Attributed Graph Transformation via Rule Schemata: Church-Rosser Theorem

Attriuted Graph Transformation via Rule Schemata: Church-Rosser Theorem Ivaylo Hristakiev and Detlef Plump (B) University of York, York, UK detlef.plump@york.ac.uk Astract. We present an approach to attriuted graph transformation which requires neither infinite graphs containing data algeras nor auxiliary edges that link graph items with their attriutes. Instead, we use the doule-pushout approach with relaelling and extend it with rule schemata which are instantiated to ordinary rules prior to application. This framework provides the formal asis for the graph programming language GP. In this paper, we astract from the data algera of GP, define parallel independence of rule schema applications, and prove the Church-Rosser Theorem for our approach. The proof relies on the Church-Rosser Theorem for partially laelled graphs and adapts the classical proof y Ehrig and Kreowski, ypassing the technicalities of adhesive categories. Introduction Traditionally, the theory of graph transformation assumed that laels in graphs do not change in derivations (see, for example, []). But in applications of graph transformation it is often necessary to compute with laels. For instance, finding shortest paths in a graph whose edges are laelled with distances requires to determine the shorter of two distances and to add distances. Graphs in which data elements of some fixed algera are attached to nodes and edges have een called attriuted graphs since [], the first formal approach to extend graph transformation with computations on laels. In that paper, graphs are encoded as algeras to treat graph structure and algera data uniformly. With a similar intention, the papers [6,9] go the other way round and encode the data algera in graphs. Each data element ecomes a special data node and auxiliary edges connect ordinary nodes and edges with the data nodes. The latter approach has ecome mainstream ut has some serious drawacks (emoaned as the akwardness of attriutes in [7]). Firstly, the way attriutes are attached to edges leads to the situation of edges having other edges as sources. This requires non-standard graphs and makes the model unusual. Secondly, and more importantly, there is typically an infinite numer of data nodes ecause standard data algeras (such as integers or lists) have infinite domains. Supported y a Doctoral Training Grant from the Engineering and Physical Sciences Research Council (EPSRC) in the UK. c Springer International Pulishing AG 6 P. Milazzo et al. (Eds.): STAF 6, LNCS 9946, pp. 45 6, 6. DOI:.7/978--9-5-4

46 I. Hristakiev and D. Plump This means that attriuted graphs are usually infinite, leading to a discrepancy etween theory and practice as graphs are stored using finite representations. In the approach of [6], even rules are normally infinite ecause they consist of graphs containing the complete term algera corresponding to the data algera. In this paper, we propose an alternative approach to attriuted graph transformation which avoids oth infinite graphs and auxiliary attriute edges. Instead of merging graphs with the data algera, we keep them separate. Host graph items simply get laelled with data elements and rule graph items get laelled with terms. To make this work, rules are instantiated y replacing terms with corresponding data values and then applied as usual. Hence our rules are actually rule schemata whose application can e seen as a two-stage process. In order to modify attriutes, it is crucial that interface items in rules can e relaelled. We therefore use the doule-pushout approach with partially laelled interface graphs as a formal asis [7]. This approach is also the foundation of the graph programming language GP [5]. The fixed data algera of GP consists of integers, character strings, and heterogeneous lists of strings and integers. In this paper, we astract from this particular algera and consider an aritrary data algera (see Susect..). In Sect., we define parallel independence of rule schema applications and prove the so-called Church-Rosser Theorem for our setting. Roughly, this result estalishes that independent rule schema applications can e interchanged and result in the same graph. Our proof nicely decomposes into the Church-Rosser Theorem for the doule-pushout approach with relaelling plus a simple extension to rule schemata (see Susect..). The Church-Rosser Theorem for the relaelling setting was otained in [8] as a corollary of an astract result for M, N -adhesive transformation systems. However, we delierately avoid the categorical machinery of adhesiveness, van Kampen squares, etc. which we elieve is difficult to digest for an average reader. Instead, we merely adapt the classical proof of Ehrig and Kreowski [5] to partially laelled graphs, essentially y replacing properties of pushouts and pullacks in the unlaelled case y properties of natural pushouts in the setting of partially laelled graphs. (A pushout is natural if it is also a pullack.) The rest of this paper is organized as follows. In Sect., we descrie the general idea of our approach. In Sect., we present the notions of parallel and sequential independence and formalize the Church-Rosser Theorem at the rule schema level. Section 4 contains the relevant proofs. A conclusion and future work are given in Sect. 6. We assume the reader to e familiar with asic notions of the doule-pushout approach to graph transformation (see []). An extended version of this paper, along with complete proofs, can e found in []. Attriuted Graph Transformation via Rule Schemata In this section, we present our approach to transforming laelled graphs y rule schemata. We egin y riefly reviewing laelled graphs and the doule-pushout approach to graph transformation with relaelling (see [7] for details).

