1 2 A 2 TG 2 1 A 2 C1

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1 Linear Ordered Graph Grammars and Their lgeraic Foundations? Ugo Montanari and Leila Rieiro Dipartimento di Informática, Universitá di Pisa, Pisa, Italy Instituto de Informática, Universidade Federal do Rio Grande do Sul, Porto legre, razil stract. In this paper we define a special kind of graph grammars, called linear ordered graph grammars, that can e used to descrie distriuted systems with moility and oject-ased systems. Then we show how to model such grammars and their semantics in terms of tiles making explicit the aspects of interactivity and compositionality, that are of great importance in distriuted systems. Introduction Wide area network programming is important and difficult to cope with using ad hoc techniques. In particular, testing in such environments has shown to e very inefficient due to the high dynamicity and possiility of failures of the environment. Thus, formal techniques to assure the correct ehavior of the system are even more needed in this framework. Graph transformation systems [8, 7] have een used in a wide variety of applications, and can provide suggestive and technically adequate models of computation, semantic foundations and verification methods. There have een many approaches to model distriuted systems using graph transformation systems [9]. Considering that our aim is to e ale to model moility of processes within a system, a natural way of representing distriuted systems graphically is to model communication channels as nodes (astracting out the physical network), agents/processes/amients as (hyper)arcs and ports as tentacles (for example, the modeling of ß-calculus agents using graph transformations defined in [6]). If we want to have a more oject-oriented view, we could see ojects as nodes and messages as (hyper)arcs, as it was done in [6]. What is important to notice is that in oth modelings, moility involves the creation and passing of nodes, while the process/message interpretation of arcs requires resource consciousness ( linearity" in Girard sense). In the graph transformation modelings discussed aove, this means that neither rules nor matches for rules may identify arcs (ecause they model resources). ctually, if we do not have this linearity,? This work was partially supported y the projects GILE (FET project), COMET (MIUR project), IQ-Moile (CNPq and CNR) and ForMOS (CNPq and Fapergs).

2 or resource consciousness, property, the construction of a suitale compositional true concurrency semantics (ased on concatenale processes, for example) is not possile (this will e further explained in Sect. ). In this paper we propose to specialize the existing single pushout (SPO) approach to graph transformations [, 8] to a kind of grammars that can e used to model oth distriuted systems with moility and oject-ased systems. These grammars will e called linear ordered graph grammars. The linearity property is achieved y ordering nodes and arcs, and allowing only injective morphisms on items that represent resources (arcs). Interactivity and astract semantics via oservations are key features of communicating systems. Compositionality is also very important for large distriuted systems. Tiles have these as their main aims, and previous work has shown their feasiility [0, 9]. In addition, they exhiit an algeraic flavor (tiles have a straightforward axiomatization as monoidal doule categories) which may allow for universal constructions, compositionality and refinement in the classical style of algeraic semantics [5]. We show that the representation of SPO graph transformations of linear ordered graph grammars within tile logic is simple and natural. In this way, we provide a asis for an algeraic theory of graph grammars where oservations and interactions are in the foreground, and this is not usually emphasized in the DPO/SPO literature, although it is a very important aspect of the kind of systems we are interested in modeling This paper is organized as follows: Sect. reviews the asic concepts of SPO graph rewriting using typed hypergraphs, Sect. presents the main concepts of tile systems, introducing the kind of arrows needed for modeling graphs and (partial) graph morphisms as tiles, Sect. introduces linear ordered graph grammars and shows how their syntax and semantics can e represented as tiles/tile rewrite systems. Section 5 rings a summary of our main results. Graph Rewriting In this section we will introduce the asic concepts of graph rewriting in the single pushout (SPO) approach [, 8]. However, instead of the standard definitions of morphisms ased on total morphisms from sugraphs, we will explicitly use partial functions and a weak commutativity requirement (instead of the usual commutativity). In [] it was shown that the two definitions are equivalent. Moreover, the definition ased on partial functions has two advantages: it allows us to define typed hypergraph categories in an easy way (as instances of generalized graph structure categories), and it makes clearer the correspondence to the tile rewriting model that will e defined in Sect.. Definition (Weak Commutativity). For a (partial) function f : with domain dom(f), let f? : ψ dom(f) and f : dom(f) denote the corresponding domain inclusion and the domain restriction. Given functions a : 0, : 0 and f 0 : 0 0, where a and are total, we write f 0 ffia ffif and say that the diagram commutes weakly iff f 0 ffiaffif? =ffif. Remark. If f and f 0 are total, weak commutativity coincides with commutativity. Note that ((f?) ;f) is a factorization of f.

