Technical Report: SRI-CSL July Computer Science Laboratory, SRI International, Menlo Park. also. Technical Report: TR-98-09

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1 Technical Report: SRI-CSL July 1998 Computer Science Laoratory, SRI International, Menlo Park also Technical Report: TR Dipartimento di Informatica, Universita di Pisa Process and Term Tile Logic Roerto BRUNI Jose MESEGUER Ugo MONTANARI Dipartimento di Informatica Computer Science Laoratory Dipartimento di Informatica Universita di Pisa SRI International Universita di Pisa Corso Italia Ravenswood Ave. Corso Italia Pisa, Italy Menlo Park, CA 94025, USA Pisa, Italy

2 Process and Term Tile Logic Roerto Bruni y runi@di.unipi.it Jose Meseguer z meseguer@csl.sri.com Ugo Montanari x ugo@di.unipi.it Astract In a similar way as 2-categories can e regarded as a special case of doule categories, rewriting logic (in the unconditional case) can e emedded into the more general tile logic, where also side-eects and rewriting synchronization are considered. Since rewriting logic is the semantic asis of several language implementation eorts, it is useful to map tile logic ack into rewriting logic in a conservative way, to otain executale specications of tile systems. We extend the results of earlier work y two of the authors, focusing on some interesting cases where the mathematical structures representing congurations (i.e., states) and eects (i.e., oservale actions) are very similar, in the sense that they have in common some auxiliary structure (e.g., for tupling, projecting, etc.). In particular, we give in full detail the descriptions of two such cases where (net) process-like and usual term structures are employed. Corresponding to these two cases, we introduce two categorical notions, namely, symmetric strict monoidal doule category and cartesian doule category with consistently chosen products, which seem to oer an adequate semantic setting for process and term tile systems. The new model theory of 2EVH-categories required to relate the categorical models of tile logic and rewriting logic is presented making use of a recently developed framework, called partial memership equational logic, particularly suitale to deal with categorical structures. Consequently, symmetric strict monoidal and cartesian classes of doule categories and 2-categories are compared through their emedding in the corresponding versions of 2EVH-categories. As a result of this comparison, we otain a correct rewriting implementation of tile logic. This implementation uses a meta-layer to control the rewritings, so that only tile proofs are accepted. Making use of the reective capailities of the Maude language, some (general) internal strategies are then dened to implement the mapping from tile systems into rewriting systems, and some interesting applications related to the implementation of concurrent process calculi are presented. Research supported y Oce of Naval Research Contracts N C-0225 and N C-0114, y National Science Foundation Grant CCR , 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 U.S. Army contract DABT63-96-C-0096 (DARPA); 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. Research carried out in part while the rst and the third authors were visiting at Computer Science Laoratory, SRI International, and the third author was visiting scholar at Stanford University. y Dipartimento di Informatica, Universita di Pisa. Corso Italia n.40, Pisa, Italia. z Computer Science Laoratory, SRI International. 333 Ravenswood Av., Menlo Park CA , U.S.A. x Dipartimento di Informatica, Universita di Pisa. Corso Italia n.40, Pisa, Italia. 1

3 Contents 1 Introduction 4 2 Tile Logic Background : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Algeraic Theories : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Rewriting Logic : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Algeraic Tile Logic : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Nave Process Tile Logic : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : The Inference Rules for Process Tile Logic : : : : : : : : : : : : : : : : : : : Proof Terms for Process Tile Logic : : : : : : : : : : : : : : : : : : : : : : : Axiomatizing Process Tile Logic : : : : : : : : : : : : : : : : : : : : : : : : Nave Term Tile Logic : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : The Inference Rules for Term Tile Logic : : : : : : : : : : : : : : : : : : : : Proof Terms for Term Tile Logic : : : : : : : : : : : : : : : : : : : : : : : : Axiomatizing Term Tile Logic : : : : : : : : : : : : : : : : : : : : : : : : : : 32 3 Doule Categories Notation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Inverse : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Diagonal Categories : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Transformations etween Doule Functors : : : : : : : : : : : : : : : : : : : : : : : Symmetric Monoidal Doule Categories : : : : : : : : : : : : : : : : : : : : : : : : Cartesian Doule Categories (with consistently chosen products) : : : : : : : : : : 44 4 Relating Doule Categories with Extended 2-Categories Partial Memership Equational Logic : : : : : : : : : : : : : : : : : : : : : : : : : Partial Algeras and Memership Equational Theories : : : : : : : : : : : : The Tensor Product Construction : : : : : : : : : : : : : : : : : : : : : : : Categories and 2VH-Categories : : : : : : : : : : : : : : : : : : : : : : : : Extended 2VH-Categories : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Monoids and Symmetries : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Cartesian Theories : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 63 5 Computads VH-computads : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Term Tile Rewriting Systems and Computads : : : : : : : : : : : : : : : : : : : : : 70 6 Dealing with Nondeterminism Nondeterministic Rewriting Systems : : : : : : : : : : : : : : : : : : : : : : : : : : Internal Strategies in Rewriting Logic : : : : : : : : : : : : : : : : : : : : : : : : : Collective Strategies in Maude : : : : : : : : : : : : : : : : : : : : : : : : : : : : : The Kernel : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Collection of Rewritings : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Nondeterminism and Term Tile Systems : : : : : : : : : : : : : : : : : : : : : : : : Non Uniform Case : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Uniform Case : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 85 7 Maude as a Semantic Framework Finite CCS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Concurrent and Located CCS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 97 A The Axioms of Process Tile Logic 114 2