Attriuted Graph Transformation via Rule Schemata 47. Doule-Pushout Approach with Relaelling A partially laelled graph G over a (possily infinite) lael set L consists of finite sets V G and E G of nodes and edges, source and target functions s G,t G : E G V G, a partial node laelling function l G,V : V G L, and a partial edge laelling function l G,E : E G L. Given a node or edge x, we write l G (x) = to express that l G (x) is undefined.graphg is totally laelled if l G,V and l G,E are total functions. The classes of partially and totally laelled graphs are denoted y G (L) andg(l), respectively. A premorphism g : G H consists of two functions g V : V G V H and g E : E G E H that preserve sources and targets: g V (s G (e)) = s H (g E (e) and g V (t G (e)) = t H (g E (e)) for all edges e. Premorphism g is a graph morphism if it preserves laels, that is, if l G (x) =l H (g(x)) for all items x such that l G (x) is defined. A graph morphism g preserves undefinedness if it maps unlaelled items in G to unlaelled items in H. We call g an inclusion if g(x) =x for all items x. Note that inclusions need not preserve undefinedness. Finally, g is injective (surjective) if g V and g E are injective (surjective), and an isomorphism if it is injective, surjective and preserves undefinedness. Partially laelled graphs and graph morphisms constitute a category (which is M, N -adhesive [8] if one picks M to e the injective morphisms and N to e the injective morphisms that preserve undefinedness). In this category, pushouts need not exist as can e oserved in Fig. (a). A rule r = L K R over L consists of two inclusions K L and K R such that L and R are graphs in G(L) andk is a graph in G (L). Definition (Direct derivation). A direct derivation etween graphs G and H in G (L) via a rule r = L K R consists of two natural pushouts as in Fig., where g : L G is an injective graph morphism. We denote such a derivation y G r,g H. The requirement that the pushouts in Fig. are natural ensures that the pushout complement D in Fig. is uniquely determined y rule r, graph G and morphism g [7, Theorem ]. Figure() demonstrates that non-natural pushout complements need not e unique. It is worth noting that in the traditional setting of doule-pushout graph transformation with totally laelled graphs, the pushouts are automatically natural y the injectivity of L K and K R. Operationally, the application of rule r to graph G proceeds as follows: () match L with a sugraph of G y means of an injective graph morphism g : L G satisfying the dangling condition: nonodeing(l) g(k) is incident to an edge in G g(l); () otain a graph D y removing from G all items in g(l) g(k) and, for all unlaelled items x in K, making g(x) unlaelled; () add disjointly to D all items from R K, keeping their laels, to otain a graph H; (4) for all unlaelled items x in K, l H (g(x)) ecomes l R (x). We do not distinguish etween nodes and edges in statements that hold analogously for oth sets. A pushout is natural if it is also a pullack.

48 I. Hristakiev and D. Plump L K R g NPO NPO G D H Fig.. A direct derivation NPO NPO PO PO c (a) () Fig.. (a) Pushouts need not exist. () A natural and a non-natural doule pushout. In [7] it is shown that if G is totally laelled, then the resulting graph H is also totally laelled. Moreover, unlaelled items in the interface graph K have unlaelled images in the intermediate graph D y the naturalness condition for pushouts.. Rule Schemata Rule schemata for attriuted graph transformation were introduced in the context of the graph programming language GP [6]. We first review signatures and algeras (details can e found, for example, in [, Appendix B]). Consider a signature Σ consisting of a set S of sorts and a family of operation symols OP =(OP w,s ) (w,s) S S. AΣ-algera A consists of a family of carrier sets (A s ) s S containing data values, and a set of functions implementing the operations of Σ. Aterm algera T Σ (X) is uilt up from terms consisting of constants and variales, where X is a family of variales that is disjoint from OP. An assignment α: X A is a family of mappings (α s : X s A s ) s S, giving a value to each variale in X. Its unique extension α : T Σ (X) A evaluates terms according to α. We assume a fixed Σ-algera A whose elements are used as host graph laels, and a corresponding term algera T Σ (X) whose terms are used as laels in rule schemata. To avoid an inflation of symols, we sometimes equate A or T Σ (X) with the union of its carrier sets. Definition (Rule schema). A rule schema r = L K R consists of two inclusions K L and K R such that L and R are graphs in G(T Σ (X)) and K is a graph in G (T Σ (X)).

