fr 1 := v 1 r 2 := n 1 n 1 := d(n 2 n 3 ) n 2 := g(n 3 ) n 3 := f (v 1 ) n 4 := f (v 2 )g, rt =(r 1 r 2 ) and var =(v 1 v 2 v 3 ). Notice that if we consider as standard the names in the lists of variales and roots (i.e. v 1 ::: v h and r 1 ::: r k ), isomorphic renaming is restricted to the names which are neither variales nor roots. Furthermore, in this case a term graph is fully specied when its type and its set of sentences are given, since the length of its list of variales can e recovered from its type. 6

7 Particularly interesting are the following term graphs which are called atomic: generators: for every f h 2 h identities: id h f h : h! 1 = fr 1 := f h (v 1 ::: v h )g id h : h! h = fr 1 := v 1 ::: r h := v h g permutations: h k h k : h + k! h + k = fr 1 := v h+1 ::: r k := v h+k r k+1 := v1 ::: r k+h := v h g duplicators: r h r h : h! 2h = fr1 := v1 ::: r h := v h r h+1 := v1 ::: r2h := v h g dischargers:! h! h : h! 0 = fg: We now introduce two operations on term graphs. The sequential composition of two term graphs is otained y gluing the list of roots of the rst graph with the list of variales of the second, and it is dened only if their numers are equal. The parallel composition instead is always dened, and it is a sort of disjoint union where variale and root lists are concatenated. Denition 2.2 (sequential and parallel composition of term graphs) Given two term graphs G1 : h! k = (S1 var1 rt1) and G2 : k! l = (S2 var2 rt2), let us take two instances in their isomorphism classes such that rt1 = var2 and that no other names are shared etween G1 and G2. Furthermore, let S e the set of clauses in S1 of the form r := n and let e the corresponding name sustitution. The sequential composition of G1 and G2 is the term graph G1 G2 : h! l =((S1nS) [ S2 var1 rt2). Given two term graphs G1 : h1! k1 =(S1 var1 rt1) and G2 : h2! k2 = (S2 var2 rt2), let us take two instances in their isomorphism classes such that no names are shared etween G1 and G2. The parallel composition of G1 and G2 is the term graph G : h1 + h2! k1 + k2 =(S1 [ S2 var1 var2 rt1 rt2). 2 We need a concise term-like notation for term graphs, without explicit lists of variales and roots. Thus we introduce a notion of presentation of a term graph which allows for several shorthands. To eliminate the need of specifying variales and roots, we consider the partial ordering on names dened y the assignments. It is denitely true that all roots are maxima and all variales which appear in the assignments are minima. However there might e: i) maxima which are not roots (like name n4 in the our example G) and ii) variales which do not appear at all (like v3). To get precisely variales as minima and roots as maxima, we introduce a ctitious mame > and we assume n v>for all names n in i) and ii) aove. To 7

8 express these new dependencies in presentations, we introduce a new sentence!(n) for each such dependency (e.g.!(n 4 ) and!(v 3 ) in our example). Moreover, we express the orderings of minima and maxima (which are needed since minima and maxima are now variales and roots) simply y the orderings of their names in V. A second shorthand consists of replacing with its denition every name which is neither a variale nor a root, and which is used just once. For instance in our example we replace r 2 := n 1, n 1 := d(n 2 n 3 ) and n 2 := g(n 3 ) with r 2 := d(g(n 3 ) n 3 ). Finally, we replace assignments like r 1 := v 1 in our example, whose only role is to mark as a root a name which is used in other sentences, with sentences consisting of single names, like v 1. However we must e careful at this point, since the ordering of r 1 among the roots may e dierent than the ordering of v 1. Since several presentations may correspond to the same term graph, we give here a reduction procedure ale to translate presentations of term graphs into the form of Denition 2.1. Denition 2.3 (presentations of term graphs and reduction procedure) Given a signature and a set V of names, a presentation P of a term graph is a set of sentences of the form n, or n := T (n 1 ::: n h ), or!(t (n 1 ::: n h )), where T is a term on the signature, possily just a name. As in Denition 2.1, a name can e assigned at most once and no cycles must e present. >From presentation P we derive as follows a triple (S var rt) dening a term graph: we replace sentence n with n 0 := n, where n 0 is anewname with n n 0 ut where no name n 00 exists in P with 4 n n 00 n 0 we decompose in the ovious way the sentences of the form n := T (n 1 ::: n h ), or!(t (n 1 ::: n h )) into asic sentences of the form n := f h (n 1 ::: n h ) and!(n), always using new names let P 0 e the resulting presentation. We dene a partial ordering v, where n v n 0 i n 0 := n 2 P 0 or n 0 := f k (n 1 ::: n ::: n k ) 2 P 0 and n v > i!(n) 2 P 0, with > an additional element in the ordering var is the list of the minima in v ordered according to the total ordering of V. Similarly, rt is the list of the maxima, excluding for > S consists of the set of sentences in P 0 which are assignments, i.e. sentences!(n) are disregarded. 2 A term-like presentation of our example G is as follows: fn 1 n 3 := d(g(n 2 ) n 2 ) n 2 := f (n 1 )!(f (n 4 ))!(n 5 )g. Our notation for presentations is consistent with the ordinary record no- 4 If this is not possile, an -conversion should e applied to P. These conditions on n 0 are dictated y the need of yielding the same root ordering for the derived term graph independently from the choice of n 0. 8