4 B The Axioms of Term Tile Logic 115 C Hypertransformations 118 C.1 The 3-fold category SqD : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 120 C.2 The 4-fold category SqSqD : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 120 C.3 Hypertransformations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 120 D Maude 122 D.1 Basic Syntax : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 122 D.2 Shorthands : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 124 D.2.1 Variale Declarations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 124 D.2.2 Susort Declarations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 124 D.2.3 Memership Assertions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 125 D.2.4 Using iff in a Conditional Sentence : : : : : : : : : : : : : : : : : : : : : : 125 D.3 Built-ins : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 126 D.3.1 Booleans : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 126 D.3.2 Machine Integers : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 126 D.3.3 Quoted Identiers : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 126 D.4 The Meta-Level : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 126 D.5 Parametric Modules and Inx Operators : : : : : : : : : : : : : : : : : : : : : : : : 128 3

5 1 Introduction The tile model [32, 35] is a formalism for modular descriptions of the dynamic evolution of concurrent systems. The idea is that a set of rules denes the ehaviour of certain asic modules, which may interact through their interfaces. Roughly speaking, we consider a module to e just an open (e.g., partially specied) conguration of the system. Then, the ehaviour of a whole system is dened as a coordinated evolution of its sumodules. The name \tile" is due to the graphic representation of such rules. Graphically, a tile has the form a s s 0 and textually it is written s?! a s 0, stating that the initial conguration s of the system evolves to the nal conguration s 0 producing an eect, which can e oserved y the rest of the system. However, such a step is allowed if and only if the sucomponents of s (which is in general an open conguration) evolve to the sucomponents of s 0, producing the trigger a. The vertices of the tile are called interfaces. Tiles can e composed horizontally (through side eects), vertically (computational evolutions of a certain component), and in parallel (concurrent steps) to generate larger steps. It is evident that the tile model extends rewriting logic [50] (in the nonconditional case), taking into account rewriting with side eects and rewriting synchronization, and can e naturally equipped with oservational equivalences and congruences ased on eects. In fact, in (non-conditional) rewriting systems, oth triggers and eects are just identities; therefore rewriting steps may e applied freely, i.e., without interacting with the rest of the system. Thus, unconditional rewriting logic is oviously emedded in the tile formalism as a special case. The main goal of this paper is to investigate this connection in the opposite direction extending the results of [58] to the case in which congurations and eects rely on common auxiliary structures (e.g., for tupling, projecting or permuting interfaces). This is useful ecause there exist several languages ased on rewriting logic, and the implementation of a conservative mapping of tiles into rewriting logic supports the execution of tile specications. The nature of such structures will e more evident after a rief survey of the motivation for the introduction of tile systems, and of the techniques and tools employed in their semantical characterization. The rich compositional nature of the tile model is the result of a progressive exploration of mathematical structures allowing for nitary descriptions of complex context-dependent transition systems. In Computer Science, (laelled) transition systems are one of the most widely used formalisms, intuitively arising from the operational understanding of a computational system. First, an astract description of the system is dened, whose set of congurations (i.e., the feasile assignments to memory cells, registers, data structures, etc.) gives the set of states S of the transition system. Then, a transition relation T S S is dened, representing the possile evolutions of the system. A set of actions (or laels) A is sometimes introduced to take into account also oservational aspects: T ecomes a ternary relation T S A S, and an external oserver may have discriminating capailities over dierent evolutions etween the same pair of states. In many cases, taking advantage of a possile compositional structure over the states, the relation T can e inductively dened according to that structure. As an example, the states of a Petri net [66] are multisets of places, an elementary evolution is a transition t that rewrites a multiset u t to a multiset v t, and a transition can re (i.e., e executed) in every state u with u t u, leading to the state v u u t v t, where,, and respectively denote multiset inclusion, dierence and union. Thus, evolutions of a multiset are dened in terms of its susets, and disjoint susets may concurrently evolve. Another signicant paradigm is given y term rewriting systems [50], where the states are terms of an algera, and elementary evolutions are rewriting steps otained (y closure under sustitution and contextualization) from a set of rewriting rules (with free variales). Also the well-known structural operational semantics approach (SOS) [65] 4