3 The compatiility condition defined aove means that everything that is mapped y the function must e compatile. The term weak" is used ecause the compatiility is just required on preserved (mapped) items, not on all items. Now we will define hypergraphs and partial morphisms. In this paper we will use undirected hypergraphs, and therefore we will only define a source function connecting each arc with the list of nodes it is connected to. We use the notation S Λ to denote the set of all (finite) lists of elements of S. Definition ((Hyper)Graph, (Hyper)Graph Morphism). (hyper) graph G =(N G ; G ; source G ) consists of a set of nodes N G, a set of arcs G and a total function source G : G NG Λ, assigning to each arc a list of nodes. (partial) graph morphism g : G H from a graph G to a graph H is a pair of partial functions g N : N G N H and g : G H which are weakly homomorphic, i.e., gn Λ ffi sourceg source H ffi g (gn Λ is the extension of g N to lists of nodes). morphism is called total if oth components are total. The category of hypergraphs and partial hypergraph morphisms is denoted y HGraphP (identities and composition are defined componentwise). To distinguish different kinds of nodes and arcs, we will use a notion of typed hypergraphs, analogous to typed graphs [5, ]. Definition (Typed Hypergraphs). typed hypergraph is a tuple HG TG = (HG;type HG ;TG) where HG and TG are hypergraphs, called instance and type graph, respectively, and type HG : HG THG is a total hypergraph morphism, called typing morphism. morphism etween typed hypergraphs HG TG and HG TG is a partial function f : HG HG such that type HG type HG ffi f. The category of all hypergraphs typed over a type graph TG, denoted y THGraphP(TG), consists of all hypergraphs over TG as ojects and all morphisms etween typed hypergraphs (identities and composition are the identities and composition of partial functions). For the well-definedness of the categories aove and the proof that these categories have pushouts we refer to [] (these categories can e constructed as generalized graph structures categories). The construction of pushouts in such categories is done componentwise, followed y a free construction to make the source and type components total (actually, as types are never changed y morphisms, this free construction is not needed in the case of typed graphs). In the rest of the paper, we will usually denote typed hypergraphs just y graphs. Definition (Graph Grammar). Given a (hyper)graph TG,a rule is a morphism in THGraphP(TG). graph grammar is a tuple GG =(T G; IG; Rules) where TG is a (hyper)graph, called type graph, IG is a (hyper)graph typed over TG,called initial or start graph, and Rules is a set of rules over TG. Note that, due to the use of partial morphisms, this is not just a comma category construction: the morphism type is total whereas morphisms among graphs are partial, and we need weak commutativity instead of commutativity. In [] a way to construct such categories was defined.

4 Definition 5 (Derivation Step, Sequential Semantics of a Graph Grammar). Given a graph grammar GG = (T G; IG; Rules), a rule r : L R Rules, and a graph G typed over TG, a match m : L G for rule r in G is a total morphism in THGraphP(TG). derivation step G =) r;m G using rule r and match m is a pushout in the category THGraphP(TG). The morphism r 0 is called co-rule. m L G r (PO) r 0 R m 0 G derivation sequence ofgg isasequence of derivation steps r i;m i G i =) G i+, i f0::ng, n IlN, where G 0 = IG, r i Rules. The composition of all co-rules ri 0 is called derived production ofasequential derivation. The sequential semantics of GG is the set of all sequential derivations of GG. n example of a rule and derivation using typed hypergraphs is shown in Figure. rcs are drawn as squares and nodes as ullets. The tentacles of the arcs descrie the source function and are numered to indicate their order. Indices on nodes and arcs here just indicate different instances of items with same types. Intuitively, arcs represent resources, for example agents or amient names, and nodes represent amients. The rule may descrie a kind of open operation of amients and the creation of a new amient: agent is preserved, the amient represented y ffl (that is inside amient ffl with name ) is opened, having the effect that all agents and amients that were inside ffl efore the application of the rule are now inside amient ffl, and a new amient inside ffl is created (amient ffl of R, named C). Note that this interpretation does not correspond exactly to a rule of the amient calculus, since the agent that triggers the execution () is preserved, instead of eing sustituted y its continuation. This rule was chosen to illustrate using one example that our framework is expressive enough to model name fusion, a feature that is needed in some calculi (for example, in the fusion calculus y Parrow and Victor, as well as in the amient calculus) and preservation of resources, needed to model, for example, replication ( operator in the pi-calculus). L r R TG m m C C G H C r C C 5 Fig.. Derivation using rule r at match m