9 tation for terms and sustitutions, and coincides with it when no sentences n are present, terms like!(t (n 1 ::: n h )) are disregarded and no names are left esides variales and roots. Term like presentations will e used throughout the paper. However we should translate our presentations to the asic notation whenever we want to check if two presentations represent the same term graph. We close this section with some asic properties of term graphs [8,9]. Theorem 2.4 (decomposition of term graphs) Every term graph can e otained y evaluating an expression containing only atomic term graphs as constants, and sequential and parallel composition as operators. 2 For instance the term graph G of our previous example can e represented as: G =(r 1 (id 1 (f r 1 g id 1 d))) (f! 1 )! 1. The following theorem gives a characterization of term graphs as gs-monoidal theories. A gs-monoidal theory is a logical theory similar to, ut weaker than, the algeraic theory of terms and sustitutions. Theorem 2.5 (characterization of term graphs) The term graphs on the signature are the arrows of the gs-monoidal theory GS() generated y. 2 In the Appendix we give the nitary axiomatization of gs-monoidal theories presented in [18,19,8,9]. 3 The Tile Model We now descrie the asic features of the tile model, in the version where oservations are term graphs and congurations are term cographs. Term cographs are term graphs where the arrows of the types are all reversed 5. The presentation follows [19], ut is simpler, since the tile sequents we have here (the at sequents) have no associated proof terms. In the following we will call them simply tile sequents. Denition 3.1 (tile sequent, tile rewrite system) Let h and v e two signatures, called the horizontal and the vertical signature respectively. A h - v tile sequent is a quadruple s! a t, where s : h! k and t : l! m are termcographs on h,whilea: h! l and : k! m are termgraphs on v. (Co)graphs s, t, a and are called respectively the initial conguration, the nal conguration, the trigger and the eect of the tile. Trigger and eect are 5 The example of Section 2, G = fn 1 n 3 := d(g(n 2 ) n 2 ) n 2 := f (n 1 )!(f (n 4 ))!(n 5 )g, can e represented as term graph with (r 1 (id 1 (f r 1 g id 1 d))) (f! 1 )! 1 :2! 3 and as term cograph with ((id 1 (d g id 1 r 1 f )) r 1 ) (! 1 f )! 1 :3! 2. 9

10 called oservations. Underlined integers h, k, l and m are called respectively the initial input interface, the initial output interface, the nal input interface and the nal output interface. A tile rewrite system (trs) R is a triple h h v Ri, where R is a set of h - v sequents called rewrite rules. 2 A trs R can e considered as a logical theory, and new sequents can e derived from it via certain inference rules. Denition 3.2 (tile logic) Let R = h h v Ri e a trs and let GS op ( h ) (resp. GS( v )) e the cographs (resp. the graphs) on the signature h (resp. v ). Then we say that R entails the class R of the tile sequents s! a t otained y nitely many applications of the following inference rules: asic rules: (generators) (h-re) s : h! k 2 GSop ( h ) composition rules: id s = s id h! s 2 R id k (p-comp) (h-comp) (v-comp) auxiliary rules: (permutations) = s a! s a! s a! t 2 R t 2 R (v-re) a : h! k 2 GS( v) a id a = id h! id k 2 R t 0 = s 0 a! 0 t 0 2 R 0 0 = s s 0 aa0! t t 0 2 R 0 = s a! c t 0 = s 0 c! t 0 2 R 0 = s s 0 a! t t 0 2 R = s a! r 0 = r! a0 0 0 = s a a0! t 2 R 0 t 2 R 0 0 = h k h k h k! id h+k 2 R 0 1 = id h+k h k k h! id h+k 2 R id h+k h k a 1 0 = id k h h k h+k! h k 2 R 1 1 = id id h+k h k h+k! k h 2 R: id h+k k h 2 Basic rules provide the sequents corresponding to rewrite rules, together with suitale identity tiles, whose intuitive meaning is that an element of GS op ( h ) can e rewritten to itself using only trivial trigger and eect. Similarly for GS( v ). Composition rules provide all the possile ways in which 10