6 is a relevant generalization of this kind of methodology. We are especially interested in SOS specications for process description algeras [2, 39, 59], where states are terms of a free algera { whose operators reect the asic composition aspects of the system { and a set of inference rules (guided y the structure of the states) inductively denes the transition relation. In recent years, the expressiveness and properties of a variety of SOS rule formats have een investigated and compared [67, 5, 37, 4]. Context systems [43], and structured transition systems [22, 26] are two interesting developments of the SOS approach. In the former, the transition relation is extended to contexts (that is, terms where free variales may occur) instead of closed terms, thus characterizing the ehaviour of partially specied components of a system. In the latter, also transitions are equipped with an algeraic structure, usually y lifting the structure dened on the states in such a way that computationally equivalent evolutions are identied in the algera of transitions. A similar methodology is also at the asis of rewriting logic [51, 53]: a logic theory is associated to a term rewriting system, in such a way that each computation represents a sequent entailed y the theory. The entailment relation is specied y means of simple inference rules, accordingly to the term algera under consideration. As an important result, equivalent computations correspond to the same sequent, and therefore deduction ecomes equivalent to concurrent computing. The tile model [32, 35] allows expressing rewrite rules with side eects, extending oth the SOS approach and also context systems to a framework where the rules have a very general format, and, as already noticed, trigger and eects extends also rewriting systems with a mechanism of rewriting synchronization. This aspect is very important when modelling process algeras via a rewrite system, ecause the ehaviour of most process algeras depends on the interaction etween agents and \the rest of the world". By analogy with rewriting logic, the tile model also comes equipped with a purely logical presentation [35], where tiles are just considered as special (proof) sequents suject to certain inference rules. Since rewriting logic can e considered as a semantic framework for the study of concurrent systems with state changes, tile logic can e thought of as a logic of concurrent systems with conditional state changes and synchronization. Given a tile system, the associated tile logic is otained y adding some auxiliary tiles and then freely composing in all possile ways (i.e., horizontally, vertically and in parallel) oth auxiliary and asic tiles. Auxiliary tiles may e necessary to represent consistent rearrangements of the interfaces due to the topological structure of the actual conguration. To give a formal denition of auxiliary structure we assume the existence of the categories of congurations and eects (e.g., states in S and actions in A of the associated transition systems are arrows of categories). The advantages of using category theory in computer science are well summarized in [36]. We just remark here the following aspects: (a) suitale classes of (structure-preserving) functors etween categories (representing transition systems) oer an immediate denition of simulation morphism etween the underlying systems; () considering categories \in the small" (i.e., ojects are states and arrows are computations), a commuting diagram may identify \computationally equivalent ehaviours", also from a concurrent viewpoint; (c) considering categories \in the large" (i.e., ojects are categorical models and arrows are simulation functors), isomorphisms may e used to characterize equivalent models; (d) universal constructions (i.e., adjunctions, (co)reections, etc.) may e used to dene a notion of optimal model; (e) (co)limits often summarize useful compositions also from a model theoretic viewpoint. Moreover, categories generalize transition systems in an ovious way: states are ojects and transitions are arrows equipped with a partial composition operator ; (associative and with identities), corresponding to the intuitive sequential composition of transitions for expressing computations (identities represent idle components of the system). As an example, monoidal categories can eectively model Petri net ehaviours [57]; in particular, for each Petri net N, there exists a freely generated strictly symmetric strict monoidal category T [N] such that the monoidal operation denes parallel composition of Best-Devillers processes, and the functoriality axiom (of tensor product ) expresses a asic fact aout the true concurrency of the model. A second example, showing that the use of categories oer a general and convenient characterization also of congurations, is given y Lawvere theories. An algeraic theory [44, 45, 40] is just a cartesian category having natural numers as ojects. The free algeraic theory associated to a (one-sorted) signature is called the Lawvere theory for, and is denoted y Th[] (also L ): the arrows from m 5

7 @? to n are in a one-to-one correspondence with n-tuples of terms of the free -algera with (at most) m variales, and composition is term sustitution. In a certain sense, a Lawvere theory is just an alternative presentation of a signature, ecause the additional structure (for tupling, projecting and permuting the elements of a tuple) is generated in a completely free way: only the operators of the signature contain information, whereas the other constructors add nothing ut auxiliary structure. From this point of view, the use of a wires and oxes notation turns out to e very useful for a visual and intuitive understanding of the role played y auxiliary structure: variales are represented y wires (we assume an implicit total order of the variales involved) and the operator of the signature are denoted y oxes laelled with the name of the operator. For instance, the term f(x 1 ; g(x 2 ); h(x 1 ; a)) over the signature fa : 0?! 1; g : 1?! 1; h : 2?! 1; f : 3?! 1g and variales x 1 < x 2 admits the following graphical representation: x 1 a h x 2 g f It should e ovious that wire duplications (e.g., of x 1 ) and crossing of wires (e.g., of x 2 and a copy of x 1 ) are auxiliary, in the sense that they elong to any wires and oxes model, independently from the underlying signature. It follows that, if we use the wires and oxes notation for congurations and eects, then this kind of operations (e.g., rearrangements of wires) elongs to oth dimensions (i.e., they are shared). Moreover, consistent rearrangements of wires on oth dimensions do not change the meaning of a rule, ut only its interface. To illustrate this point, let us consider a simple tile system where the aove signature is the signature of congurations, and 0 fs : 1?! 1; t : 2?! 1g is the signature of eects, having the following asic tiles: a y x 2 1 g 2 y z 1 w 1 t x 1 U UUUUUUUUUUUUUUU 7 7x ~ 7 ~ 7 ~ 7 ~ ~ 7 7 x t z 1 f y 1 s Q QQQQQQQQQ( z 2 h w 1 Then, it should e clear that the conguration f(a; x 1 ; g(x 2 )) should e ale to evolve to h(x 1 ; x 2 ), producing an eect s (as a result of the horizontal composition, or synchronization, of the two tiles). However, we cannot compose the tiles in the ovious way without rearranging the interfaces, ecause the arguments of trigger t are separated y a variale in the initial (input) interface of the second tile (notice the crossing of wires), while the rst tile applies only to adjacent arguments (notice that it is always possile to put an idle component in parallel with the rst tile to model the second argument of f). Thus we have the following nave characterization of auxiliary tiles: Auxiliary tiles coincide with the consistent rearrangements of interfaces in oth dimensions, where consistency means that the composition of the wire transformations induced y the initial conguration and the eect of the tile is equivalent to the composition of the wire transformations due to the trigger and the nal conguration. 6