Attriuted Graph Transformation via Rule Schemata 49 L x y K y R x+y y L α K α R α NPO NPO G D H Fig.. Example of a rule schema direct derivation To apply a rule schema r to a graph, the schema is first instantiated y evaluating its laels according to some assignment α: X A. Definition (Rule schema instance and direct derivation). Consider agraphg in G (T Σ (X)) and an assignment α: X A. Theinstance G α is the graph in G (A) otained from G y replacing each lael l with α (l). The instance of a rule schema r = L K R is the rule r α = L α K α R α. A rule schema direct derivation via r etween graphs G and H in G (A) is a direct derivation G rα,g H via the instance r α according to Definition. We write G r,g,α H if there exists a direct derivation from G to H with rule schema r, graph morpshism g and assignment α. Note that we use for the application of oth rule schemata and rules. Figure shows an example of a rule schema direct derivation, where we assume that algera A contains the integers with addition (+). The variales x and y are of sort int and are mapped y assignment α to and, respectively. This allows for the relaelling of node to. Note that this rule schema gives rise to infinitely many instances ecause the carrier set of integers is infinite. Given an injective premorphism g : L G and an assignment α: X A, a graph morphism g : L α G is induced y g and α if g V = g V and g E = g E.In other words, the application of α to L must turn g into a lael-preserving graph morphism. The following proposition gives a necessary and sufficient condition for a rule schema with left-hand side L to e applicale with a morphism induced

5 I. Hristakiev and D. Plump y g and α. The proof relies on a result in [7] aout the existence and uniqueness of direct derivations in the doule-pushout approach with relaelling. Proposition (Existence and uniqueness of direct derivations). Consider a rule schema r = L K R, an injective premorphism g : L G with G in G (A), and an assignment α: X A. Then there exists a direct derivation G r,g,α H such that g is induced y g and α, ifandonlyifg satisfies the dangling condition and each item x in L satisfies l G (g(x)) = α (l L (x)). Moreover, in this case H is determined uniquely up to isomorphism. Proof. If : By assumption, each item x in L satisfies l G (g(x)) = α (l L (x)) = l L α(x) and hence g : L α G with g V = g V and g E = g E is a graph morphism. Moreover, it is clear that g satisfies the dangling condition with respect to r α ecause g satisfies the dangling condition with respect to r. Thus, y [7, Theorem ], there is a direct derivation G r α,g H where H is determined uniquely up to isomorphism y r α and g. Since r α and g are uniquely determined y r, α and g, it follows that H is uniquely determined y r, α and g, too. Only if : Suppose that G r,g,α H where g is induced y g and α. Then, y definition, G rα,g H. Hence, y [7, Theorem ], g satisfies the dangling condition. Since g V = g V and g E = g E, it is clear that g satisfies the dangling condition with respect to r. Moreover, since g is lael-preserving, each item x in L satisfies l G (g(x)) = l G (g (x)) = l L α(x) =α (l L (x)). As indicated aove, a rule schema r = L K R may have infinitely many instances. Even if one restricts to instances that are compatile with a given premorphism g : L G, there may e infinitely many instances to choose from. For example, consider a premorphism that maps a node in L laelled with x + y to a node in G laelled with the integer (assuming the conventions of Fig. ). There are infinitely many assignments meeting the laelling condition of Proposition ecause the equation x + y = has infinitely many solutions over the integers. Example (GP rule schemata). Laels in the graph programming language GP [, 5] are integers, character strings or heterogeneous lists of integers and character strings. Lists are constructed y concatenation: given lists x and y, their concatenation is written x:y. Expressions in the left-hand side L of a GP rule schema are syntactically restricted to ensure that at most one instance of the schema is compatile with a given premorphism g : L G. To this end, left-hand expressions must neither contain arithmetic operators (except unary minus) nor repeated list variales, and all variales occurring on the right-hand side of a rule schema must also occur on the left-hand side. Figure 4 shows the declaration of a GP rule schema inc. Its left-hand laels contain typed variales which are instantiated with concrete values during