11 sequents can e composed, while auxiliary rules are the counterpart of the atomic permutation graphs discussed aove for term graphs. For instance the tile denoted 0 1 consists of a horizontal permutation on h k the initial conguration (notice the character 0 as the rst upper index) of the tile, and of the inverse permutation on the eect oservation (notice the character 1 as the second upper index). The remaining sides are identities, and similarly for the other permutation tiles. While for reasons of symmetry we include four permutation rules, it is easy to see that one would e enough, since the remaining three could e derived y concatenating one of them with identity tiles. The role of permutation tiles is to permute the names on one vertex of the tile (the initial output interface in the example), still mantaining the same connections etween the adjacent term (co)graphs. For instance, given any tile = s! a t with s and having h + k as target and source respectively, the composition (id s 0 1 ) ( id k h ) produces the tile 0 a = s h k! t. Here k h two permutations have eenintroduced, ut the connections etween the two involved term (co)graphs, represented y the composition s h k k h, are still the same as s, since h k k h = id h+k. It is easy to see that, y horizontal and vertical composition of the asic 3 permutation tiles, it is possile to otain permutation tiles 1 with arritary permutations 1, 2, 3 and 4 on the four sides, provided that 1 2 = 3 4.! 2 4 It is straightforward to extend the notion of isimilarity to deal with tile rewrite systems. Denition 3.3 (tile isimilarity). Let R = h h v Ri e a trs. A symmetric equivalence relation GS op ( h ) GS op ( h ) is a tile isimulation for R if, whenever s t for generic s t elements of GS op ( h ), then for any sequent = s! a s 0 entailed y R there exists a corresponding one = t! a t 0 with s 0 t 0. The maximal tile isimulation equivalence is called tile isimilarity, and denoted y t. 2 Notice that this notion of isimilarity is more general than the ordinary one, since it applies to pairs of system components which are open, while the ordinary notion applies only to closed agents. 4 Concurrent and Located Semantics for CCS There are many concurrent models for process calculi. Some of them focus on the operational aspects, dening certain concurrent machines for the calculi. Other models are equipped with notions of oservation ale to capture causal 11

12 dependencies or localities, and dene astract semantics, usually via isimulation. In this section we try to comine oth aspects y presenting a version of concurrent CCS with locations, of which we provide oth the concurrent operational and the astract location semantics. This calculus is supposed to e close to those presented in the literature [16,5,4,22,10,28]. The asic idea of location semantics [4] is to associate a dierent location with each process, to allow the external oserver to see an action together with the location where it takes place. Hence, processes a::nil + :a:nil and a:nil j :nil are distinguished as the second process can perform a and in dierent places, while the rst process cannot. To dene the concurrent operational semantics, we follow the approach of [16] which associates an n-ary operator to each SOS rule for CCS with n- premises, and then imposes certain axioms on the resulting algera of transitions and computations. Concurrent computations are equivalence classes in this formal system. In [21] it is proved that the same equivalence is induced y mapping CCS into Petri nets. It is usually conjectured that the same equivalence can also e derived y following the approach ased on proved transition systems and residuals [5]. We rst show a strong version of the operational semantics, where locations are visile also in the case of synchronization. We then present a dierent synchronization rule which hides locations. Besides eing closer to the tile version, this presentation of the locality transition system allows us to avoid dening two dierent kinds of transitions (i.e. standard and location transitions) as in [4]. Syntax Let e the alphaet for asic actions, (which is ranged over y ) and the alphaet of complementary actions ( = and \ = ) the set =[ will e ranged over y. Let 62 e a distinguished action, and let [fg (ranged over y ) ethe set of CCS actions. Let Loc e a totally ordered denumerale set of locations (ranged over y l). Laels of transitions consist of actions and strings of locations, denoted y u. A synchronization transition is laelled y two strings (in the strong version) or none (in the weak case). The generic denotation is k. In lk, location l is concatenated with each string in k. The syntax of nite CCS agents with locations is dened y the following grammar. p := nil :p l :: p p + p p j p In the following we will consider only agents where p in :p does not contain locations. For the sake of simplicity we do not include restriction, although there is no prolem with it. Recursion could also e handled introducing specialized tiles, as in [17]. 12