8 Algeraic theories provide a clear mathematical representation of auxiliary constructors as suitale natural transformations, whose components are called symmetries, duplicators, and dischargers. This result will e very useful to relate our nave denition with a more formal denition. Lawvere theories introduce a very general notion of model (i.e., chosen functor from Th[] to a cartesian category with chosen products C) and model morphism (i.e., natural transformation etween two models). This fact has een well-exploited in the categorical semantics of rewriting systems. In fact, in the eld of term rewriting, the states are terms over a certain signature (i.e., arrows of the associated Lawvere theory), and rewriting steps are transitions etween two terms (with variales). It has een shown in [50], that a rewriting theory R yields a cartesian 2-category 1 L R, which does for R what a Lawvere theory does for a signature (i.e., models can e dened as 2-product-preserving 2-functors). Gadducci and Montanari pointed out in [33], that if also side-eects are to e taken into consideration during the rewriting process, then doule categories [25, 1, 41] should e considered as a natural model. A doule category can e informally descried as the superposition of a horizontal and a vertical category of cells, the former dening eect propagations, and the latter descriing state evolutions. Then, in the same way as the term algera is freely generated y a signature, and the initial model of rewriting logic is freely generated from the rules of the rewriting system, the tiles freely generate a (monoidal) doule category which constitutes the natural operational characterization 2 in the spirit of initial model semantics. In this paper we consider two main interesting cases of shared auxiliary structures. In particular the notions of Process Tile Logic and Term Tile Logic are introduced: Flat (e.g., any two sequents having the same \order" are identied, thus no emphasis is given upon the axiomatization of logic proofs) versions of process tile logic have een shown to e especially useful for dening compositional models of computation of moile calculi, and causal and located concurrent systems [27, 28]. The auxiliary tiles of process tile logic express consistent permutations of interfaces along the horizontal and vertical structures. Term tile logic should represent the ovious extension of term rewriting logic. Connections etween the two logics are particularly interesting ecause in oth logics the underlying cartesian category structure manifests itself at the level of syntax, allowing the use of the standard term notation with term sustitution as composition. The auxiliary tiles of term tile logic allow consistent permutations of interfaces along the horizontal and vertical structures (as for process tile logic), consistent free copying, and consistent projections on sucomponents. The natural semantics of process and term tile logics are given in terms of suitale classes of doule categories whose equational axioms identify intuitively equivalent tile computations. For this purpose, we introduce the notions of Symmetric strict monoidal doule categories and Cartesian doule categories (\with consistently chosen products"). As far as we know these denitions are new, ecause all the previous attempts (ased on internal constructions) for analogous notions have led to asymmetric models, where the auxiliary structure (i.e., symmetries, duplicators, and dischargers) is fully exploited in one dimension only. We elieve that this should not e the case, oth conceptually and for the kind of applications we have in mind; therefore we propose a roader notion of doule cartesianity y developing an alternative approach, following the idea of hypertransformations [25] for many-fold categories, and exploiting the results for doule categories. In particular, we dene the notion of generalized transformations, which act in oth dimensions, and assert the coherence of the two ways of transforming the structure. Then, we instantiate the denition to the special cases of symmetries, duplicators, and dischargers, in a similar way as it 1 A 2-category [41, 46] is a category C such that, for any two ojects a, and, the class C[a; ] of arrows from a to in C, forms a (vertical) category. The arrows of these hom-categories are called cells and satisfy particular composition properties. As an example, the category Cat of categories and functors is a 2-category. Actually, Cat[C; C 0 ] is the category having the functors from C to C 0 as ojects, and the natural transformations etween such functors as arrows. 2 The tiles are cells, the contexts are arrows of the 1-horizontal category, the side-eects are the arrows of the vertical 1-category, and 0-ojects model connections etween the somehow syntactic horizontal category and the dynamic vertical evolution. 7

