ARITA, Tomokazu

TSUCHIDA, Kensei

YAKU, Takeo

Kyoto University Research Information Repository

TitleSyntactic Characterization of Two-Dimensional Grid Graphsby a Context-Sensitive Graph Grammar(New Trends in Theoryof Computation and Algorithm)Author(s)ARITA, Tomokazu; TSUCHIDA, Kensei; YAKU, TakeoCitation数理解析研究所講究録 (2006), 1489: 78-84Issue Date2006-05URL http://hdl.handle.net/2433/58226RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversitySyntactic Characterization of Two-Dimensional Grid Graphs by aContext-Sensitive Graph Grammar *$\mathrm{n}$ (Tomokazu ARITA)Department of Languages and Information Studies, Obirin University(Kensei TSUCHIDA)Department of Information and Computer Sciences, Toyo University(Takeo YAKU)Department of Computer Science and System Analysis, Nihon UniversityAbstractGraph grammar can characterize several type of graphs. Vigna and Ghezzi showed that thelanguage of grid graph8 could not be constructed by their context-free graph grammars [4]. In thispaper, we give a graph grammar for two-dimensional grid graphs. The graph gramamr for grid graph8is context-sensitive.keywords: context-sensitive graph grammar, grid graphs.1 IntroductionOur interest is in a graph grammar for two-dimensional grid graphs.Graph grammar has succeeded in characterizing graph structures and generating several types ofgraphs [3, 9, 8]. In [5] and [6], there are graph grammars that can derive ladders. Vigna and Ghezzishowed that a language of grid graphs could not be constructed using their context-free graph grammars[4]. In [1], Pavlidis also described that the set of grid graphs could not be generated using their context-free graph grammars.On the other hand, Burosch and Laborde characterized grid graphs using the products of paths andshowed that two-dimensional gnids can be recognized in linear time by applying a combinatorial algorithm[7].Graph grammar characterization for grid graphs possibly derives characterization of grid graphs withmultiple labels, such as business document tables [10] or symbol tables in program documents.In this paper, We show that there is a context-sensitive graph grammar for two-dimensional gridgraphs. This grammar derives grid graphs by synchronizing using state propagation among nodes ingraph rewriting rules. Label propagation is based on state propagation of cellular automata (see e.g.,[2] $)$ .2 Notations and DefinitionsFirst, we begin by reviewing some graph notations. For this paper, we consider directed graphs withnode and edge labels. We also consider undirected graphs by extending the directed graphs.Deflnition 1 [9] Let $\Sigma$ be an alphabet of node labels and $\Gamma$ an alphabet of edge labels. A graph over$\Sigma$ and $\Gamma$ is $G=(V,E,\psi)$ , where $V$ is the finite set of nodes, $E\subseteq\{(v,l,w)|v,w\in V,l\in\Gamma\}$ is the set ofedges, and th: $Varrow\Sigma$ is the labding function.’The result in this paper appeared in T. Arita, K. Tsuchida, and T. Yaku, “Syntactic Characterization of the Two-Dimensional Grid Graphs”, IEICE Ttansactions on Information and Systems, vol.E89-D, no.2, pp.771-778, February 