Attriuted Graph Transformation via Rule Schemata 5 inc(a,x,y:list; i:int) x:i a y:i x:i a y:i+ Fig. 4. Declaration of a GP rule schema graph matching. By convention, the interface of the rule schema consists of two unlaelled nodes. The effect of inc is to increment the rightmost element in the list of node. In this paper we are not concerned with implementation issues and do not impose any restrictions on rule schemata. Astracting from GP s lael algera, other possile data types for laels include (multi)sets, stacks, queues and records. Church-Rosser Theorem In this section, we present the notion of parallel independence for direct derivations with relaelling and then extend it to applications of rule schemata.. Independence of Direct Derivations with Relaelling Let each of the diagrams in Fig. 5 represent two direct derivations according to Definition. Definition 4 (Parallel and sequential independence). Two direct derivations H r,m G r,m H as in Fig. 5(top) are parallel independent if there exist morphisms i : L D and j : L D such that f i = m and f j = m. Two direct derivations G r,m H r,m H as in Fig. 5(ottom) are sequentially independent if there exist morphisms i : R D and j : L D such that f i = m and f j = m. It will turn out that parallel and sequential independence are related: two direct derivations H r,m G r,m H are parallel independent if and only if the direct derivations H r G r,m,m H are sequentially independent, where r denotes the inverse rule of r and m is the comatch of m. Lemma (Characterization of parallel independence). Two direct derivations H r,m G r,m H are parallel independent if and only if for all items x L and x L such that m (x )=m (x ), x K and x K,and l K (x ) and l K (x ).

5 I. Hristakiev and D. Plump R K L L K R NPO NPO NPO NPO j m m i H D f G f D H m L K R L K R NPO NPO NPO NPO j m m i G D f f D H H Fig. 5. Parallel and sequential independence (top and ottom, respectively) The first condition states that every common item is an interface item. The second condition states that no common item is relaelled y either derivation. Example (Counterexample to parallel independence). Figure6 shows two direct derivations H G H that use instances of the rule schema of Fig.. The derivations are not parallel independent: there are no morphisms L D and L D with the desired properties. The prolem is that node gets relaelled, reaking the second independence condition. R K L L K R 4 4 H D G D H Fig. 6. Counterexample to parallel independence Our main result (Theorem ) will show that rule schema direct derivations that are parallel independent can e interchanged to otain a common result graph. First, we state the Church-Rosser theorem for plain rules in the sense of Definition. This has een proved in [8] as a corollary of the Church-Rosser theorem for M, N -adhesive transformation systems. However, we otain the result directly without using the notions of adhesiveness and van Kampen square. The proof follows the original Church-Rosser proof of [5]. At specific points it will e necessary to show that the results for NPO decomposition apply to the given setting. See Sect. 4 for the complete proof.

Attriuted Graph Transformation via Rule Schemata 5 Theorem (Church-Rosser theorem for plain rules). Given two parallel independent direct derivations G r,m H and G r,m H, there are a graph H and direct derivations H r,m H and H r,m H. Moreover, G r,m H r,m H as well as G r,m H r,m H are sequentially independent.. Church-Rosser Theorem for Rule Schema Derivations This susection lifts the previous independence result to rule schema applications. The main idea is to simply add instantiation on top of plain direct derivations. Definition 5 (Parallel independence of rule schema derivations). Two rule schema direct derivations G r,m,α H and G r,m,α H are parallel independent if the plain derivations with relaelling G α H r,m and G α H r,m are parallel independent according to Definition 4. R K L L K R R α K α L α L α K α R α m m j i H D f G f D H Theorem (Church-Rosser theorem for rule schemata). Given two parallel independent rule schema direct derivations G r,m H and G r,m H, there is a graph H and rule schema direct derivations H r,m H and H r,m H. Moreover G r,m H r,m H r,m H are sequentially independent. H as well as G r,m Proof. From Theorem, we know that independence of the plain derivations with relaelling G α H r,m and G α H r,m implies the existence of a graph H and direct derivations H r α,m H and H r α,m H. This is illustrated in Fig. 7. The direct derivations G α H r,m and G α H r,m use instances of the rule schemata r and r, and therefore there are rule schema direct derivations H r,m H r,m H and H r,m H and G r α,m H r,m 4 Proof of Theorem H. With Theorem follows that oth G α r H,m are sequentially independent. The proof follows the original Church-Rosser proof of [5]. At specific points it will e necessary to show that the results for NPO decomposition apply to the