13 (Act) [ l p >: :p! l l :: p (Loc) t : p! q l 62 loc(k) k l :: t : l :: p! l :: q lk (Sum) t : p! p 0 k t<+q : p + q! k p 0 t : p! p 0 k q+ >t: q + p! k p 0 (Comp) t : p! p 0 loc(k) \ loc(q) = k tcq : p j q! p 0 j q k t : p! p 0 loc(k) \ loc(q) = k qt : q j p! q j p 0 k t 1 : p 1 (Synch)! q 1 t 2 : p 2 u1! q 2 loc(u 1 ) \ loc(q 2 )= = loc(u 2 ) \ loc(q 1 ) u2 t 1 j t 2 : p 1 j p 2 Tale 1 Transition Algera! q 1 j q 2 u1 u2 We useloc(p) and loc(k) to indicate the set of location names occurring in process p or in lael k. A process p is called pure if loc(p) =. Throughout the paper we assume that a process contains at most one occurrence of each location. This restriction does not appear in [4]. In [22], however, it has een pointed out that no discriminating power is added if we are allowed to choose a location twice in a computation and that our denition is equivalent to the one in [4]. Operational Semantics The Transition Algera (TA) of CCS with locations is displayed in Tale 1. For instance, the term ([a l 1 p >cp 0 ) j [a l 2 q > descries the proof of the transition: (a:p j p 0 ) j a:q! (l 1 :: p j p 0 ) j l 2 :: q: l1 l2 As another example, a synchronization of process l 1 :: a:nil j l 2 :: l 3 :: a:nil is descried y the following transition: l 1 :: [a l 4 nil >j l 2 :: l 3 :: [a l 5 nil >: l 1 :: a:nil j l 2 :: l 3 :: a:nil We can now introduce the concurrency relation. Denition 4.1 (concurrency relation ) 13! l 1 :: l 4 :: nil j l 2 :: l 3 :: l 5 :: nil: l1l4 l2l3l5

14 t 1 cp 2 then q 1 t 2 p 1 t 2 then t 1 cq 2 t 1 then t 2 t 3 then t 4 l :: t 1 then l :: t 2 l:: t 3 then l :: t 4 t 1 then t 2 t 3 then t 4 t 1 < +p then t 2 t 3 < +p then t 4 t 1 then t 2 t 3 then t 4 p+ >t 1 then t 2 p+ >t 3 then t 4 t 1 then t 2 t 3 then t 4 t 1 cp 0 then t 2 cp 0 t 3 cp 0 then t 4 cp 0 t 1 then t 2 t 3 then t 4 p 0 t 1 then p 0 t 2 p 0 t 3 then p 0 t 4 t 1 then t 2 t 3 then t 4 t 1 j t then t 2 cqt 3 cp then t 4 j t t 1 then t 2 t 3 then t 4 t j t 1 then qt 2 pt 3 then t j t 4 t 1 then t 2 t 3 then t 4 t 0 1 then t0 2 t0 3 then t0 4 t 1 j t 0 1 then t 2 j t0 2 t 3 j t0 3 then t 4 j t0 4 Tale 2 The Concurrency Relation Let ( then then ) e a quaternary relation on transition proof terms, dened as the least commutative 6 relation dened y the strucural rules i of Tale 2, with t i : p i! q i, t 0 : 0 i! q 0, t : p! i p0 i i k i k 0 k i The concurrency relation identies the diamonds in the transition algera. q. 2 The axiom denes the asic diamonds, while the inductive rules reproduce the diamonds in all possile contexts. identication in the algera of computations 7 : (t 1 then t 2 t 3 then t 4 ) implies t 1 t 2 = t 3 t 4. A diamond then forces an Astract Semantics The transition algera for the astract semantics is otained y replacing rules (Act) and (Synch) in Tale 1 with the following rules: (Act 0 ) [ l p >: :p! l (Synch 0 ) l :: p [ p >: :p! p t 1 : p 1! q 1 t 2 : p 2! q 2 u1l1 u2l2 t 1 j t 2 : p 1 j p 2! d(q 1 l 1 ) j d(q 2 l 2 ) where d(p l) deletes l, i.e. replaces l :: p 0 with p 0 in p. 6 Namely, (t1 then t2 t3 then t4) i (t3 then t4 t1 then t2). 7 A transition is a computation, and composition _ _ of computations is associative and has identities. 14

