This paper is concerned with hierarchical graph models and graph transformation rules, specifically with the problem of transforming a part of graph which may contain subordinated nodes and edges. Meta-rules are proposed as a formal way of representing transformations which remove or duplicate a node along with its contents. The paper discusses the behaviour of meta-rules when applied to different types of hierarchical graphs, possible failure cases, and concludes by introducing a type of hierarchical graphs in which meta-rules can always be successfully expanded

Palacz, Wojciech

English

University of Maria Curie-Skłodowska (UMCS): Scientific e-Journals / Uniwersytet Marii Curie-Skłodowskiej: e-czasopisma naukowe

Annales UMCS Informatica AI VIII, 2 (2008) 89–96DOI: 10.2478/v10065-008-0027-zHierarchical graph transformations with meta-rulesWojciech Palacz∗Department of Computer Design and Graphics, Faculty of Physics, Astronomy andApplied Computer Science, Jagiellonian University,ul. Reymonta 4, 30-059 Krako´w, PolandAbstractThis paper is concerned with hierarchical graph models and graph transformation rules, specif-ically with the problem of transforming a part of graph which may contain subordinated nodesand edges. Meta-rules are proposed as a formal way of representing transformations whichremove or duplicate a node along with its contents. The paper discusses the behaviour ofmeta-rules when applied to different types of hierarchical graphs, possible failure cases, andconcludes by introducing a type of hierarchical graphs in which meta-rules can always besuccessfully expanded.1. IntroductionOur research group is interested in computer support systems for the prelim-inary phase of the design process. In this phase, the designer works on a fairlyhigh level of abstraction, considering the decomposition of the whole object intosubcomponents. Hand–drawn sketches similar to the one displayed in Fig. 1are traditionally used to capture design decisions.∗e-mail address: wojciech.palacz@uj.edu.plPobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.plData: 04/08/2020 21:53:04UMCS90 Wojciech PalaczFig. 1. Structural model of a houseIn order to store such sketches in computer memory, an appropriate datamodel must be utilized. Graphs are the most obvious choice. Our group hasdefined over the years several formalized graph models meant for this purpose;experiences gained suggest that the model used should be hierarchical, so thatit can directly represent hierarchical (sometimes even recursive) nature of real-world objects. Additionally, used formalism should provide some kind of graphtransformation rules, which allow for automatic generation and modification ofdesigns.A generic framework for hierarchical graphs was developed and presented in[1, 2]. It provides a way for constructing graphs where graph atoms (nodes,edges, and hyperedges) can be nested inside any other atom, and extends thewell–known double–pushout graph transformation rules so that they can beapplied to the hierarchical graphs.Our example uses directed edges and labeled nodes; thus, the formal defini-tion of the hierarchical graphs used by the floor layout design support systemshould be as follows:Definition 1. Graph G is a tuple (V,E, s, t, lab, par), where:• V and E are the finite and disjoint sets of nodes and edges;• s : E → V and t : E → V are the edge attachment functions (source andtarget);• lab : V → L is a node labeling function (L is a set of labels);• par : V ∪ E ∪ {⊥} → V ∪ E ∪ {⊥} is a parent assigning function;• par(⊥) =⊥, ∀a ∈ V ∪ E ∀k ∈ N+ : park(a) 6= a.Pobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.plData: 04/08/2020 21:53:04UMCSHierarchical graph transformations with . . . 91The special symbol ⊥ is used to denote that a given node or edge has noparent (i.e. is at the top level of the graph). Conditions specified in the defini-tion ensure acyclicity of the parent function, which means that the parent-childrelations between graph atoms are a tree with ⊥ as its root.In addition to graphs we also need to define graph morphisms. They aremappings between two graphs, mapping nodes to nodes and edges to edgeswhile preserving all their properties (edge attachments, labels, etc.). For graphsspecified by Definition 1, a corresponding morphism definition looks like this:Definition 2. Morphism between two graphs G and H is a functionf : VG ∪ EG ∪ {⊥} → VH ∪ EH ∪ {⊥}, where:• ∀v ∈ VG : f(v) ∈ VH , ∀e ∈ EG : f(e) ∈ EH ;• ∀v ∈ VG : labG(v) = labH(f(v));• ∀e ∈ EG : f(sG(e)) = sH(f(e)) and f(tG(e)) = tH(f(e));• ∀a ∈ VG ∪ EG : f(parG(a)) = parH(f(a)).Morphisms are usually denoted simply as f : G→ H. If f is both surjectiveand iniective, then f is an isomorphism; in many practical applications twoisomorphic graphs are considered to be the same graph.It can be demonstrated that graphs from Definition 1 and morphisms fromDefinition 2 constitute an algebraic category (see [1] or [2]). Thus, it is possibleto define double–pushout rules. This paper will introduce them in an informalway; see [3] for the classical definition and exhaustive discussion of their proper-ties, and [1, 2] for the definition of hierarchical rules and corresponding proofs.Fig. 2. A double–pushout rule which duplicates a nodePobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.plData: 04/08/2020 21:53:04UMCS92 Wojciech PalaczA double–pushout rule consists of three graphs (known as the left–hand side,the interface, and the right–hand side, usually denoted by L, K, R) and twomorphisms. Morphisms (denoted as l : K → L, r : K → R) map the interfaceto the left– and right–hand side graphs, respectively. It can be said that theinterface is the intersection of both sides.Fig. 2 displays an example rule. In order to apply it to the graph presentedin Fig. 1 (let us denote it by G), an occurrence of the left–hand side must befound in graph G, i.e. a morphism m : L→ G must be found (there is exactlyone match possible). Next, atoms present in L but not in K are deleted fromG (in our example there are none), and then atoms present in R but not in Kare added to G (there is one such node). The obtained result graph will containa new living room on the 1st floor.2. Meta–rulesSimple rules can transform only these graph atoms which were matched tothe left–hand side. Operations like removal or duplication of a given nodealong with its unknown subordinated nodes and edges are beyond their power.This is unfortunate, because such operations are often necessary – for example,a typical user of a design support system for floor layouts will expect to be ableto copy, move and delete rooms along with their contents. Meta–rules are theproposed solution. They contain markers, which are matched to subordinatednodes and edges; by deleting or duplicating markers these atoms (unknown inadvance) are deleted or copied.Definition 3. Graph with markers is a tuple (V,E, s, t, lab, par,mark),where:• (V,E, s, t, lab, par) is a graph (see Definition 1);•mark : V ∪E → PR(M) is a marker assigning function (M is a set of markers,PR(M) is the set of subsets with repetitions of M).For a given atom a, mark(a) may be empty. Every graph with markerscan be trivially converted to the corresponding simple graph by discarding themarker function.Definition 4. Graph transformation meta–rule consists of three graphswith markers (L,K,R) and two morphisms (l, r) such that:• after discarding markers L← K → R is a transformation rule;• markL(a) is empty or contains exactly one marker for all a ∈ VL ∪ EL;• all markers used in L are different;Pobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.plData: 04/08/2020 21:53:04UMCSHierarchical graph transformations with . . . 93• markK(a) ⊂ markL(l(a)) for all a ∈ VK ∪ EK ;• markers used in R must be also present in L.Fig. 3. A meta-rule for duplicating rooms with their contentsMeta–rules are not applied directly. Instead, their markers are expandedto obtain derived rule specific to the graph being transformed, and then thisderived rule is applied.Definition 5. If m : L → G is a morphism matching a meta–rule toa graph being transformed, and α is a marker nested in an atom a from L,then expansion set of this marker is defined as exp(α) = ch∗G(chG(m(a)) −m(chL(a))). Function ch : VG ∪ EG ∪ {⊥} → P (VG ∪ EG) is the children func-tion, i.e. the inversion of the parent function.Please note that exp(α) is called ”expansion set”, not ”expansion subgraph”.A set of nodes and edges from graph G is a subgraph of G if and only if thefollowing conditions are met:• if edge e is in exp(α), then s(e) and t(e) are in exp(α);• if atom a and its ancestor park(a) are in exp(α), then par(a) is in exp(α).A rule can be derived from a meta–rule if and only if all marker expansion setsare subgraphs. The derivation algorithm works in a loop, eliminating markersone by one. While there are markers in L, the following steps are executed:1. Choose a marker from L – let us assume it is α, nested in atom a.2. Delete α from L and insert a copy of exp(α) in its place, extend morphismm with identity function so that atoms of this copy are mapped onto theoriginal atoms in G.Pobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.plData: 04/08/2020 21:53:04UMCS94 Wojciech Palacz3. Check if α is present in K. If yes, then it has to be in l−1(a). Replaceit with exp(α), extend morphism l with identity mapping so that newlyinserted copy of exp(α) in K is mapped onto the copy in L.4. If the condition in the previous step was true, then check R to see if α ispresent in r(l−1(a)). If yes, replace it and extend morphism r as in theprevious step.5. Replace all occurrences of α in R with exp(α), do not modify r.The correctness of this algorithm can be proved by observing that the resultsproduced by every step are valid meta–rules; that because of conditions inDefinition 4 eliminating all markers from L means that all markers in K and Rwere eliminated also; and that meta–rule without any markers can be consideredequivalent to a simple double–pushout rule.3. Performance in practical applicationsExpression power of meta–rules depends heavily on hierarchical structure ofgraph models being transformed. Definition 1 places no constraints on whereatoms can be nested; it would be possible, for example, to create a graph whichhas several nodes subordinated to an edge, which, in turn, is nested insideanother edge, which is inside a node. This flexibility is a distinguishing featureof our hierarchical graphs framework, but it can be overwhelming.In a given practical application, the models used will usually