54 I. Hristakiev and D. Plump L L K K R L α L α R R α K α m G m K α R α D D H H m m D D L α L α K α K α H R α R α Fig. 7. Church-Rosser theorem for rule schemata given setting. This is ecause for partially laelled graphs, pushouts need not always exist, and not all pushouts along injective morphisms are natural. These facts have een oserved in Fig.. Using the definition of parallel independence (Definition 4), we start y decomposing the derivations as shown in Fig. 8. K L L K () i j () m m D D D D () () D G D Fig. 8. First decomposition diagram The graph D is otained as a pullack of (D G D ). The universal property of pullacks gives us that K D and K D decompose into K D D and K D D respectively. We also have that ( + ) and ( + ) are NPOs ecause they are left-hand sides of derivations. Furthermore, D G and D G are injective and jointly surjective which makes () an NPO ([, Lemma 4]).

Attriuted Graph Transformation via Rule Schemata 55 D G and D G injective imply that D D and D D are also injective. The susequent parts of the proof contain four claims which are proven afterwards. Claim. The squares () and () are NPOs. Next, the pushouts D of (D K R ) (5) and D of (D K R ) (6) are constructed. These exist y the following claim: Claim. In Fig. 9, the pushouts D of (D K R ) (5) and D of (D K R ) (6) exist. Again using uniqueness, the morphisms R H and R H decompose into R D H and R D H. We also have that (5 + 7) and (6 + 8) are NPOs ecause they are right-hand sides of derivations. R K L L K R (5) () i j () (6) m m D D D D D D (7) () () (8) H D G D H Fig. 9. Second decomposition diagram Also, (7) and (8) ecome NPOs y the NPO Decomposition Lemma [, Lemma 5.]. Claim. In Fig. 9, the squares (7) and (8) are NPOs. The graph H is constructed as a pushout of (D D D ) (4). (See square (4) in Fig..) Claim 4. The pushout of (D D D ) exists. This pushout ecomes NPO y [, Lemma ] and the arguments in the proof of Claim 4. Furthermore, the graph H is totally laelled due to the way D, D and D are constructed - D can contain unlaelled items only from D K which are laelled in D, and vice versa. The pushouts can e rearranged as in Fig. to show that G r,m H r,m H as well as G r,m H r,m H are sequentially independent. Note that the graph H is totally laelled. This concludes the proof of the Church-Rosser theorem. Next, we present the proofs of the aove claims.

56 I. Hristakiev and D. Plump L K R L K R i () (5) j () (6) m D D m D D D D () (7) (7) (4) G D H D H Fig.. Rearranged pushouts ProofofClaim. We need to show that the conditions of the NPO Decomposition Lemma [, Lemma 5.] hold for the following diagrams. K D D () () K D D () () L D G L D G D D is injective ecause D G is injective y definition and () is PB. () has already een proven to e NPO (at the start of this section). Pushout exists over (L K D )asl is totally laelled, oth morphisms are injective and K D preserves undefinedness, all y the definition of direct derivation with relaelling. The square K L D D commutes ecause ( + ) and () are NPOs. Therefore, all conditions of the NPO Decomposition Lemma [, Lemma 5.] hold. The proof for the second diagram is analogous. This concludes the proof that the squares () and () are NPOs. Proof of Claim. As in the previous proof, R and R are totally laelled, all morphisms are injective and oth K D and K D preserve undefinedness, all y the definition of direct derivation with relaelling. Therefore, the pushouts (5) and (6) exist y [, Lemma.]. Proof of Claim. In the context of Fig., we need to show that the conditions of the NPO Decomposition Lemma [, Lemma 5.] hold. D D has already een estalished as injective. We need that there exists unique NPO complement of K R D. We have that K R is injective y definition. R D is injective ecause K D and L D are injective. The pushout (5+7) is a right-hand side of a derivation and R H = R D H. Consequently, R D satisfies the dangling condition w.r.t. K R, thus the existence of a unique NPO complement is given y [, Lemma.]. The proof for the second diagram is analogous. This concludes the proof that the prerequisites for the NPO Decomposition Lemma [, Lemma 5.] hold. Hence, squares (7) and (8) ecome NPOs.

