Theory of Graph Transformations

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1 Theory of Graph Transformations Helge Wiemann University of Bremen studies of computer science Abstract This paper shall give an introduction into the theory of rule-based graph transformation which is itself a wellstudied area in the context of computer science. It begins with a short introduction to the topic of graphs and continues with the description of some quite important notions which are essential for the understanding of the theory of graph transformation. The topic of the algebraic approach as the central method for graph transformations is presented as the main topic in this paper, strengthened by illustrations of the two sub approaches SPO (single-pushout approach) and DPO (double-pushout diagram) which is more frequently used in practice than the SPO. All theoretical aspects are often accompanied by an application example to illustrate them related to practice. At the end of this paper you will find a comparison of the two sub approaches as a final conclusion and a special application example of the DPO. 1. Introduction Die grossartige Schönheit eines Graphen besteht darin, dass er unklare, schwer durchschaubare Rätsel auf ein wunderbares Ding aus Linien und Knotenpunkten reduzieren kann. (D. Olivastro. Das chinesische Dreieck: die kniffligsten mathematischen Rätsel aus Jahren. Droemer Knauer, München, 1995, S.273) When we talk about graphs, we talk about a powerful and well-established tool to represent various data structures as well as states of concurrent and distributed systems. More generally speaking you can describe whole sets of objects including the relations between them [2]. Famous examples and various kinds of graphs are the following ones [2] : flow diagrams Petri nets UML diagrams Figure 1. Petri net Entity-Relationship diagrams finite automata maps (e.g. of roads or countries) In many cases you do not only want to build up graphs as a static structure with a static behavior of its objects. If you need to have a dynamical structure you need a graph which is able to transform. For instance you are firing transitions in Petri Nets or UML state diagrams or you want to transform a graph by deleting nodes or edges in the case of other UML object diagrams. Graphs are quite generic structures and therefore graph transformations combine the concepts of graphs and the application of rules to transform them. Graphs are well-suited and often used structures and because of their quite different applications, you can find graphs in many different application areas as various kinds. They can be directed or undirected, maybe labeled or unlabeled. They can be simple or multiple and they are equipped with binary edges or hyperedges [2]. Sometimes,

2 2.1. category theory Figure 2. crossover as a graph [3] in the case of Petri nets, graphs even provide two different types of nodes, just called either condition or transition. By the way a graph with this attribute is called bipartite graph. Because of their possible attribute to be a directed graph, the graph is a quite famous data structure for transportation and tour planning problems (maps, Königsberger Brückenproblem, e.g. in the area of theoretical computer science and mathematics). With their directed edges from one node to another one, you can also easily represent various control flows, e.g. test results with a switch between TRUE and FALSE. In this paper we only focus on directed, edge-labeled and multiple graphs with binary edges. Furthermore we concentrate on the double-pushout approach as the one of the most frequently used approaches for a graph transformation. This kind of graph is a closed system G = (V,E,s,t,l). The capital letter V describes in this case a finite set of nodes whereas the capital letter E describes a finite set of edges. These two sets of objects build up the graph together. The letters s, t and l qualify three different mappings. s: E V is a mapping which assigns a source s(e) node to every edge in the set of E. The other analog mapping t: E V assigns a target t(e) node to every edge in E so that you have the confirmation that every edge is suspended on two nodes [3]. If there is an edge in the graph with s(e) = t(e) it is also called a loop. Of course you can also have a node with more than one entry or exit edge. The last mapping l: E V is needed to assign a label (l(e)) to every edge in E. The description of these kinds of graphs is absolutely flexible and has the possibility to cover also other types of graphs. 2. graph rewriting Before we will go further to the explicit approaches of graph transformations, we first have to introduce some important notions around the topic graph rewriting for the better understanding. The category theory occupies a central position not only in contemporary mathematics, but also in the area of the theoretical computer science and is therefore also the basis for the algebraic approach of graph transformations. It can roughly be described as an abstract way to deal with different algebraic mathematical structures and the relationships between them. Furthermore it is at the very least a quite powerful language and conceptual framework which allows to see how structures of different kinds are related to one another as well as the universal components of a family of structures of a given kind. In our case and context we will have a look at the category of graphs, with its components nodes and edges. Generally speaking a category C can be described as a class ob(c) of objects, with the following additional auxiliary attributes: There also exists an other class mor(c) of morphisms for every pair of objects (a,b) with a unique source object a and target object b, we write f: a b. Furthermore, for a triple a, b and c of objects, there is also a partial operation from pairs of morphisms (f: a b and g: b c) to a final morphism h: a c, which is also called the composition of morphisms in the category C. Finally, for every single object a, there is a morphism 1x: x x, called the identity morphism of a, often written as id(a). To sum it up, all morphisms in the category C have to satisfy the two axioms associativity and identity [4]. Therefore an interesting feature of the category theory is that it provides a uniform treatment of the notion of structure. Especially the algebraic approach with its methods of pushouts as graph morphisms which will be described later on, based upon this category theory (category of graphs) subgraphs Formally a graph G is a real subgraph of another graph H, if it satisfies the following conditions: all nodes of G also consist in H (V G in V H ), all edges of G also consist in H (E G in E H ), every edge in G has the same source node in H (s(e) G = s(e) H ), analog has every edge in G the same target node in H (t(e) G = t(e) H ) and finally every edge in G must have the same label in the other graph H. With a given graph, you can simply obtain a subgraph by removing some nodes and edges, with the essential condition that the removal of a node is accompanied by the removal of all its adjacent edges. More formally speaking: the given graph, as is generally known, is a system G = (V,E,s,t,l). You want to remove nodes and edges, so that you have in this case additionally a pair of nodes and edges, written as X = (V X,E X ) in (V,E).