belong to somespecialized subclass of hierarchical graphs. Our running example is about floorlayouts; as Fig. 1 displays, nodes represent subcomponents of a house, and edgesrelations between these components. This means that nodes are containersand will have other nodes nested inside, in order to represent the structure ofa house. So, hierarchy between nodes mirrors the hierarchy of components inthe object being designed. But what about edges?There is no obvious answer in their case the programmer who implementsa floor layout design support system may make a somewhat arbitrary decisionabout their placement. For example, it would be possible to decide that alledges should be placed on the top level of a graph, because they don’t playany role in the hierarchy of house components. Or it could be decided thateach edge should have the same parent as its source node. This decision hasconsequences when it comes to transformation rules which modify edges – thesecond graph subclass is much easier to work with, because edge and at leastone of the attached nodes are on the same hierarchy level.The decision about edges influences results produced by meta–rules, too.Please recall that meta–rules were introduced in order to meet expectations ofusers. An average user, when presented with our running example, would sayPobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.plData: 04/08/2020 21:53:04UMCSHierarchical graph transformations with . . . 95that duplicated room should contain all subcomponents present in the origi-nal room, and that the original relations between these components (i.e. theconnection between the video recorder and the TV set) should be also copied.Some percentage of users would also say that the newly created TV set shouldbe connected to the antenna, because the old one was.If edges representing relations are to be copied when applying the meta–rule from Fig. 3, then they must be included in exp(α), which means that the”living room” node must be their parent. But the expansion set containing twonodes (V CR, TV ) and two edges won’t be a subgraph, and we won’t be ableto successfully complete the derivation algorithm. As it turns out, the optimalcase has exp(α) consisting of two nodes and their connecting edge. The derivedrule is displayed in Fig. 4.Fig. 4. A derived rulePlease note that we have managed to preserve edges representing relationsinternal to the expansion subgraph, and omitted edges which were crossing theboundary of exp(α). It is possible to consistently obtain such results for othermeta–rules and graphs if graphs used have the following properties:• only nodes may contain subordinated atoms;• if edge e connects nodes v and w, then par(e) is equal to the closest commonancestor of v and w.These two conditions guarantee that all expansion sets will be valid sub-graphs, which means that it will be always possible to derive a rule. Edgeswhich connect nodes in exp(α) are included, those which cross the boundary ofexp(α) are not.Pobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.plData: 04/08/2020 21:53:04UMCS96 Wojciech PalaczEven if derivation is guaranteed to succeed in this graph subclass, the derivedrule may fail to transform the graph. It is a well–known property of double–pushout rules – they cannot be applied if they delete atoms, and those deletionswould leave some edges or children dangling.On one hand, meta–rules are more susceptible to dangling edges than simplerules; unknown node contents represented by markers may be connected todistant parts of the graph, which would prevent the rule from deleting thesecontents. On the other hand, simple rules have no way of dealing with danglingchildren, and meta-rules can use markers to represent these children and deletethem en masse.4. ConclusionsThe meta–rules presented in this paper can be used to formally describetransformations operating on graph nodes along with their (unknown in ad-vance) contents. Best results can be obtained when graphs being transformedfulfill specific requirements on atom hierarchy; in such subclass of graphs, de-rived rules can always be constructed, and will include all edges representinginternal relations between nodes in the same marker expansion set.The problem of duplicating external relations has not had an elegant solutionso far. Also, meta–rules cannot remove nodes if it leaves some edges dangling– this is an inherent limitation of the double–pushout graph transformationapproach. These two problems merit further investigation because of theirimportance in practical applications.References[1] Palacz W., Algebraic hierarchical graph transformation, Journal of Computer and SystemSciences 68(3) (2004).[2] Palacz W., Hierarchical graphs in computer-aided design systems, Ph.D. thesis, Jagiel-lonian University, (2006).[3] Rozenberg G. (ed), Handbook of graph grammars and computing by graph transformation,Volume 1: foundations, World Scientific Publishing (1997).Pobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.plData: 04/08/2020 21:53:04UMCSPowered by TCPDF (www.tcpdf.org)

Hierarchical graph transformations with meta-rules

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Hierarchical graph transformations with meta-rules

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