Attriuted Graph Transformation via Rule Schemata 57 K D D K D D (5) (7) (6) (8) R D H R D H Fig.. Pushouts (5), (6), (7) and (8). K K R (5) (6) D R D (4) D H ProofofClaim4. For a pushout to exist, D and D have to agree on the laels of the unlaelled nodes of D ([, Lemma.]). D is constructed as the pushout of (R K D ) with R eing totally laelled. Its node and edge sets and laelling function is as defined in [, Lemma.]. Moreover, R D and D D are injective and jointly surjective. There are main cases for an item x to e laelled in D : the item is created y the first derivation x R K. This means it does not exist in D, D or G. Consequently, this item does not have a preimage in D y pullack construction ([, Lemma.]). Therefore, it cannot e a source of conflict for pushout existence. the item is relaelled y the first derivation, meaning its preimage in D (and D ) is unlaelled x K and l K (x) =l D (x) =l D (x) =. By the definition of parallel independence, no common items are relaelled making the item not have a preimage in R. Therefore it is unlaelled in D (y definition of pushout), making it a non-conflict w.r.t. pushout existence. the item is in D R, i.e. a laelled item of D K. We have that D D and D D are lael preserving, so the lael of x in D is the same as in D. In all cases, the laels of D are preserved y the second derivation. The argument for the laelled items of D is analogous. This concludes the proof that the pushout (4) over (D D D ) exists.

58 I. Hristakiev and D. Plump 5 Related Work We have adapted the classical Church-Rosser proof of Ehrig and Kreowski [5] to partially laelled graphs and extended the result to rule schemata, essentially y replacing properties of pushouts and pullacks in the unlaelled case y properties of natural pushouts in the setting of partially laelled graphs. In [6], the theory of attriuted graph transformation is developed in the framework of so-called adhesive HLR categories. Among other results, the Church-Rosser Theorem is proved in this setting. The approach is further studied in [4] y adding nested application conditions and proving the previous results for this more expressive approach. Both are a generalized version of the Church- Rosser Theorem of [9]. So-called symolic graphs are attriuted graphs in which all data nodes are variales, comined with a first-order logic formula over these variales. In [] it is shown that this approach can specify and transform classes of ordinary attriuted graphs that satisfy the given formula. The underlying graph structure is the same as in [6], hence the approach shares the issues descried in the introduction. Recently, a generalised Church-Rosser theorem for attriuted graph transformation has een proved in [] y using symolic graphs. A notion of parallel independence is used that takes into account the semantics of attriute operations, in order to reduce the numer of false positives in conflict checking. 6 Conclusion In this paper, we have presented an approach to attriuted graph transformation ased on partially laelled graphs and rule schemata which are instantiated to ordinary rules prior to application. We have defined parallel independence of rule schema applications and have proved the Church-Rosser theorem for our approach. The proof relies on the Church-Rosser theorem for graph transformation with relaelling and adapts the classical proof y Ehrig and Kreowski, ypassing the technicalities of adhesive categories. Future work includes estalishing other classical graph transformation results in our setting, such as emedding and restriction theorems. Furthermore, we aim at studying critical pairs and confluence oth for the particular case of the GP language and for attriuted graph transformation over aritrary lael algeras. In particular, we plan to give a construction of critical pairs (laelled with expressions) that guarantees the set of critical pairs is oth finite and complete. Completeness would mean that all possile conflicts of rule schema applications can e represented as emeddings of critical pairs.