15 Using (Synch 0 ),thesynchronization of process l 1 :: a:nil j l 2 :: l 3 :: a:nil of the previous example ecomes: l 1 :: a:nil j l 2 :: l 3 :: a:nil! l 1 :: nil j l 2 :: l 3 :: nil as dened in [4]. We now introduce the notion of location isimilarity. We adopt the standard notation for weak transitions: )= (!) and ) = )! ). Denition 4.2 (location isimilarity) A inary relation R is a location simulation if prq implies: for each p ) ul p 0, with loc(l) \ loc(q) =, there exists some q ) ul 0 0 p Rq for each p ) 0 p there exists some q ) 0 q with 0 0 p Rq. u u q 0 with Arelation R is called alocation isimulation if oth R and R 1 are location simulations. Two processes p and q are location isimilar (p l q) if prq for some location isimulation R. 2 5 A Tile Rewrite System for Concurrent Located CCS The aim of this section is to show how the framework provided y the tile model can e applied to provide natural concurrent, located semantics for CCS. In what follows we introduce the components of the tile rewrite system, i.e. horizontal signature, vertical signature and rewrite rules. Horizontal Signature. The symols of the signature h, and their arities, are as follows: :1(P ref ix) + :1(Choice) + :1(Lef t Choice)! + :1(Right Choice) :0(Codischarger): Congurations of the tile rewrite system are term cographs over the signature h. 15

16 Vertical Signature The symols of v are as follows :1(Action) T :2(Synchronization) T :1(Lef t T )! T :1(Right T ) Oservations are term graphs over the signature v. In the aove signature, symols +, + and! + are used to translate the CCS operator +. Similarly, T, T and! T model the action otained via synchronization. Symol is used for making inactive processes refused in a choice. Since congurations and oservations of rules are term (co)graphs 8, we use for them the notation developed in Section 2. We denote names as e, e 0, etc. Strong Rewrite Rules Tomatch the SOS notation as closely as possile, from now on a tile s! 0 a s will e represented as: a s! 0 s Following the SOS convention, the antecedent (trigger) a will e omitted when it is the empty term graph. The rewrite rules are displayed in Tale 3. Notice that roots (variales) of term cographs representing congurations elong to input (output) interfaces of the tiles, while variales (roots) of term graphs representing oservations elong to initial (nal) interfaces of the tiles. We can now comment on the denition of the rules. The application of the rule Prefix causes the rewriting of the initial conguration 0 e := (e) into the nal conguration 0 e!(e) where a new variale has een created. The intuition is that this new variale corresponds to the name of the event associated to the ring of the prex. Such a name is never cancelled y further rewriting steps. The rule (Comp ) asically descries the asyncronous evolution of parallel processes (those associated to e 1 and e). To illustrate the application of this rule let us consider the rewriting of process a:nil j :nil. 8 Employing gs-monoidal theories equipped with hypersignatures would allow for replacing +, +, and! + with a unique symol + 0. Similarly, asymol T 0 could replace T, T and! T. 16

17 (Prefix ) e 0 e 0 := (e) e 0 :=(e) e! e 0!(e) (Suml ) e 1 e 2 e 0 := (e 1 )! e 1 :=+ (e) e 2 := +(e) e := +(e) e 0 1:=e e :=(e)! e 0 e 1 e 2 := (Comp ) e e 1 := e e e 1 e 0 := (e) e e 0 :=(e)! 0 e e e 1 := e (Synch ) (Twin) e 1 e 2 e 0 1 := (e 1) e 0 2 := (e 2) 0 0 e 1 :=T (e ) e 0! 2 := T (e0 ) e 0 :=T (e1 e2) e1 e2 e 1 e 2! 0 e0 e 1 e 1 e 2 2 e 1 := e e 2 := e e 0 1 :=T (e0 ) e 0 2 :=! T (e 0 ) e 0 := T (e 1 e 2 ) e 1 e 2 e0 1 := T (e0 ) e 0 2 :=! T (e0 ) e 0 :=T (e e) e! e 1 := e e 2 := e e 0 1 e0 2 Tale 3 Strong Rewrite System We start with the Prefix a rule: e e := a(e) e :=a(e) e! e 0!(e) we compose horizontally id!1 with it, and the result in parallel with id!1. We thus get:!(a(e))!((e 1 )) e 0 :=a(e) e e1!!(e)!(e 0 )!((e1 )): We are now ready to compose this tile horizontally with Comp a : =!(a(e))!((e)) e 0 :=a(e) e!!((e))!(e 0 ): We can now compose horizontally id!1 with Prefix, and the result in parallel with id!1 :!((e))!(e 0 ) e 00 :=(e) e e 0!!(e)!(e 0 )!(e 00 ): Finally, y composing vertically with the latter tile:!(a(e))!((e)) e 0 :=a(e) e 00 :=(e) e!!(e)!(e 0 )!(e 00 ): The eect of this tile (the term graph e 0 := a(e) e 00 := (e) e) tells us that the two actions can e executed in parallel. Rule Synch accounts for synchronizations at dierent locations, like a::nil j c::nil ac! nil j nil. Rule Twin, when Synch is composed horizontally with it, 17