2006[13]. and part of the result in this paper appeared in Section 4 of T. Arita et al., “Syntactic Processing of Diagrams byGraph Grammars“, the Proceedings of 16th IFIP World Computer Congress 2000 ICS2000, pp.145-151, 2000. [12].1489 2006 78-84 78In this paper, we always assume that $G$ is connected. Graph $G=$ ( $V,$ $E$ , Cb) is an undirected graph if,for every $(v, l, w)\in E$ , there is also $(w, l, v)\in E[9]$ .Deflnition 2 [7] A (undirected) graph is called $a$ (two-dimensional) grid graph iff the graph consists ofthe product of two path graphs [7]. Let $P_{n1}$ and $P_{n2}$ be path graphs, then $P_{n1}\square P_{n2}$ denotes the productof $P_{n1}$ and $P_{n2}$ . If $P_{n1}$ is a path graph with $n1$ nodes and $P_{n2}$ is a path graph with $n2$ nodes, then $T_{n1,n2}$denotes a grid graph for the product of the $P_{n1}$ and $P_{n2}$ .Figure 1 shows a grid graph $T_{3,4}$ .Figure 1: A grid graph.We now consider that two graphs are “isomorphic.” Two graphs $G_{1}=(V_{1},E_{1},\psi_{1})$ and $G_{2}=$(V2, $E_{2},\psi_{2}$ ) are isomorphic if there is abijection $f:V_{1}arrow V_{2}$ such that $E_{2}=\{(f(v),l, f(w))|(v,l,w)\in E_{1}$$\}$ and, for all $v\in V_{1}$ , zb2 $(f(v))=\psi_{1}(v)[9]$ .3 Context-Sensitive Graph GrammarsNext we consider graph transformations. Graphs are transformed based on production rules. We dealwith the following productions, in which one production replaces a subgraph in one graph with othergraph.In this section, we modify \’eNCE graph grammars [9] and introduce context-sensitive graph gram-mars. The left-hand side of the production in the $\mathrm{e}\mathrm{d}\mathrm{N}\mathrm{C}\mathrm{E}$ graph grammar is defined using a single nodelabel. But the left-hand side of the production in our graph grammar is defined using a connected graph.Deflnition 3 (cf. [9]) A Production $p$ is of the form $Marrow(D,C)$ , where $M$ is a graph over $\Sigma$ and $\Gamma,$ $D$is a graph over $\Sigma$ and $\Gamma$ called an embeded graph, and $C\subseteq\Sigma \mathrm{x}\Gamma \mathrm{x}V_{M}\mathrm{x}\Gamma \mathrm{x}V_{D}\mathrm{x}$ {in, out}, cW\’e aconnection relation, is a set of connection $in\epsilon$tructions.A production $p:Marrow(D,C)$ removes a subgraph $M’$ of its applied graph $G$ , such that $M$ and $M’$ areisomorphic, and replaces $M’$ with $D’$ , such that $D’$ and $D$ are isomorphic and the set of nodes in $G$ and$D’$ are pairwise disjoint. Then the connection instructions of $C$ replace neighboring edges around $M’$ . Aconnection instruction $(\alpha,\beta,x,\gamma,y,\sigma ut)$ of $C$ means that if there was an edge labeled $\beta$ from the node $x$in $G$ for which $(D,C)$ is substituted to a node $w$ with label $\alpha$ , then the embedding process will establishan edge labeled 7 from $y$ to $w$ . Similarly, a connection instruction ($\alpha,\beta,x,\gamma,y$ , in) of $C$ means that ifthere was an edge labeled $\beta$ from a node $w$ with label a to the $x$ in $G$ for which $(D, C)$ is substituted,then the embeddin$\mathrm{g}$ process will establish an edge labeled $\gamma$ from $w$ to $y$ .That is, let $G=(V_{G}, E_{G}, \psi_{G})$ and $H=(V_{H},E_{H},\psi_{H})$ be graphs on $\Sigma$ and $\Gamma$ . Let $p:Marrow(D,C)$ bea production such that $V_{G}$ and $V_{D}$ are pairwise disjoint. Then $G\Rightarrow_{p}H$ holds for $p$ if there is a subgraph$M’$ of $G$ such that $M’$ and $M$ are isomorphic. That is we obtain $H$ from $G$ as follows: $V_{H}=(Vc-V_{M’})$$\cup V_{D},$ $E_{H}=\{(x,l,y)|x,y\in V_{G}-V_{M’}\}\cup E_{D}\cup\{(w,l,x)|\exists m\in\Gamma,$ $\exists v\in V_{M’}$ : $(w,m,v)\in E_{G}$ ,$(\psi_{G}(w),m,v,l,x,in)\in C\}\cup$ { $(x,$ $l,w)|\exists m\in\Gamma,$ $\exists v\in V_{M^{\prime:}}(v,m,w)\in E_{G},$ $(\psi_{G}(w),m,v,l,x$ , out) $\in C$},$\psi_{H}(x)=\psi_{G}(x)$ if $x\in(V_{G}-V_{M’})$ , and $\psi_{H}(x)=\psi_{D}(x)$ if $x\in V_{D}$ (cf. [9]).Deflnition 4 ([9]) A graph grammar is $GG=(\Sigma, \Delta, \Gamma,\Omega, P, S)$ , where $\Sigma$ is the finite set of node labels,$\Delta\subseteq\Sigma$ is the set of terminal node labels, $\Gamma$ is the finite set of edge labels, $\Omega\subseteq\Gamma$ is the set of final edgelabds, $P$ is the set of $\mathrm{p}$roductions, and $S$ is the start graph.We call context-sensitive graph grammar, in which the left-hand side of a production is a graph.For productions $M_{1}arrow(D_{1},C_{1})$ and $M_{2}arrow(D_{2},C_{2})$ , Two connection relations $C_{1}$ and $C_{2}$ are calledisomorphic if there are isomorphism $f$ from the node set of $M_{1}$ to the node set of $M_{2}$ and $f’$ from thenode set of $D_{1}$ to the node set of $D_{2}$ such that $C_{2}=\{ (\sigma,\beta, f(v),\gamma, f’(x),d)|(\sigma,\beta,v,\gamma,x,d)\in C\mathrm{i}\}.$79Figure 2: An example of an application of a production.Two productions $M_{1}arrow(D_{1}, C_{1})$ and $M_{2}arrow(D_{2},C_{2})$ are called isomorphic if $M_{1}$ and $M_{2}$ are isomor-phic graphs, $D_{1}$ and $D_{2}$ are isomorphic graphs, and $C_{1}$ and $C_{2}$ are isomorphic.Let $GG=(\Sigma, \Delta,\Gamma, \Omega,P, S)$ be a graph grammar. Let $G$ and $H$ be graphs on $\Sigma$ and $\Gamma$ and let $p$:$Marrow(D, C)$ be a production in $P$ . Then $G\Rightarrow_{p}H$ is called a derivation step, and a sequence of derivationsteps is called a derivation (cf. [9]).We assume that $P$ does not contain distinct isomorphic productions in this grammar.Figure 2 shows an example of a production and a derivation of a graph. Node labels $A,B,$ $C,D$ arenon-terminal, and label $\#$ is terminal. Every edge in the graphs shown in Figure 2 is unlabeled. Fur-thermore, a box is a nonterminal node, a black circle is a terminal node, an arrow is a directed edge,and a dotted arrow is part of a connection instruction. We use edge label $\#$ for unlabeled edges. Pro-duction $p$ is $Marrow(D, C)$ , where $M=(V, E,\psi)$ such that $V=\{x_{1},x_{2}\},$ $E=\{(x_{1}, \#, x_{2})\},$ $\psi(x_{1})=A$ ,and $\psi(x_{2})=B$ , and $D=(V_{D}, E_{D},\psi_{D})$ such that $V_{D}=\{x_{3},x_{4},x_{5}\},$ $E_{D}=\{(x_{8}, \#,x_{4}),$ $(x_{4}, \#,x_{5})$ ,$(x_{5}, \#,x_{3})\},$ $\psi_{D}(x_{3})=\#,$ $\psi_{D}(x_{4})=\#,$ $\psi_{D}(x_{6})=D$ , and $C=\{c_{1},c_{2}\}$ such that $c_{1}=$ ( $\#,$ $\#,x_{1},$ $\#,x_{5}$ , in)and $c_{2}=$ ($C,$ $\#,$ $x_{2},$ $\#,$ $x_{4}$ , out).Graph $H$ in Figure 2 is generated by applying $p$ to graph $G$ .Deflnition 5 The language of graph grammar $GG=$ ( $\Sigma,$ $\Delta,\Gamma$ , St, $P,$ $S$) is $L(GG)=\{g|g$ is derived from$S$ by $GG$ , all node labels of $g$ are terminal node labels, and all edge labels of $g$ are final edge labels. }4 A Grid Graph GrammarIn this section, we construct a graph grammar for two-dimensional grid graphs and describe some graphgrammar characteristics.Deflnition 6 A grid gmph grammar $GG_{G}$ is defined as follows: $GG_{G}=(\Sigma_{G}, \Delta_{G}, \Gamma_{G}, \Omega_{G}, P_{G}, S_{G})$,where $\Sigma_{G}=$ { $S,$ $H,$ $L,$ $A,$ $B,$ $R,$ $C,$ $X,$ $D,$ $R’,$ $E,$ $P_{h}$ , END, $\#$ }, $\Delta_{G}=\{\#\},$ $\Gamma_{G}=\{\#\},$ $\Omega_{G}=\{\#$$\}$ , $P_{G}$ is defined in Figure 3, and $S_{G}=(\{v\}, \phi, \psi_{Sc})$ such that $\psi_{S_{G}}(v)=S$ .Here we explain the labels in $P_{G}$ . Label $S$ is the start node label. Label $L$ is the leftmost node labelof grid graphs. Label $H$ is the label for making grid graphs that are not path graphs. Labels $R$ and $R$‘are the rightmost node labels of grid graphs. Labels $A,$ $B,$ $P_{h}$ are for growing grid graphs to the left.Labels $C$ and $X$ are for turning right. Label END is the last nonterminal node label, and label $\#$ is theterminal node label.Note that the unlabeled nodes and edges are labeled $\#$ .We show an example of a derivation of $GG_{G}$ by Figures 4, and 5 in the Appendix. This derivationis for the grid graph $T_{3,4}$ . Figure 3 illustrates the productions of $GG_{G}$ . In Figure 3, we denote theconnection instructions of the productions using dotted lines.Theorem 1. Graph $T$ is a grid graph iff $T$ is an element of $L(GG_{G})$ .80for a oath$\mathrm{P}16:_{\square }^{\mathrm{S}}arrow\square \mathrm{P}\mathrm{h}$$\mathrm{P}17:\#\underline{1}\square arrow \mathrm{P}\mathrm{h}*$ $!_{--\bullet \mathrm{R}}*\mathrm{P}\mathrm{h}$ P18: $\#\underline{1}\square arrow \mathrm{P}\mathrm{b}\#.1.\#-.arrowarrow \mathrm{E}\mathrm{N}\mathrm{D}$for a top linefor $\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{P}\mathrm{a}\mathfrak{g}\mathrm{a}\mathrm{u}\mathrm{Q}\mathrm{n}$for a bottom lineFigure 3: Productions of $GG_{G}$ .81We can extend this graph grammar to graph grammars with multiple node labels.Burosch and Laborde proposed a decision algorithm for two-dimensional grid graphs that runs inlinear time for the number of nodes [7].There are several applications of labeled grid graphs (e.g. symbol tables, bitmap image data etc.) thatnaturally relate labeled grid graphs. For example, the colors of pixels in bitmap images are representedby node labels.Our graph grammar $GG_{G}$ can be extended to several applications.5 ConclusionWe constructed a graph grammar for $\mathrm{t}\mathrm{w}\triangleright \mathrm{d}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{d}$grid graphs.In the future, we win apply this grammar to grid graphs with multiple node labels such as financialstatements. The results could possibly be applied to the parsing of grid graphs with multiple labels suchas business document tables (e.g. [10, 11]) or symbol tables in program documents.AcknowledgmentsThe authors would like to thank Professor Kimio Sugita of Tokai University and Associate ProfessorKoichi Yamazaki of Gunma University for their valuable discusIions and helpful comments.References[1] T. Pavlidis, “Linear and Context-free graph grammars”, J. ACM, vol. 19, no. 1, pp.11-22, 1972.