9 0. B happens for the 1-dimensional case. Moreover, y doing that, we give evidence for the usefulness of axiomatizing the resulting doule categories, thus allowing for the denition of more signicant models than the at ones. Actually such models could also take into account the structure of proofs. This approach motivates the following formal characterization of auxiliary tiles: Auxiliary tiles for process and term tile logic are suitale generalized transformations respecting some coherence equations, where coherence means that they are uniquely dened. The comparison etween tile logic and rewriting logic is carried out y emedding their corresponding categorical models in a recently developed, more general framework, called partial memership equational logic [54, 56, 10]. In doing so, we extend the result of [58], y dening an extended version of 2-categories, called 2EVH-categories, providing a systematic connection etween models of tile logic and of rewriting logic. The idea is to \stretch" doule cells into ordinary 2-cells as pictured elow, mantaining the capaility to distinguish etween congurations and eects, whereas the auxiliary structure ecomes shared, i.e., it elongs to oth classes. s s a s 0 a + s 0 Doing this, 2EVH-categories are ale to simulate { in the sense that the algeraic structure of the original doule categories is recoverale in terms of operations on 2-cells { the structure of doule categories, where oth the horizontal and vertical 1-categories share some non-trivial structure other than ojects. In this attening process we must e careful aout two issues, namely, the possile identication of distinct doule cells, and the possile existence of 2-cells having correct horizontal-vertical partition of the source and vertical-horizontal partition of the target, ut which do not represent any doule cell. From the facts that: (1) each arrow of a 2-category can e viewed as an identity 2-cell, (2) each auxiliary operator is a shared arrow, and (3) auxiliary tiles are consistent (in the sense that the composition of s with is equivalent to the composition of a with s 0 ), it follows that 2EVH-categories allow for a third characterization of auxiliary tiles: Auxiliary tiles coincide with the possile square-shaped decompositions of the identity 2-cells associated to auxiliary constructors. We will show that the three dierent denitions of auxiliary tiles that we have sketched in this introduction coincide. Partial memership equational logic is particularly suitale for the modelling and the emedding of categorical structures, rstly ecause the sequential composition of arrows is a partial operation (e.g., it is dened if and only if the target of the rst argument is equal to the source of the second argument), and secondly ecause memership predicates over a poset of sorts allow modelling the ojects as a suset of the arrows and arrows as a suset of cells (as it is usually done in category theory). Moreover, the tensor product construction illustrated in [58] can e easily formulated in partial memership equational logic and this allows for a convenient denition of monoidal doule categories as the tensor product of the theory of categories (twice) with the theory of monoids. Though the results are very satisfying from a theoretical perspective, they cannot e applied directly to rewriting implementations of tile systems, ecause we are interested only in correct computations. Indeed, we need suitale meta-strategies to control the possile nondeterminism contained in a tile specication and in its translation. This could e summarized y saying that \the rewriting engine must e ale to lter rewriting computations". To overcome this diculty, we make use of the reective capailities [17, 18] of the rewriting logic language Maude [15] to dene suitale internal strategies [19], which help the user control the computation and collect (some of) the possile (correct) results. The key point is that the internal strategies dened here 8

10 for simulating tile systems can also e thought of as general meta-strategies for rewriting systems in general. We have experimented with Maude some executale tile specications of interesting CCS-like process calculi, and have successfully developed and applied general internal strategies to lter and collect tile computations. The structure of the paper is as follows. In Section 2 we recall some asic facts aout algeraic theories, rewriting logic, and tile logic (Section 2.1), and then we introduce the new tile models ased on process-like and term structures of congurations and eects. Each model is presented in its at version rst, then is equipped with an algera of proofs, and then naturally equivalent proof terms are equated to characterize the natural semantic framework of the logic. In Section 3, we introduce suitale categorical models for process and term tile logic, developing the notion of generalized transformation and diagonal categories to deal with symmetries, duplicators and dischargers. As a result, we propose a precise characterization of symmetric strict monoidal doule categories and cartesian doule categories with chosen products. In Section 4 and 5 we present the full comparison etween tile logic and rewriting logic through partial memership equational logic, then showing how to map tiles into ordinary rewrite rules. As a result of this comparison, we otain a correct rewriting implementation of tile logic, in which dierent tile sequents having the same \order" cannot always e distinguished. This implementation requires a meta-layer to control the rewritings, so that only tile proofs are accepted. In Section 6 we present some general meta-strategies (written in the Maude language) fullling this last requirement. In Section 7 we apply the previous results to show how Maude { thanks to its reective capailities and, in particular, to the possiility of dening internal strategy languages { can in fact e used to prototype and execute tile rewriting systems. In particular, we dene executale implementations of some CCS-like process calculi (namely, nite CCS and located CCS), preserving their original semantics. 9

11 @ 2 Tile Logic Tiles are rewrite rules with side-eects, extending the SOS approach to open systems and also to heterogeneous systems. A generic tile has the form s?! a s 0, stating that the partially specied conguration s may evolve to s 0 producing an oservale eect, ut this rewriting step is allowed if and only if the sucomponents of s evolve to the sucomponents of s 0 producing the oservation a, which is the trigger of the rule. The notions of conguration and oservation are very general here, the only requirement is that they come equipped with operations of parallel and sequential composition. In fact, tiles can e comined y means of three composition operators, extending those dened on their order: parallel ( ), horizontal ( ), and vertical ( ) composition. Parallel composition intuitively corresponds to the concurrent rewriting of disjoint components of the system. Vertical composition models successive rewriting, i.e., computations. Horizontal composition synchronizes evolutions of a conguration and its sucomponents. Although tile systems are essentially monoidal doule categories [25], the tile model allows for a purely logical presentation, where tiles are considered as sequents (suject to certain inference rules and normalization axioms), in the style of rewriting logic. Then, deduction in the tile logic exactly corresponds to computing in the tile model (i.e., applying composition rules in all possile ways, starting from a set of asic tiles), and the axioms of tile logic identify equivalent proofs of a sequent entailed y the logic. The simplest possile interpretation of structured congurations and oservations is considered in [11, 12], consisting of PT net markings. As an important result, horizontal composition in the tile model yields a notion of transition synchronization, an important feature for compositionality, missing in ordinary nets (where only token synchronization is provided), and usually achieved through complex constructions. As an another example, tile models for most process algeras [35] have process terms as congurations, and elements of the free monoid on oservale actions (which are unary symols) as oservations. However, when either causality aspects or ound names are taken into account, it is possile to consider more general horizontal and vertical structures, dealing with (local and gloal) names. Since models of computation ased on the notion of free and ound names are widespread, the notion of name sharing is essential for several applications, ranging from logic programming, -calculus and process algera with restriction (or name hiding mechanisms) to moile processes (where local names may e communicated to the external world, thus ecoming gloal names). We can think of names as links to communication channels, or to ojects, or to locations, or to remote shared resources, or, also, to some cause in the event history of the system. In general, names can e freely -converted, ecause the only important information they oer is sharing. The wires and oxes notation presented in the introduction can give an intuitive understanding of a name sharing mechanism. Let us consider a certain signature with constants 0, 1 and 2, and inary operators f and g. Then the congurations c 1 and c 2 in the picture elow can model quite dierent systems. 0 c 1 f 0 c 2 0 > > > > >???? f In a value-oriented interpretation, oth c 1 and c 2 yield the same term f(0; 0). Instead, in a reference-oriented interpretation, c 1 and c 2 dene dierent situations: in the former the two sucomponents of the f ox are uncorrelated, while in the latter they point to the same shared location. The dierence ecomes even more clear, if we assume a tile system in which the conguration 0 may e rewritten either to 1, producing an eect e 1, or to 2, with eect e 2 6 e 1, and the conguration f(x 1 ; x 2 ) may e rewritten to g(z 1 ; z 2 ) only if x 1 yields e 1 and x 2 yields e 2 as 10