5 Tiles and lgeraic Semantics In this section we will define the asic notions of tile rewrite systems. Each tile can e seen as a square consisting of four arrows: the horizontal ones descrie states, and the vertical ones oservations. First we define some monoidal theories that will e used to uild suitale arrows of tiles that correspond to graphs and graph morphisms. Then we define a tile rewrite system ased on such arrows. The following definition is a slight restriction of the definition in [7] to consider only undirected hypergraphs. Definition 6 (One-sorted (hyper-)signatures). hyper-signature ± = (S; OP ) consists of a singleton S = fsg and a family OP = fop h g hiln of sets of operators with h arguments and no result (no target sort), where h is a natural numer. Given an operation op h OP, h is the arity of op h, and the domain of op h is denoted ys h. ± n denotes the set of operators of OP with arity n. Now we define an extension of a signature ± that will add as sorts all operation names in ± and as target of each operation the corresponding sort. Definition 7 (Extended (hyper-)signatures). Given a signature ± = (S; OP), its extension is a signature ± E =(S E ;OP E ), where S E = fsg[fopjop : n 0 OPg and OP E = fop E h : sh op h jop h : s h 0 OPg.. Monoidal Theories We will define algeraic theories to descrie graphs and graph morphisms. In particular, the gs-monoidal theory defined here is an extension of the one defined in [] ecause we consider different arrows for sorts of an extended hyper signature: for the only sort coming from the underlying hyper signature, we use the usual definition of gs-monoidal theories, ut for the other ones, we do not allow the duplicator and discharger operators. In addition, we define a new theory: the partial graph morphism (pgm) theory, allowing discharger and new operators for sorts that are not in the underlying signature only (the rest is the same as in the gs-monoidal theory). The reason for this will ecome clear in the next section, where we will use gs-monoidal theories to model (hyper)graphs and pgm-monoidal theories to model special (hyper)graph morphisms. Definition 8 (Connection diagrams). connection diagram G is a -tuple ho G ; G ;ffi 0 ;ffi i: O G, G are sets whose elements are called respectively ojects and arrows, and ffi 0 ;ffi : G O G are functions, called respectively source and target. connection diagram G is reflexive if there exists an identity function id : O G G such that ffi 0 (id a ) = ffi (id a ) = a for all a O G ; it is with pairing if its class O G of ojects forms a monoid; it is monoidal if it is reflexive with pairing and also its class of arrows forms a monoid, such that id, ffi 0 and ffi respect the neutral element and the monoidal operation. In the following we will use the free commutative monoid construction over a set S, denoted y (S Ω ; Ω; 0). The elements of S Ω can e seen as lists of elements of S, and will e referred to using underlined variales. If S is a singleton set, we will denote elements of S Ω y underlined natural numers (indicating the numer of times the only element ofs appears in the list, for example s Ω s Ω s will e denoted y ).

6 id x Ω id y for all x;y S Ω E. Tale. Inference Rules Definition 9 (Graph theories). Given a (hyper)signature ± =(S; OP ) and its extension ± E =(S E ;OP E ), the associated graph theory G(±) is the monoidal connection diagram with ojects the elements of the free commutative monoid over S E ((S E ) Ω ; Ω; 0) and arrows generated y the inference rules generators, sum and identities shown in Tale satisfying the monoidality axiom id xωy = (generators) f ± n f : n f G(±) (sum) s : x y;t: x 0 y 0 s Ω t : x Ω x 0 y Ω y 0 G(±) (identities) x S Ω E id x : x x G(±) (composition) s : x y;t: y z s; t : x z M(±) (duplicators) n S Ω r n : n n Ω n GS(±) (dischargers) n S Ω n : n 0 GS(±) (permutations) x;y S Ω E ρ x;y : x Ω y y Ω x GS(±) Definition 0 (Monoidal theories). Given a (hyper)signature ± =(S; OP ) and its extension ± E =(S E ;OP E ), the associated monoidal theory M(±) is the monoidal connection diagram with ojects the elements of the free commutative monoid over S E ((S E ) Ω ; Ω; 0) and arrows generated y the following inference rules: generators, sum and identities (analogous to the rules in Def. 9), and the rule composition in Tale. Moreover, the composition operator ; is associative, and the monoid of arrows satisfies the functoriality axiom (s Ω t); (s 0 Ω t 0 )=(s; s 0 ) Ω (t; t 0 ) (whenever oth sides are defined) and the identity axiom id x ; s = s = s; id y for all s : x y. Definition (gs-monoidal theories). Given a (hyper)signature ± =(S; OP ) and its extension ± E =(S E ;OP E ), the associated gs-monoidal theory GS(±) is the monoidal connection diagram with ojects the elements of the free commutative monoid over S E ((S E ) Ω ; Ω; 0) and arrows generated y the following inference rules: generators, sum, identities and composition (analogous to the rules in Def. 0), and the rules duplicators, dischargers and permutations in Tale. Moreover, the composition operator ; is associative, and the monoid of arrows satisfies the functoriality axiom (see Def. 0); the identity axiom (see Def. 0); the monoidality axioms: for all n;m S Ω and x;y;z S Ω E ρ xωy;z =(id x Ω ρ y;z ); (ρ x;z Ω id y ) 0 = r 0 = ρ 0;0 = id 0 ρ 0;x = ρ x;0 = id x xωy = x Ω y r nωm =(r n Ωr m ); (id n Ω ρ n;m Ω id m )