[2] T. Yaku, “The Constructibility of a Configuration in a Cellular Automaton”, J. Comput. Syst. Sci.,vol. 7, no. 5, pp.481-4SC, 1973.[3] R. Franck, “A Class of Linearly Parsable Graph Grammars,” Acta Infomatica, 10, pp.175-201, 1978.[4] P. D. Vigna and C. Ghezzi, “Context-Free Graph Grammar”, Inf. Control, 37, 207-233, 1978.[5] J. Engelfriet, G. Leih, and G. Rozenberg, $u_{\mathrm{A}\mathrm{p}\mathrm{e}\mathrm{x}}$ Graph Grammar”, Lecture Notes in ComputerScience, vvol. 291, pp.167-185, 1986.[6] J. Engelfriet and G. Leih, “Linear Graph Grammars: Power of Complexity”, Information and Com-putation, vol. 81, pp.88-121, 1989.[7] G. Burosch and J.M. Laborde, “Characterization of grid graphs”, Discrete Math. 87, pp.85-88, 1991.[8] K. Yamazaki and T. Yaku, “A Pumping Lemma and the Structure of Derivation in the BoundaryNLC Graph Language”, Information Sciences 75, pp.81-97, 1993.[9] G. Rozenberg (Ed.), Handbook of Graph Grammar and Computing by Graph Transformation, WorldScientific Publishing, New Jersey, vol. 1, 1997.[10] M. Shimizu, K. Tsuchida, and T. Yaku, “A Formalization of Business Documents by Graph Gram-mars”, IPSJ SIG Technical Report, 2004-IS-88, 10, PP.6&73, 2004 (in Japanese.)[11] T. Kirishima, T. Motohashi, T. Arita, K. Tsuchida, and T. Yaku, uSyntax for Tables”, Proc. 21stIASTED Applied Informatics 2003, pp.1185-1190, 2003.[12] T.Arita, K.Tomiyama, T. Yaku, Y. Miyadera, K. Sugita, and K. Tsuchida, “Syntactic Processingof Diagrams by Graph Grammars”, the Proceedings of 16th IFIP World Computer Congress 2000ICS2000, pp.145-151, 2000.[13] T. Arita, K. Tsuchida, and T. Yaku, “Syntactic Characterization of the Two-Dimensional GridGraphs”, IEICE Transactions on Information and Systems, vol.E89-D, No.2, pp.771-778, 2006.82Appendix A: A Derivation Based on $GG_{G}$This part shows the derivation process of the grid graph with three rows and four columns.An Example of a Derivation for Grid Graph T3,4$\square \mathrm{S}$$\Rightarrow \mathrm{P}3$$\Rightarrow \mathrm{P}5$$\Rightarrow \mathrm{P}7$$\Rightarrow \mathrm{P}8$$\Rightarrow \mathrm{P}10$$\Rightarrow \mathrm{P}5$$\Rightarrow \mathrm{P}7$Figure 4: A derivation based on $GG_{G}(1)$ .83An Example of a Derivation for Grid Graph T3,4$\Rightarrow \mathrm{P}12$$\Rightarrow \mathrm{P}13$$\Rightarrow \mathrm{P}15$Figure 5: A derivation based on $GG_{G}(2)$ .84

Syntactic Characterization of Two-Dimensional Grid Graphs by a Context-Sensitive Graph Grammar(New Trends in Theory of Computation and Algorithm)

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Syntactic Characterization of Two-Dimensional Grid Graphs by a Context-Sensitive Graph Grammar(New Trends in Theory of Computation and Algorithm)

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