Attriuted Graph Transformation via Rule Schemata 59 References. Bak, C.: GP : efficient implementation of a graph programming language. Ph.D. thesis, Department of Computer Science, University of York (5). http://etheses. whiterose.ac.uk/586/. Ehrig, H.: Introduction to the algeraic theory of graph grammars (a survey). In: Claus, V., Ehrig, H., Rozenerg, G. (eds.) Graph Grammars 978. LNCS, vol. 7, pp. 69. Springer, Heidelerg (979). doi:.7/bf574. Ehrig, H., Ehrig, K., Prange, U., Taentzer, G.: Fundamentals of Algeraic Graph Transformation. Monographs in Theoretical Computer Science. Springer, Heidelerg (6) 4. Ehrig, H., Golas, U., Hael, A., Lamers, L., Orejas, F.: M-adhesive transformation systems with nested application conditions. Part : parallelism, concurrency and amalgamation. Math. Struct. Comput. Sci. 4(4) (4) 5. Ehrig, H., Kreowski, H.-J.: Parallelism of manipulations in multidimensional information structures. In: Mazurkiewicz, A. (ed.) MFCS 976. LNCS, vol. 45, pp. 84 9. Springer, Heidelerg (976). doi:.7/-54-7854- 88 6. Ehrig, H., Prange, U., Taentzer, G.: Fundamental theory for typed attriuted graph transformation. In: Ehrig, H., Engels, G., Parisi-Presicce, F., Rozenerg, G. (eds.) ICGT 4. LNCS, vol. 56, pp. 6 77. Springer, Heidelerg (4). doi:. 7/978--54-- 7. Hael, A., Plump, D.: Relaelling in graph transformation. In: Corradini, A., Ehrig, H., Kreowski, H.-J., Rozenerg, G. (eds.) ICGT. LNCS, vol. 55, pp. 5 47. Springer, Heidelerg (). doi:.7/-54-458-8 8. Hael, A., Plump, D.: M, N -adhesive transformation systems. In: Ehrig, H., Engels, G., Kreowski, H.-J., Rozenerg, G. (eds.) ICGT. LNCS, vol. 756, pp. 8. Springer, Heidelerg (). doi:.7/978--64-654-6 5 9. Heckel, R., Küster, J.M., Taentzer, G.: Confluence of typed attriuted graph transformation systems. In: Corradini, A., Ehrig, H., Kreowski, H.-J., Rozenerg, G. (eds.) ICGT. LNCS, vol. 55, pp. 6 76. Springer, Heidelerg (). doi:.7/-54-458-8 4. Hristakiev, I., Plump, D.: Attriuted graph transformation via rule schemata: Church-Rosser theorem (long version) (6). http://www.cs.york.ac.uk/plasma/ pulications/pdf/hristakievplump.6.full.pdf. Kulcsár, G., Deckwerth, F., Lochau, M., Varró, G., Schürr, A.: Improved conflict detection for graph transformation with attriutes. In: Proceedings of Graphs as Models (GaM 5). Electronic Proceedings in Theoretical Computer Science, vol. 8, pp. 97 (5). Löwe, M., Korff, M., Wagner, A.: An algeraic framework for the transformation of attriuted graphs. In: Sleep, R., Plasmeijer, M., van Eekelen, M. (eds.) Term Graph Rewriting: Theory and Practice, pp. 85 99. Wiley, New York (99). Orejas, F., Lamers, L.: Symolic attriuted graphs for attriuted graph transformation. In: Graph and Model Transformation. Electronic Communications of the EASST, vol. () 4. Plump, D.: Confluence of graph transformation revisited. In: Middeldorp, A., Oostrom, V., Raamsdonk, F., Vrijer, R. (eds.) Processes, Terms and Cycles: Steps on the Road to Infinity: Essays Dedicated to Jan Willem Klop on the Occasion of His 6th Birthday. LNCS, vol. 88, pp. 8 8. Springer, Heidelerg (5). doi:. 7/6548 6

6 I. Hristakiev and D. Plump 5. Plump, D.: The design of GP. In: Proceedings of the Workshop on Reduction Strategies in Rewriting and Programming (WRS ). Electronic Proceedings in Theoretical Computer Science, vol. 8, pp. 6 () 6. Plump, D., Steinert, S.: Towards graph programs for graph algorithms. In: Ehrig, H., Engels, G., Parisi-Presicce, F., Rozenerg, G. (eds.) ICGT 4. LNCS, vol. 56, pp. 8 4. Springer, Heidelerg (4). doi:.7/978--54-- 7. Rensink, A.: The edge of graph transformation graphs for ehavioural specification. In: Engels, G., Lewerentz, C., Schäfer, W., Schürr, A., Westfechtel, B. (eds.) Graph Transformations and Model-Driven Engineering: Essays Dedicated to Manfred Nagl on the Occasion of His 65th Birthday. LNCS, vol. 5765, pp. 6. Springer, Heidelerg (). doi:.7/978--64-7-6