18 allows for synchronizations at one location only, like a:nil j a:nil! nil j nil. Rule Suml puts a codischarger constant on the refused ranch. Refused alternatives will appear as inactive factors in congurations. In addition to Comp, we also have rules TlComp, TrComp and TwinComp which take care of composition for T-moves (left and right side) and twin T-moves. We also have a rule Sumr. We do not show these rules. According to our denition of term graph given in Section 2, an ordering of variales and roots musteprovided. For instance, it is essential in computing the sequential composition of two term graphs. Thus it would appear as necessary to specify the ordering of names in all the interfaces of our tiles. For instance, what is the ordering etween e and 0 e 1 in the nal input interface of rule Comp? However it is easy to see that name ordering is immaterial for rules. In fact, given a rule it is always possile to otain tiles with dierent orderings of names in the interfaces y composing with suitale auxiliary (permuation) tiles. Thus any consistent renaming in the presentation 9 of the rules of a tile rewrite system R does not change the tiles entailed y R. Astract Semantics We would like to handle weak location isimulation with the uniform notion of tile isimulation presented in Denition 3.3. To this purpose, it is necessary to add rewrite rules ale to transform eects with and T oservations into identities. More precisely, we need an additional symol in the horizontal signature: F :1 (Filter) and the following three additional weak rules: (Filter ) e 1 e 0 1 := (e 1) e 1 := F (e) e e 0 :=(e)! e 1 := F (e) e 0 1 := F ) (e0 (Filter ) e 1 := F (e) e 1 e 0 1 := (e 1)! e e 1 := e 0 e 0 1 := e 0 := F (e) e0 0 00! (Filtersynch) e00 1 := T (e ) e2 := T (e ) e := T (e e0 ) e0 e e1 := F (e 1) e2 := F (e 2) e 1 e2! G where G = fe 0 1 := e 1 e 00 1 := e 1 e 0 2 := e 2 e 00 2 := e 2 e 1 := F (e 1 ) e 2 := F (e 2 )g. The weak rewrite system consists of the strong rules and of the aove three weak rules. 9 Rememer that names appear only in our presentations for term graphs and tiles. Tiles themselves contain no names. 18

19 The lters work as follows. If we start from a conguration with a lter for every variale, oservations containing are ale to go through as they are, while and T oservations are transformed into identities. Notice that in the latter case the new name(s) generated y the transition (one for and two for T ) are merged with the names efore the transition, i.e. eliminated. Notice also from Filter that lters reproduce themselves on every newly generated variale. Location isimilarity for our congurations is thus dened for the weak rewrite system as tile isimulation (see Denition 3.3). 6 Comparing SOS and Tile Semantics We rst show the correspondence for the concurrent operational semantics and then for the astract location semantics. Operational Semantics To show the correspondence of our tile rewrite system with the transition algera normalized with the concurrency relation, we rst translate located agents p into congurations [p ]. It is convenient to dene inductively the auxiliary function ind[p] as returning a pair of congurations, the rst translating the sequential components of p which do not contain a location, the second referring to the located components. Function [p ] is the parallel composition of the two congurations, where the variales corresponding to the located components are ordered according to the total ordering of Loc. Denition 6.1 (from located CCS agents to congurations) Let p e a located CCS agent. Then [p ] : 0! jloc(p)j +1 is the con- guration corresponding to agent p, where function [ ] is dened elow. the inductive denition we assume ind[p] = hf gi, where f g :0! jloc(p)j, and similarly for p 1 and p 2. In : 0! 1 and ind[nil] = h! 1 id 0 i ind[:p] = hf id 0 i ind[p 1 + p 2 ] = hf 1 f 2 +! + r 1 + id 0 i ind[l :: p] ind[p 1 j p 2 ] = h! 1 f gi = hf 1 f 2 r 1 g 1 g 2 i [p ] = f (g p ) where p = fr 1 := v i1 ::: r n := v i ng, with i 1 ::: i n the list of locations of p ordered according to the prex tree walk of p, and v l <v l 0 i l<l 0. 2 Maye the only surprising clause is that for l :: p. It states that, when an agent is prexed with a location, the component which is nonlocated ecomes 19