7 the coherence axioms: for all n S Ω and x;y S Ω E r n ;(id n Ωr n )=r n ;(r n Ω id n ) r n ; ρ n;n = r n r n ;(id n Ω n )=id n ρ x;y ; ρ x;y = id x Ω id y the naturality axiom: for all s : x y;t: z w and Ω t); ρ y;w = ρ x;z ;(t Ω s) (s of gs-monoidal theories that are constructed without the generators axiom rrows called asic gs-monoidal arrows. The theory otained only with asic are is denoted ygs(±). arrows [] it was shown that term graphs (a particular kind of hierachical hy- In could e suitaly represented as arrows of a gs-monoidal theory. pergraphs) we are rather interested in non-hierachical graphs, and the restriction is Here y selecting signatures where operators have output types which are achieved from input types, i.e. operators cannot e nested. Ojects represent disjoint of variales (nodes) and an arrow n m represents the application of a lists of operators (arcs) to the variales n generating the results (roots) (multi)set For example, assuming that the sort of nodes of L is fsg, the graph L in m. represents the union of term-graphs (ffl ; ffl ) and (ffl ) sharing the Figure ffl. This term-graph can e modeled as the following gs-monoidal arrow: variale s;s ;(id s Ωr ); ( Ω ) :s Ω s Ω. Graphically, this arrow is represented ρ Figure ( and are the first and second occurrences of sort s in s Ω s). in In the pictures, the operator Ω and the underlines in the domain Notation: codomain of the arrows will e omitted. Moreover, we will draw indexed and (ffl) in the graphical representation to descrie the instances of sorts. ullets theory defined elow will e used to model a special kind of graph The morphisms that are total on nodes (and may identify nodes), and are morphism: on arcs (and may not identify arcs). Identification of nodes is achieved partial the r operator, as in gs-monoidal theories. The possiility of deleting arcs is y (pgm-monoidal theories). Given a (hyper)signature ± =(S; Definition and its extension ± E =(S E ;OP E ), the associated pgm-monoidal the- OP) PGM(±) is the monoidal connection diagram with ojects the elements of ory free commutative monoid over S E ((S E ) Ω ; Ω; 0) and arrows generated ythe the inference rules: sum, identities, composition and permutations (analogous following to the rules in Def. ), and x (S E S) Ω x S Ω E L : ss > Fig.. rrow of a gs-monoidal theory achieved y the operator, yielding partial functions (only for arcs). (new) x :0 x PGM(±) S Ω n n : n n Ω n PGM(±) r (duplicators) (dischargers) x : x 0 PGM(±)