12 triggers, ecoming z 1 and z 2, i.e., the asic tiles of the system are as follows: x 1 0 y 1 e 1 1 w 1 0 y 1 e 2 2 w 1 e 1 z 1 P PPPPPPPPP' x 2 Q e Q 2 QQQQQQQQ( f y 1 z 2 g w 1 Then, c 1 may e rewritten, while c 2 cannot; in fact, if we try to rewrite 0 with the rst tile, the same eect e 1 is propagated to oth arguments of f, and the conguration is stuck, ecause we cannot apply the third tile, and similarly if we try to rewrite 0 with the second tile. Term graphs [24] are a reference-oriented generalization of the ordinary (value-oriented) notion of term, where the sharing of suterms can e specied also for closed (i.e., without variales) terms 3. The distinction is made very precisely y the axiomatization of algeraic theories: terms and term graphs dier y two axioms, representing, in a categorical setting, the naturality of transformations for copying and discharging arguments [20]. Term graphs have een shown useful in [27] to dene a tile model for the (asynchronous) -calculus [60] (one of the most studied moile calculi), and in [28] to represent oth the operational and the astract semantics of CCS [59] with locations [9] within the tile model. In oth cases, at versions of the tile model are used, and the general notion of tile isimilarity [35] is employed to quotient out congurations, thus recovering the ordinary astract semantics. In this section we introduce two versions of tile logic, called Process Tile Logic, and Term Tile Logic. They model two specic situations in which the structure of congurations and oservations are quite similar, and a set of auxiliary tiles seems to capture precisely their similarity. Conguration and oservation in process tile logic are dened in terms of a suclass of directed, acyclic hyper-graphs, where each node has at most one entering (exiting) arc. The \process" terminology is taken from net theory, due to the characterization of concatenale (deterministic) processes of PT nets via symmetric strict monoidal categories [23]. Here congurations may model states of a great variety of distriuted systems (at a certain level of astraction), and oservations may exactly model causal dependencies etween the resources consumed and generated y concurrent and cooperative evolutions of distriuted agents. Models proposed in [27, 28] are essentially at process tile logic modelsa equipped with \ad-hoc" notions of sharing and garage collection. Auxiliary tiles for process tile logic are essentially tiles for consistent permutations of interfaces. Term tile logic is the natural generalization of term rewriting logic. Here, oth congurations and oservations are term algeras. Thanks to the work of Lawvere relating algeraic theories and cartesian categories, and to classical results on cartesianity (with chosen products) as enriched monoidality, the auxiliary structure which allows the generation of the term algera starting from a signature is characterized y three natural transformations called symmetries, duplicators, and dischargers. Similarly, auxiliary tiles of term tile logic are the consistent generalization of such transformations w.r.t. the two dimensions of tile systems. Intuitively, in process and term tile logic, congurations and oservations have in common the auxiliary structure, i.e., the possiility of re-arranging the interfaces as explained in the introduction. Moreover, auxiliary tiles model exactly the consistent re-arrangements, in the sense that given any auxiliary tile s?! a s0, the composition of the transformation induced y s followed y the one induced y should yield the same result as the transformation induced y a followed y the one induced y s 0. An important requirement is that there should e a unique auxiliary tile for each possile idimensional transformation, i.e., all the possile decompositions of the proof terms of auxiliary tiles yielding the same order should e equivalent. 3 Terms can share variales, ut shared suterms of a closed term can e freely copied, always yielding an equivalent term. 11