20 the rst located component, while the new nonlocated component is empty. Notice that this ordering of located components (as the one for p 1 j p 2 ) is consistent with the prex tree walk of p. Some examples, where we use our notation for term graphs, should make the mapping clear. We start with some pure agents. [a::nil ] =!((a(e))) [a:nil j :nil ]=!(a(e))!((e)) [(a:nil j :nil) j (c:nil + d:nil) ]=!(a(e 1 ))!((e 1 )) e 1 := e!(c(+ (e 2 )))!(d( +(e 2 ))) e 2 := +(e) =!(a(e))!((e))!(c(+ (e 2 )))!(d( +(e 2 ))) e 2 := +(e): Considering now alocatedagent p = l 1 :: (l 7 :: a:nil j l 5 :: :nil) we have: or, using our term graph notation:! [p ]=! 1! 1 (! 1 a) (! 1 ) id :0! 4! [p ]=!(e 0 )!(e 1 )!((e 2 ))!(a(e 3 )): Notice that variale e 0 is not used in [p ], since there is no sequential component in p which is nonlocated, while we also have!(e 1 ) since no sequential component is prexed only y l 1. As another example, we have: [a:nil j (:nil j l :: c:nil) ]=!(a(e 0 ))!((e 0 ))!(c(e 1 )): In what follows, we will consider congurations dened up to factors s = [p ]. Such components are inactive, in the sense that any tile having s s 0 as initial conguration can e decomposed as = id s 0, where 0 is a tile having s 0 as initial conguration. It is possile to dene a translation fj jg from transitions to tiles derivale in the strong rewrite system, using induction on transition proof terms. The tiles otained in this way are complete, in the sense that, when composed vertically, they yield essentially all derivale tiles. Furthermore, given any diamond of the concurrency relation of Tale 2, the translated computations commute. Proposition 6.2 (from proved transitions to tiles) It is possile to dene inductively a function fjtjg from terms t of the transition algera in Tale 1 to tiles such that the following properties hold. (i) Tile fjtjg is derivale in the strong tile rewrite system. (ii) Any derivale tile with initial conguration [p ] can e otained y repeatedly composing vertically tiles in the range of fj jg and tiles composed of identity and auxiliary tiles. (iii) A diamond (t 1 then t 2 t 3 then t 4 ) implies fjt 1 jgfjt 2 jg = fjt 3 jgfjt 4 jg: 20

21 We now riey outline the proof of Proposition 6.2. As hinted aove, function fj jg is dened y induction on transition proof constructors. Clauses for (Act), (Loc) and (Sum) are easy: fj [ l p > jg = id [[p]] Prefix fjl :: tjg = id!1 fjtjg fjt <+pjg =(fjtjgid [[p]] ) Suml. We can also hint at the form of the clauses for (Comp) and (Synch), even if writing them in detail may require the development of suitale notation. For fjtcqjg we have several cases. If q has no nonlocated component, or if the action of t takes place at located components only (either one or two according to the action eing or T ) then an identity is enough. Otherwise, if t performs an action then rule Comp is used. If t performs a synchronization etween two nonlocated components, rule TwinComp is needed. Finally, if t performs a synchronization etween a non located and a located component, rules TlComp or TrComp are required. For fjt 1 j t 2 jg wealways use Synch. We also use Twin if the synchronization involves nonlocated components for oth p 1 and p 2, TlComp or TrComp if only one of them is located, nothing else otherwise. In allcases we need permutation tiles uilt from auxiliary tiles i j to ensure suitale wire twisting. For h k instance, using permutation tiles it is possile to ring close the components we want to synchronize, to synchronize them, and to ring them ack to the original positions. It is possile to see that derivale tiles starting from [p ] can e roken down vertically into one-step tiles, i.e. tiles computed y fj jg, and perm-tiles of the form s! s, where is any permutation. This decomposition is essential for the correctness of the tile semantics shown in the next paragraph. The validityofthe axiomatization in the tile model can e easily checked on the inductive denition of fj jg. It would e interesting if the axiomatization were also complete, i.e. enough to equate all computations in the CCS algera yielding the same tile via fj jg. This is not the case here, since e.g. the two transitions of a:nil + a:nil would yield the same tile ut would not e equated y. A natural option to try for capturing exactly the operational concurrency of located CCS would e to consider tiles equipped with a proof term as originally dened in [19]. 2 Astract Semantics Location isimilarity and tile location isimilarity coincide for pure CCS agents. Proposition 6.3 (tile semantics is correct) Given two pure CCS agents p and q, we have 10 p l q i [p ] F t [q ] F.2 10 Rememer that F is the lter needed to convert strong eects into weak eects. 21