8 composition operator ; is associative and the monoid of arrows satisfies The functoriality axiom (Def. 0); the identity axiom (Def. 0); the monoidality the y, for all x; y; S Ω E ); the coherence axiom (Def. ); and the naturality axiom (Def. ). z theory otained using the same ojects and arrows generated y the rules The except the composition is called short pgm-graph theory, denoted y aove, The theory otained using all rules except new and with discharger spgm(±). that in pgm-monoidal theories we did not use the generators axiom. Note has the effect that arrows of these theories do not correspond to graphs as This theories (ecause arcs are not allowed), they rather descrie relationships gs-monoidal etween the ojects (that are lists of nodes and arc laels). Moreover, arrow ofpgm(±) is also an arrow ofgs(±). an the following definition, we use the fact that asic gs-monoidal arrows In one sorted signatures correspond to total functions in the inverse direction, of is, each gs-monoidal arrow n m corresponds to a total function m n that [] for the proof). s asic pgm-monoidal arrows are also asic gs-monoidal (see they also correspond to total functions. This will e used to construct arrows, squares of such arrows ased on pushouts of functions. These pushouts pullack model the node-component morphisms of rule applications. will (Derivation Pair). Given one-sorted signatures ± h and ± v, Definition arrows s : n m GS(± h ) and t : q m PGM(± v ). The deriva- and pair of s and t, denoted yderip air(s; t), isapair of a asic gs-monoidal tion a asic pgm-monoidal arrows (s 0 : p q;t 0 : p n) such that the inverse and Tile Rewriting Systems. (Tile sequent, Tile rewrite system). Let ± h and ± v e two Definition arrows of GS(± h ) while a : x p and : y q are arrows of PGM(± v ). are l, r, a and are called respectively the initial configuration, the final rrows the trigger and the effect of the tile. Trigger and effect are called configuration, Underlined strings x, y, p and q are called respectively the initial oservations. interface, the initial output interface, the final input interface and the final input interface. output a r where oservations a and are arrows of short pgm-graph theory spgm(± v ) (i.e. no sequential composition is allowed the uild them). to tile rewrite system (trs) R is a triple h± h ;± v ;Ri, where R is a set of h -± v short tile sequents called rewrite rules. ± trs R can e considered as a logical theory, and new sequents can e from it via certain inference rules. derived x Ω 0 = ρ 0;0 = id 0 and xωy = axioms (all of Def. plus 0 = only for x S Ω is called asic pgm-graph theory, denoted ypgm(±). square is a pushout in the category of sets and total functions. a (one sorted) signatures, called the horizontal and the vertical signature respec- ± h -± v tile sequent is a quadruple l tively. r, where l : x y and r : p q short tile sequent is a tile l

9 Definition 5 (pgm-monoidal tile logic). Let R = h± h ;± v ;Ri e a trs. Then we say that R entails the class R of the tile sequents s a t otained y finitely many applications of the inference rules depicted in Tale. Tale. pgm-monoidal Tile Logic asic rules: (generators) s a s a Composition rules: (h-comp) t R t R (h-refl) s : x y GS(± h) id s = s idx s R ff = s a c (p-comp) ff = s a t; ff 0 = s 0 c t 0 R t; t 0 R idy t; ff 0 = s 0 a 0 t 0 R 0 ff Ω ff 0 = s Ω s 0 aωa0 t Ω t 0 R Ω 0 (v-comp) ff Λ ff 0 = s; s 0 a uxiliary rules: (perm) a : x y;: x0 y 0 PGM(± v) ρ a; = ρ x;x 0 (v-refl) a : x y PGM(±v) id a = id a x id y R ; a ff = s a aω ρ y;y 0 R Ωa u; ff 0 = u a0 0 ff ff 0 = s a;a0 t R ; 0 t R ; (Pnodes) s : n m GS(± h);a: q m PGM(± v) (s 0 ;a 0 )=derip air(s; a) s a0 s 0 R a Linear Ordered Graph Grammars Now we will define a restricted type of graph grammar: the nodes and arcs of each graph are ordered, the rules do not delete nodes and do not collapse arcs, and the matches do not collapse arcs. With this kind of restriction, the derivation steps can e otained componentwise as a pushout of nodes and pushout of arcs (ecause no nodes are deleted). The motivation for the definition of this kind of graph grammar is mainly ased on the applications we have in mind: moile code and oject-ased systems. The idea is to model static components (places, communication channels, ojects) as nodes and resources (processes, agents, amients, messages) as arcs. In this view, it is natural to require that we have a different way to handle these two kinds of entities, and that is why wehave a distinct treatment of nodes and arcs in a linear ordered graph grammar. The fact that nodes and arcs are ordered was imposed to achieve a suitale true concurrency semantics. In the unordered case, the description of concurrency is analogous to the collective view of tokens in a Petri net, that leads to a different semantical model as the individual token philosophy. The latter is the standard way of considering the semantics of graph rewriting. good comparison of the approaches for the case of Petri nets can e found in []. To illustrate this situation, consider the grammar depicted in Figure. There is only one rule, that deletes and creates arcs of type. The start graph of the grammar has two -arcs. Some possile ehaviors of this grammar are processes P and P, that replace, using rule r, the and -arcs of IG respectively y other -arcs. If we consider that arcs are not ordered, these processes are isomorphic. Let process P e the concatenation of P and P. This