3 Now you optain the subgraph H formally in that way: H = G-X = (V-V X,E-E X,s,t,l ) with s (e)=s(e), t (e)=t(e) and l (e)=l(e) for all e E-E X. [2] Here are three published examples of subgraphs G1, G2 and G3 of the given graph G: the dangling condition This condition will be satisfied if an edge e of M - g(l) neither has its source node nor its target node in g(l) - g(k). That means that, if you delete one node, you have to delete all edges that are adjacent to this one node. The addition of these two sub conditions forms the gluing condition [2] identification condition + dangling condition = gluing condition Figure 3. real subgraphs 2.3. graph morphisms A graph morphism in the category of graphs assigns the nodes and edges of a given graph G to the nodes and edges of another graph H while the structure of the original graph G is preserved. Of course in our case with the use of labeled graphs, the node and edge labeling is preserved too. More formally a graph morphism g: G H is a pair of functions, where one of these functions is a node mapping g V : V G V H and the other one is an edge mapping g E : E G E H [3] If G is an image in H, then it is called a match of G in H. The match of G in H according to the graph morphism g is moreover the subgraph g(g) in H, induced by the pair of mappings (g(v), g(e)). If you have an additional third graph F, then you will have a possible sequence of morphisms: f: F G and g: G H then yields to the morphism g o f: F H 2.4. gluing condition Within the algebraic approach and the application of the double-pushout approach which will be described later on, you will only have a valid graph transformation if the match of the left graph L of the rule of transformation in the given graph G satisfies the gluing condition. The gluing condition is divided into the following two sub conditions: the identification condition This condition will be satisfied if two different elements x and y of the left graph L either are mapped injectively (none two different elements of the definition quantity are mapped to the same element of the target quantity) or may only not be mapped injectively if these two elements are not deleted by the transformation rule. More formally speaking: mappings O(x) = O(y) only if x=y or x,y L R 3. algebraic approach There are several approaches of graph rewriting, one of them is the algebraic approach which based on the just mentioned category of theory. Actually the algebraic approach is divided into sub approaches, in this paper we will discuss two of them, namely the single-pushout approach (SPO) and the more frequently used double-pushout approach (DPO). Both approaches have the same intention to perform rule-based transformation of graphs as the method to execute local changes on graphs for the use of them in dynamical environments. Generally speaking the algebraic approach should specify formally and visually the semantics of rule-based systems. The rules have to express which part of graph is to be replaced by another graph. The general production is always described as p: L R [1] If you want to transform a graph, you can have many problems, such as there are some edges left after the deletion of nodes and these edges do not have any source or target node. These transformations are forbidden. To find out which transformations on a graph are allowed, we use the important upper mentioned gluing condition. If the match satisfies the gluing condition, the graph transformation is valid and be can performed without any problems later on. The action of gluing two graphs is a construction in the context of the category of graphs and graph morphisms and is called pushout double-pushout approach (DPO) The double-pushout approach, shortly called DPO, is a sub approach of the algebraic approach and is the frequently used approach for graph transformations. The DPO adopts a specific rule for the graph transformation which describes on the one hand which parts are replaced by which other ones and on the other hand side the DPO replies to the important question which kinds of transformations are allowed. From the perspective of the DPO a graph rewriting rule is a pair of morphisms in the category of graphs with total graph morphisms as arrows, specified by the formal rule r =