22 We sketch the proof of Proposition 6.3. We dene a transition algera with a slightly more informative lael. Instead of p! q we require transitions p o! q = p! q, exposing also k loc(p) k the locations of the agent. It is easy to see that this modication does not change isimilarity. We dene a function 11 [o ] on transition laels as follows: [ L ] = id jlj [ ] fl1 ::: lng l = fl := (l 0 ) l 0 l 1 ::: l n g [ ] = fl := (l fl1 ::: lng ulil i ) l 0 l 1 ::: l n g: Notice that our function depends only on the last two locations (i.e. it does not depend on u). In fact, it has een noticed [28] that the same isimilarity relation is otained relying on an incremental oservation, where the string of locations is truncated to the last two items. We dene modied functions [ ] and fj jg (for which however we will use the same denotations as in the strong case) to replace SOS rules (Act) and (Synch) with (Act 0 ) and (Synch 0 ) and to include lters and weak rewrite rules. Taking advantage of Proposition 6.2, we then show that if the initial conguration and the eect ofatileare [p ] and [o ] for some p and o, then the tile is the vertical composition of one-step tiles. a a We show that if p l q and [p ]! s 1, [q ]! s 2 are perm-tiles, then there exist p 0 and q 0 with p 0 l q 0 such that [p 0 ] = s 1 and [q 0 ] = s 2. Informally, this is true ecause permuting the variales in the oservation corresponds just to apply some injective sustitution to agent locations, which doesnot aect isimilarity. To show the only if part, given a isimulation S for congurations, we prove that if we take prq for agents whenever [p ]S [q ], then R is also a isimulation. To this purpose, we show that given a pair prq and a transition t : p! o p 0, we have fjtjg : [p ]! [[o]] [p 0 ]. Thus there is a tile [q ]! [[o ] s with [p 0 ]Ss. This tile consists of the vertical composition of one-step tiles. As a consequence we have q ) o q 0 with [q 0 ]=s and p 0 Rq 0. To show the if part, we prove that if we take [p ] S [q ] for every permutation whenever p l q,thens is also a isimulation. In fact, given a pair a [p ]S [q ], atile[p ]! s can e either a one-step tile or a perm-tile (or a vertical composition of several of these tiles, ut this case is as usual susumed y the shorter moves). If it is a perm-tile with permutation, then also [q ] has a perm-tile with the same and the nal congurations are in S since a [p ] S [q ] y construction. If [p ]! s is a one-step tile, then there exist o and p 0 with p! o p 0, [o ] = a and [p 0 ] = s. Thus we have a computation 11 Weoverload the notation used for mapping congurations, since the meaning is analogous. 22

23 q ) o q with 0 p 0 l q. As a consequence, there is a tile 0 a [q ]! [q ], which 0 is the vertical composition of the tiles corresponding to the transitions in the computation. Finally the proof oligation for [p ] S [q ] is already fullled, since y a property previously proved there are agents p and 00 q, 00 with p 00 l q 00 and [p 00 ]= [p ] and [q 00 ]= [q ]. 7 Conclusions In the paper we have presented a version of the tile model aimed at providing a uniform operational and astract framework for concurrent process calculi. As a case study, we have presented tile rewrite systems for strong and weak versions of located CCS and we have shown their correctness. An advantage of the tile approach is the full compositionality of the underlying logic, which is ale to handle computations of open system components as they were new rewrite rules specifying complex ehaviors. Another innovative aspect is related to the use of term graphs for representing distriuted congurations and partial ordering oservations. The case study in the paper concerns located CCS, ut it is easy to see that the strong version presented here is actually the same that is needed for causal CCS [12]. The weak causal version however is more complex, since an event can have any numer of immediate causes. To handle this case, we should consider more complicate structures then term graphs as oservations. For instance we should have, esides a r for sharing causes, also another atomic graph r to share eects. 8 Acknowledgments We would like to thank Roerto Bruni, Faio Gadducci, Narciso Marti-Oliet and Marco Pistore for their comments. References [1] Z.M. Ariola, J.W. Klop, Equational Term Graph Rewriting, Fundamenta Informaticae 26, , [2] E. Badouel, P. Darondeau, Trace NetsandProcess Automata, Acta Informatica 32 (7), pp [3] H.P. Barendregt, M.C.J.D. van Eekelrn, J.R.W. Glauert. J.R. Kennaway. M.J. Plasmeijer, M.R. Sleep, Term Graph Reduction, Proc. PARLE, Springer LNCS 259, , [4] G. Boudol, I. Castellani, M. Hennessy and A. Kiehn, Oserving Localities, Theoretical Computer Science, 114: 3161,