10 process has two applications of rule r, ut we can not distinguish whether these applications are dependent or independent, that is, the process in which the second application of r depends on the first is equivalent to the one in which they are independent. This contradicts the asic idea of a true concurrency semantics, in which dependencies etween actions are essential. IG L r R Fig.. Start Graph IG and Rule r Definition 6 (Linear Ordered Graph Grammars). linear ordered (hyper)graph over a type graph TG =(fsg; TG ; source TG ) is a typed hypergraph HT TG with HG =(N; HG ; source HG ) where N = f0::ng, with n IlN, and HG = S a TG a HG, with a HG = fa iji f0::mgg, where m is the cardinality of fha HG jtype HG (ha) = ag. morphism etween linear ordered hypergraphs is simply any morphism typed hypergraph morphism. linear ordered graph grammar is a graph grammar GG =(T G; IG; Rules) where TG has only one type of nodes, IG and all graphs in the rules of Rules are linear ordered graphs and for each rule r =(r N ;r ):L R of Rules, r is injective and r N is total. linear ordered match m =(m N ;m ) is a total graph morphism where the m component is injective. The semantics of a linear ordered graph grammar is defined as for a usual graph grammar, taking into account that only linear ordered matches are allowed, that is, a match is resource conscious on arcs, ut may identify nodes. In the following sections, we will show how to descrie the syntax and semantics of such a graph grammar using tiles and tile rewriting systems.. Syntax: Graphs and Rules Consider the linear ordered derivation step shown in Figure. The graphs used in this derivation are hypergraphs having only one type of node and three types of (hyper)arcs, having the following arities (consider the sort of nodes of TG to e s): : s s; : s; C : s s. Our idea of representing graphs and rules as tiles is as follows: Graphs: Graphs will e modeled as the horizontal components of the tiles. Let TG =(fsg; ; source TG )eatype graph. ased on this graph we can uild a siganture ± h =(fsg; ± ), where ± contains all arcs of as operations (the information aout the arity is given y the source TG function). graph will e then an arrow x y of the corresponding gs-monoidal theory, where x;y (fsg]) Ω. This means that the mapping from x to y is constructed y using, esides the arcs of the graph, the operations of identity and permutation for all sorts, and for elements of sort s (nodes), we additionally allow duplication and discharging, to allow that the same node may e used as source of many hyperarcs, and that it may not e used y any hyperarc. For the hyperarcs of the graph we include a target function that assigns to each arc an occurrence of its type (indexed y a natural numer). For example,

11 graph L can e descried y the gs-monoidal arrow illustrated in Figure the Usual (closed) graphs correspond to the special case of G : x y when we. x fsg Ω and y Ω. Other graphs are called open graphs, and will e have as auxiliary components to allow the modeling of the direct derivation. used rule r : L R is a (partial) graph morphism r = (r N ;r ). Such Rules: rule can e represented as a tile having as horizontal arrows the graphs a and R, and as vertical arrows mappings that allow to glue nodes, delete L preserve and create arcs and nodes. These vertical arrows are arrows arcs, the pgm-monoidal theory for the signature ± v = ± h. Rule r of Figure of corresponds to the tile of Figure (vertical mapping is shown as dashed ). Note that at the west side of the tile r we have modeled the component y N of the rule, whereas in the east side we model the component r. r 7 (Signature of a Type Graph). Given a type graph TG = Definition ; source TG ), the corresponding hypergraph signature is ± TG =(N; TG ), (N; characterization elow descries which arrows of a gs-monoidal theory The to graphs. Other arrows will e called open graphs. The proof was correspond due to space limitations (similar proofs can e found in [] and [7]). omitted (Characterization of graphs). The graphs typed over TG= Proposition ; source TG ) with N eing a singleton are the arrows of kind n a of a (N; theory with signature ± TG, where n N Ω and a (N E N) Ω, gs-monoidal characterization elow descries a kind of graph morphism that may The nodes ut not arcs, and may e partial on arcs, ut not on nodes. The identify of a linear ordered graph grammar are exactly this kind of morphisms. rules (Characterization of graph morphisms). Graph morphisms Proposition over TG are tiles having as horizontal and vertical signature ± TG where typed north and south sides are the arrows corresponding to (closed) graphs (the the and right-hand sides of the rule, respectively), and the vertical arrows are leftof PGM(± TG ). arrows n SPO-rule consists of a left-hand side L, a right-hand side R and a Proof. graph morphism r =(r N ;r ):L R. Consider the gs-monoidal arrows partial L : n a and s R : n 0 a 0 corresponding to graphs L and R, respectively. s s N is a total function, there is a corresponding gs-monoidal arrow N : n 0 n r arrows, creation - new - is denoted y the, and deletion - ang - is denoted L : ss > R : ss > C C C Fig.. Modeling a rule as a tile where TG = fa : source TG (a) 0ja g. with ± E TG =(N E ; E ). mapping the nodes of R into the nodes of L. rrow N PGM(± TG ) ecause