4 (L K R). The graphs L and R are respectively called the left-hand side and the right-hand side of the rule. The graph K is often called gluing graph or interface graph. A rewriting step with the application of the DPO-production is defined as a pair (L K R) or (L K R) of two graph morphisms as arrows in the category of graphs with an interface graph K, where K L is injective. Because of that the interface graph K is a real subgraph of L as well as of R [1,4]. A rule application r = (L K R) can be depicted by the following diagrams. They will illustrate the formal step of a graph rewriting and will describe why this approach is called double-pushout approach. The gluing graph K is rightly described in the diagrams as L R. Figure 6. construction of the final graph temporary graph D (Figure 6). More formally: The single graph morphisms O : L R D and O R : R H are given by O (v) = O L (v) for all v V, O (e) = O L (e) for all e E, O R (v) = O (v) if v V, O R (v) = v if v V R - V, O R (e) = O (e) if e E, O R (e) = e if e E R - E [2] Figure 4. finding the match The graph morphism O L in the shown diagram models an occurrence of L in G and is called the match. Practical understanding of this is that L is a subgraph that is matched from G and after a match is found, the left side of the rule (L) is replaced by the right side of the rule (R) in the host graph G where K as L R serves as some kind of interface double-pushout approach: formal application example The following diagrams should illustrate the graph rewriting step of the DPO on a real graph example. Given are two elements: the hostgraph G (Figure 7) and the transformation rule includes the left side (Figure 8) and the right side (Figure 9). Figure 7. hostgraph [3] Figure 5. building the temporary graph If a match is found (Figure 4) there are two steps to achieve the graph rewriting. First you have to build a temporary graph (D) as a subgraph of the host graph G by deleting the matching elements (Figure 5. Note: the dangling condition within the gluing condition ensures that D is graph). Finally you have to build the final transformed graph H by adding the elements of R - L R to the built Figure 8. left side of the rule [3]

5 Figure 9. right side of the rule [3] Figure 12. final transformed graph [3] Step 1: finding a match L in G which maintains the gluing condition (Figure 10) for the graph rewriting step. You can use the graphical rule representation without an interface graph by depicting only the graphs L and R. Thus a rewriting step is only defined by a single pushout diagram with a single graph morphism as arrows as a the formal rule (production) r: L R [1] Figure 10. found match [3] Step 2: building a temporary graph Z (Figure 11) Figure 13. SPO diagram [1] This diagram illustrates the practical understanding of the SPO. You perform the rewriting step with only a single derivation (morphism) as a single-pushout from the host graph L to the target graph R. Figure 11. temporary graph Z [3] Step 3: construction of a final transformed graph N (Figure 12) 3.3. single-pushout approach (SPO) In contrast to the recently mentioned DPO, a graph rewriting rule of the SPO approach is only a single morphism and therefore only a single derivation of the host graph G with context again to the category of graphs. The SPO is often used in cases where the interface graph K as in the DPO is only a set of nodes but without any adjacent edges. Then you do not have to look at the edges 4. comparison between DPO and SPO When you take a closer look on these two sub approaches of the general algebraic approach, you will see that they are quite similar to each other. Because of the fact, that the DPO performs more secure transformations with the condition that the DPO operates structure-preserving, this approach is more frequently used than the SPO. The main and important difference between the two approaches is that the SPO performs the graph derivation without an interface graph K between the host graph G and the target graph H. Therefore the SPO operates against the DPO with no gluing condition, which means without an identification and following dangling condition. This is the reason why the SPO is said to be more powerful than the DPO because it has no restrictions within the graph rewriting step. But big problems can raise if there are edges after the rewriting step left which has no source or

6 target node. This consequence of a rewriting is strictly forbidden because the transformed graph is not a valid one. Therefore the SPO includes the danger that the graph could be destroyed after the transformation with some dangling edges. The DPO operates real structure-preserving with the help of the gluing graph and is therefore used more often because it guarantees the right and allowed transformation on the graph. Figure 16. transformation: killpacman() 5. double-pushout diagram: Pacman as an application example We all know the computer game Pacman from former times. If you remember the game field of Pacman contains different players, like Pacman and a number of ghosts, and also a finite number of positions where all players can move to. Pacman has to eat points and has to watch the ghosts who want to eat him also. With context to this topic you can also represent the game field as well as the movements of the players as a closed graph system (Figure 14) with essential transformations on this graph if a player changes its position on the field (Figure 15) or an action is fired, like for instance if one ghost eats Pacman (Figure 16) and therefore the game is over. 6. References 1. Grzegorz Rozenberg, editor: Handbook of Graph Grammars and Computing by Graph Transformation. Vol.1: Foundations. World Scientific,Singapore, H.-J. Kreowski, R. Klempien-Hinrichs, S. Kuske: Some Essentials of Graph Transformation. University of Bremen, Department of Computer Science. Bremen 3. H.-J. Kreowski: Formale Sprachen: Graphtransformation. Universitt Bremen, Sommersemester Figure 14. game field of Pacman Figure 15. transformation: movepacman()

theory of graph transformations

theory of graph transformations University of Bremen 2005-11-25 speaker : Helge Wiemann email : hwiemann@tzi.de theory of graph transformations 1 theory of graph transformations Die großartige Schönheit