13 Notice that, although auxiliary tiles for process and term tile logic are introduced in this section, their characterization, and in particular the axioms we propose, are ased on the research concerning generalized transformations, which is the suject of Section 3. However, for the sake of an easier presentation, and to aord a etter intuitive understanding of the main ideas with the minimum machinery possile, we have chosen to reverse the \mathematically natural" order of the two formalizations. 2.1 Background Algeraic Theories We recall here some asic denitions from graph theory, used to recast the usual notion of term over a signature in a more general setting, where suitale equivalence classes of monoidal (hyper)graphs equipped with auxiliary arrows are considered. Denition 2.1 [(Hyper)Signatures] A many-sorted hyper-signature over a set S of sorts is a family f w;w 0g w;w 02S of sets of operators. A many-sorted signature is just a hyper-signature such that w;w 0 6 ; ) w 0 2 S, i.e., a family f w;s g w2s ;s2s. If S is a singleton, we denote the hypersignature (signature) is called one-sorted and is simply denoted y the family f n;m g n;m2lin (f n g n2lin ). 2 Denition 2.2 [Graphs] A graph G is a 4-tuple (O G ; A G 0 1 ), where O G is the set of ojects, A G is the set of arrows, 0 1 : A G?! O G are functions, called respectively source and target. We use the standard notation f : a?! to denote an arrow f with source a and target. A graph G is reexive if there exists an identity function id : O G?! A G such that 8a 2 O 0 (id(a)) 1 (id(a)); it is with pairing if O G is a monoid; it is monoidal if it is reexive, oth O G and A G are monoids, and the 0,@ 1, and id are monoid homomorphisms (i.e., preserve the monoidal operator and the neutral element). 2 It is immediate that a many-sorted hyper-signature over S may e seen as a graph with pairing G such that its ojects are strings on S (i.e., O G S, string concatenation :: is the monoidal operator, and the empty string is the neutral element), and its arcs are laelled with operators of the signature (i.e., f : w?! w 0 2 A G i f 2 w;w 0). For simplicity, throughout the paper we will consider one-sorted hyper-signature only, ut the results extend immediately to the many-sorted case. Denition 2.3 [Graph Theories] Given a one-sorted (hyper)signature, the associated graph theory G() is the monoidal graph with ojects the elements of the additive monoid of natural numers (i.e., 0 is the neutral element, and the monoidal operation is dened as nm n+m), and arrows those generated y the following inference rules: (generators) f 2 n;m f : n?! m 2 G() (pairing) t : n?! m; t0 : n 0?! m 0 2 G() t t 0 : n n 0?! m m 0 2 G() n 2 lin (identities) id n : n?! n 2 G() Monoidality implies that is associative on arrows, id 0 is the neutral element of the monoid of arrows, and that the monoidality axiom id nm id n id m holds for all n; m 2 lin. 2 This view is very useful to dene a chain of further structural enrichments on graphs, nally leading to the usual algeraic notion of terms over a signature. We are particularly interested in this nal level, and also in the intermediate level corresponding to symmetric theories. For the sake of simplicity, we treat here one-sorted signatures only, ut the extension to the many-sorted case should follow immediately. 12

14 Denition 2.4 [Monoidal Theories, Symmetric Theories] Given a (hyper)signature, the associated monoidal theory M() is the monoidal graph with ojects the elements of the additive monoid of natural numers (i.e., 0 is the neutral element, and the monoidal operation is dened as n m n + m), and arrows those generated y the following inference rules: (generators) f 2 n;m f : n?! m 2 M() (pairing) t : n?! m; t0 : n 0?! m 0 2 M() t t 0 : n n 0?! m m 0 2 M() (identities) n 2 lin id n : n?! n 2 M() (composition) t : n?! m; t0 : m?! k 2 M() t; t 0 : n?! k 2 M() Moreover, is associative on arrows with identity id 0, the composition operator ; is associative, and the arrows of M() satisfy the identity axiom (8t : n?! m), id n ; t t t; id m, and the functoriality axiom (s t); (s 0 t 0 ) (s; s 0 ) (t; t 0 ) (whenever compositions s; s 0 and t; t 0 are dened). The symmetric theory S() associated to the (hyper)signature is the monoidal graph generated y the same inference rules and axioms given for M(), together with the following inference rule: n; m 2 lin (symmetries) n;m : n m?! m n 2 S() Moreover, the arrows of S() satisfy the naturality axiom (8t : n?! m; t 0 : n 0?! m 0 ), and the coherence axioms (8n; m; k 2 lin), (t t 0 ); m;m 0 n;n 0; (t 0 t); nm;k (id n m;k ); ( n;k id m ); and n;m ; m;n id nm : 2 Actually, a (symmetric) monoidal theory is just a particular (symmetric) strict monoidal category [46], namely the free such category generated y the signature. Denition 2.5 [Algeraic Theories] Given a signature, the associated algeraic theory A() is the monoidal graph generated y the same inference rules and axioms given for S() together with the following inference rules: (duplicators) n 2 lin r n : n?! n n 2 A() (dischargers) Moreover, the arrows of A() verify the naturality axioms (8t : n?! m), and the coherence axioms (8n; m 2 lin), t; r m r n ; (t t); and t;! m! n ; n 2 lin! n : n?! 0 2 A() r nm (r n r m ); (id n n;m id m ); r 0 id 0! 0 ;! nm! n! m ; r n ; (1 n r n ) r n ; (r n 1 n ); r n ; n;n r n ; and r n ; (1 n! n ) id n : 2 It can e considered categorical folklore that a cartesian category can actually e decomposed into a symmetric monoidal category, together with a family of suitale natural transformations, usually denoted as diagonals and projections. Then, Def. 2.5 can e proved equivalent to the classical Lawvere theory construction Th[], dating ack to the early work of Lawvere [44]. A classical result states the equivalence of these theories with the usual term algera. 13