12 r N is just a mapping of nodes, and this can e modeled just y using identities, duplicators, discharger and permutation operators for the sort of nodes, and the corresponding rules may e used to uild PGM(± TG ). The function that maps arcs of L into arcs of R does not allow the identification of arcs, ut allows to delete and create arcs. Therefore, we need identities, dischargers, new and permutation operators for the sorts of arcs, and corresponding rules are allowed to uild PGM(± TG ). The other direction can e proven analogously.. Semantics Now we will show how a direct derivation of a grammar GG can e modeled y a suitale composition of tiles of the pgm-monoidal tile logic R otained y the TRS h± h ;± v ;Ri, where ± h = ± v = ± TG (as discussed aove), and R is the set of tiles representing the rules of GG. derivation G =) r;m H using rule r : L R at match m : L G can e otained as a composition of tiles that will give raise to a tile r 0 : G H. This can e done in steps:. Context Graph: Construct the (context) graph G 0, that contains all nodes of G and all arcs that are in G and are not in the image of the match m (that is, G 0 is G after removing the deleted and preserved arcs). Note that G 0 must e a graph ecause we donot allow rules that delete nodes (and therefore G 0 can not contain dangling edges). For the example derivation in Figure we would have the graphs corresponding to G and G 0 in Figure 5 (note that the arc of G 0 corresponds to the arc of G that name in G 0 must have the index ecause it is the first occurrence of sort in C). G : ssss > C C C G : ssss > C C C Fig. 5. Graph G and context graph G 0. Tile : Construct the tile r Ωid G 0, that is the parallel composition of the tile corresponding to rule r and the identity tile of graph G 0. Note that, as L and G are graphs, the west side of this tile has only nodes, whereas the east side has only arcs. This tile elongs to R ecause rule r R (rule generators), id 0 G R (rule h-refl) and thus the composed tile also elongs to R (rule p-comp). In the example we will get the tile (a) in Figure 6.. Tile : Construct the tile corresponding to the match and derivation on the node component: the north side of this tile corresponds to m N (the component of the match morphism that maps nodes), the east side is the west side of the tile otained in step, and the remaining sides will e mappings of nodes such the the resulting square commutes and has no nodes that are not in the north or east sides already. Considering that the algeraic structures used to uild these mappings are functions in the inverse direction, this square will e a pushout in the category of sets and partial functions (in this case, all functions are actually total ecause deleting of nodes is not allowed in our setting). This tile elongs to R due to rule Pnodes, that says

13 that all such pushouts (considered as pullacks in the opposite direction) are auxiliary tiles of R. In our example, the tile otained y this construction is shown in Figure 6 (). L + G : ssssss > C 5 6 C C ml : ssss > ssssss 5 6 (a) () C 5 6 C R G : ssssss > CC + C C mr : ssss > ssssss 5 6 Fig. 6. (a) Tile () Tile. Resulting Tile: The result of the application of rule r at match m is then given y the sequential composition of the tiles otained in the last steps: Tile Λ Tile (otained y the application of rule h-comp of R). In the example, this is shown in Figure C C 5 6 C C C C Fig. 7. Resulting tile Note that we did not need to construct an auxiliary tile that corresponds to the pushout of arcs, as we did for the nodes (in tile ). This is due to the fact that we do not allow to identify arcs neither in the rule nor in the match, and therefore the east-south component of tile gives us already the arcs of the resulting graph of the derivation. In the classical SPO-setting, the corresponding pushout of arcs would have exactly the same effect: it would put all arcs that are in the right-hand side together all arcs that are in G and not in the image of the left-hand side of the rule (ecause the rule and the match are injective on arcs and the match is total). In case we would allow identification of arcs we would need a step for arcs corresponding to step.