15 Denition 2.6 [-Algera] Given a signature f n g n2lin, a -algera is a set A, together with an assignment of a function A f : A n?! A for each f 2 n. 2 As usual, we write T to denote the -algera of ground -terms, and T (X) to denote the -algera of -terms with variales in a set X. Proposition 2.7 Let e a signature. Then, for all n; m 2 lin, there exists a one-to-one correspondence etween the set A()[n; m] of arrows from n to m in A() and the m-tuples of elements of the term algera T (X) over a set X of n variales. We elieve that the constructive denition of algeraic theories separates very nicely the auxiliary structure from the -structure (etter than the ordinary description involving the metaoperation of sustitution). Moreover, the naturality axioms of r and! allow a controlled form of duplication and discharging of information Rewriting Logic Rewriting logic [50, 51, 53] is an elegant and expressive semantic framework for the specication of languages and systems, and it is a good candidate as a logical framework in which many other logics can e represented [48, 49]. A workshop [55] has een recently dedicated to a great miscellany of dierent aspects of rewriting logic, relating many dierent sujects (oject-oriented programming, reection, external and internal strategies, dierent categorical interpretations of rewriting logic, semantic asis for language implementations, actor systems). Here we just sketch an introductory description of the suject and the original 2-algeraic semantics as proposed y Meseguer in [50]. A short summary of the reective capailities of rewriting logic will e given in Section 6.2. Let e a signature. Given a set E of -equations (i.e., sentences of the form t t 0 with t; t 0 2 T (X)), T ;E (resp. T ;E (X)) denotes the -algera of equivalence classes of ground - terms modulo the equations in E (the -algera of equivalence classes of -terms with variales in X modulo the equations in E). We denote the congruence modulo E y E, and the E-equivalence class of a -term t y [t] E, or just [t]. Denition 2.8 [Rewrite Theory] A laelled rewrite theory R is a 4-tuple (; E; L; R) where is a signature, E is a set of -equations, L is the set of laels, and R L T ;E (X) T ;E (X) is the set of laelled rewrite rules. For (r; [t]; [t 0 ]) 2 R we use the notation r : [t] ) [t 0 ]. 2 Rewrite rules in R may e understood as asic sequents entailed y R. More complex deduction in the logic of R can e otained y a nite application of four simple rules. Denition 2.9 [Rewriting Sequents] Let R (; E; L; R) e a rewrite theory. We say that R entails a at sequent [t] ) [t 0 ], written R ` [t] ) [t 0 ] i [t] ) [t 0 ] can e otained y a nite numer of applications of the following rules of deduction. Reexivity Congruence Replacement [t] 2 T ;E (X) [t] ) [t] [t 1 ] ) [t 0 1]; : : : ; [t n ] ) [t 0 n]; f 2 n [f(t 1 ; : : :; t n )] ) [f(t 0 1 ; : : :; t0 n )] [w 1 ] ) [w 0 1]; : : :; [w n ] ) [w 0 n]; r : [t(x 1 ; : : :; x n )] ) [t 0 (x 1 ; : : :; x n )] 2 R [t(~w~x)] ) [t 0 (~w~x)] 14

16 Transitivity [t 1 ] ) [t 2 ]; [t 2 ] ) [t 3 ] [t 1 ] ) [t 3 ] where t(~w~x) denotes the simultaneous sustitution of w i for x i in t. 2 A rewrite theory is just a static description of \what a system can do". The meaning of the theory should e given y computational models of its actual ehaviour. Taking advantage of the correspondence etween deductions in rewriting logic and (concurrent) computations, it is natural, in the spirit of initial model semantics, to dene the initial model T R of R as a system whose states are E-equivalence classes of -terms, and whose transitions are equivalence classes of terms representing proofs in rewriting deduction, i.e., concurrent rewritings using the rules in R. The rules for generating such proof terms are otained from the rules of deduction of Def. 2.9 y decorating the sequents with appropriate proof terms. Denition 2.10 [Proof Terms of Rewrite Logic] Let R (; E; L; R) e a rewrite theory such that each rewrite rule has a dierent lael. We say that R entails the proof term : [t] ) [t 0 ], written R ` : [t] ) [t 0 ] (or just R ` ), i the proof term is generated y a nite numer of applications of the following decorated rules of deduction. Identities -structure Replacement Composition [t] 2 T ;E (X) [t] : [t] ) [t] 1 : [t 1 ] ) [t 0 1 ]; : : : ; n : [t n ] ) [t 0 n]; f 2 n f( 1 ; : : :; n ) : [f(t 1 ; : : :; t n )] ) [f(t 0 1; : : :; t 0 n)] 1 : [w 1 ] ) [w 0 1]; : : :; n : [w n ] ) [w 0 n]; r : [t(x 1 ; : : :; x n )] ) [t 0 (x 1 ; : : :; x n )] 2 R r( 1 ; : : :; n ) : [t(~w~x)] ) [t 0 (~w~x)] : [t 1 ] ) [t 2 ]; : [t 2 ] ) [t 3 ] : [t 1 ] ) [t 3 ] Each of the rules presented aove denes a dierent operation, taking certain proof terms as arguments and returning a resulting proof term. In other words, proof terms form an algeraic structure P R (X) consisting of a graph with nodes T ;E (X), with identity arrows, and with operations f (for each f 2 ), r (for each rewrite rule), and (for composing arrows). 2 Notice that we use diagrammatic order for the sequential composition of proofs, and that the composition operator is denoted y the same symol of vertical composition of natural transformations to enhance the relations with the categorical semantics descried at the end of this section. Denition 2.11 [Model T R (X)] Given a rewrite theory R, the model T R (X) of R is the quotient of the algera of proof terms P R (X) modulo the following equations (when composition of arrows is involved, we always implicitly assume that the corresponding source and target match): Category Associativity: 8; ; ; ( ) ( ) 15