14 Definition 8 (lgeraic Semantics of a Linear Ordered Graph Grammar). Given agraph grammar GG =(T G; IG; Rules), its algeraic semantics, denoted ylg(gg), is given y the tile rewrite system R = h± TG ;± TG ;Ri, R is the set of tiles corresponding to the rules of GG. r Proposition. ;m r n;r m Let ff = G 0 =) G ) =) G n easequential derivation of a graph grammar GG. Then there is a tile s a t in lg(gg) such that s and t are the graphs corresponding to G 0 and G n,respectively, and (a; ) corresponds to the derived rule of the sequential derivation ff. Proof sketch. s discussed aove, all derivation steps of a grammar can e constructed using the tile corresponding to the rule, identity tiles and pullack tiles of nodes, composed using parallel and sequential composition. ccording to the definition of tile rewrite system, these tiles and operations can e used to uild composed tiles, and therefore a tile corresponding to the derived rule must e in this tile rewrite system. Moreover, vertical composition of tiles is allowed, yielding the composition of co-rules. Note that the vertical composition of tiles is done y composing the west and east sides, and these sides correspond to the mapping of nodes and arcs of the involved graphs, respectively. The derived rule otained y the composition rn 0 ffi ffir 0 is also constructed componentwise (composition of partial graph morphisms). Proposition. Given a graph grammar and a tile rewrite system lg(gg), the horizontal tiles of lg(gg) that use one rule of GG (one generator tile) and have graphs as north and south sides correspond to derivation rules of GG. Remark: Horizontal tiles are the ones that are uilt without using the rule for vertical composition. If we allow more than one rule, the resulting tile would correspond to a parallel application of rules. Proof sketch. ssume that a tile s a t uses one rule of GG and has graphs as north and south sides. If this tile is not otained y composition of other tiles, it must e a generator, that is, a rule of GG. Generators correspond to rules of GG and thus such tiles correspond to derivations of kind L r;idl =) R, with r : L R. ll tiles using tile Tr corresponding to rule r can otained y composition can e rearranged (using the axioms) to the form T ; (T Ω TrΩT); T, where T has only nodes, T and T have graphs as north and south sides, and T has only arcs (elements of sorts i). Thus, T must e a Pnode tile ecause it must e the composition of identities and Pnode tiles (due to compositionality of colimits, the resulting tile must e a Pnode). T ΩT can e seen as the context graph to which the rule is applied to. s T and T must e closed graphs, they can only e uilt y parallel composition of other closed graphs, and therefore must e compositions of identities (they can not e rules ecause Tr is the only rule in this tile, and can not e Pnodes or permutations of nodes/arcs ecause all these tiles have een rearranged to the component T /T ). If T is not empty, it is just a permutation of arcs, depending on which permutation we take, we get all different isomorphic results of a direct derivation (on the arc component, on the node component this correspond to using permutation tiles to uild T ).

15 5 Conclusion and Future Work In this paper we defined a special kind of grammars, called linear ordered graph grammars, that can e used to descrie distriuted systems with moility and oject-ased systems. We showed how to model such grammars and their semantics in terms of tiles. In [] graph grammars in the DPO approach have een modeled using tiles. The idea was that horizontal arrows correspond to graph morphisms, vertical arrows are pairs of an identity (on the gluing graph) and the name of a production and tiles correspond to DPO rules. In this way, the oservations (vertical components of a tile) are not put into foreground (they are just names of rules). In our approach, we have modeled a graph morphism as vertical component of a tile. We could easily extend our approach to consider different kinds of arcs (oservale and non-oservale) corresponding to different horizontal and vertical signatures. This way we could model, for example, transactions, analogously to what have een done for Petri nets in []. The main aspects of our approach are: ffl Types in graphs correspond to signatures in tiles: typed graphs correspond to gs-monoidal arrows of a restricted type on a signature. Graph isomorphism is directly represented y gs-monoidal axioms. Moreover, the set of all arrows has a natural meaning in terms of open graphs (due to space limitations, this topic was not further detailed here, see [7]). ffl Graph morphisms of the special kind we need are otained ychoosing the vertical structure of tiles adequately. It is immediately apparent the correspondence etween the vertical wire structure" (again a variation of symmetric monoidal categories) and the properties of morphisms. ffl SPO can e represented y a horizontal tile construction. gain, the pushout property ecomes concrete in terms of certain tiles which can e constructed starting from a finite numer of auxiliary tiles []. ffl The effect of a derivation can e represented y a (flat) tile. We expect that canonical derivations are exactly represented y tiles equipped with their proof terms (in the Curry-Howard style). Note that we did not restrict to injective rules (as it is done in almost all true concurrency models for graph grammars). ffl The oservale effect of a derivation (which is not usually emphasized in the SPO/DPO literature) is very important in the interactive interpretation of our models of computations []. It represents the creation/fusions of channels and creation/termination of processes which took place in the computation. Typically, a useful way of connecting systems is via composition of their states, that can e done through shared channels. Therefore it is important to have information aout these channels in interfaces (that is, make them oservale) to otain a compositional semantics. ffl The model would e more satisfactory introducing also restriction/hiding of channels and processes, which would e straightforward (as shown y other papers [9]) in the tile model. This would correspond to new and interesting classes of graphs, making even more explicit the advantage of estalishing a connection etween graph transformations and tiles.

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