BAUDERON, M., AND COURCELLE, B. Graph expressions and graph rewritings. Mathematical Systems Theory 20 (1987), 83--127.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... of GOTO free C programs have tree width at most 6 [33] and recent empirical study shows that control flow graphs of most Java programs have tree width at most 3 (though in general it can be arbitrary large) 16] Once a graph has bounded tree width, we can construct a graph in an algebraic way [3, 4]. This suggests that finding a fixed point would be computed by recursive traversals on the algebraic structure, and the optimal solution would be obtained with a dynamic programming. Unfortunately, the existing results are not sufficient for our purpose. For instance, the algebraic construction ....

.... ourselves to graphs with bounded tree width, in which many NPhard graph problems are solved in linear time [11, 9] The concept of a graph with bounded tree width [25] independently appeared from early 80 s; partial k tree in terms of cliques, some algebraic construction of k terminal graphs [4, 3], and in terms of separators, and they are all equivalent. The class of graphs with bounded tree width is quite restrictive; but the significant treadoff is: The class of graphs with bounded tree width frequently has a linear time algorithm for graph problems. For graphs with bounded tree width, ....

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M. Bauderon and B. Courcelle. Graph expressions and graph rewritings. Mathematical System Theory, 20:83--127, 1987.

....in linear time; consequently, its size is linear to the size of a program. 1 : read X; 2 : while X 1 do 3 1; 4 : C : X 2; 5 : if C = 0 then 6 : X : X 2; else 7 : X : Z; fi; 2; od; 9 : write X; CFG SP term 4 7 8 9 [1, 9] 1, 2] e [9, 2] 2 [9, 3] [3, 4] [4, 5] e [2, 8] P [5, 8] 5, 6] 8, 6] 5, 7] 8, 7] Fig. 4. An example of control flow graph and its transformation to SP Term 3.2 Checking on SP Terms Now we show how the idea in Section 2 can be brought here for analyzing on SP Terms by checking on marked SP Term (a SP ....

....time; consequently, its size is linear to the size of a program. 1 : read X; 2 : while X 1 do 3 1; 4 : C : X 2; 5 : if C = 0 then 6 : X : X 2; else 7 : X : Z; fi; 2; od; 9 : write X; CFG SP term 4 7 8 9 [1, 9] 1, 2] e [9, 2] 2 [9, 3] 3, 4] [4, 5] e [2, 8] P [5, 8] 5, 6] 8, 6] 5, 7] 8, 7] Fig. 4. An example of control flow graph and its transformation to SP Term 3.2 Checking on SP Terms Now we show how the idea in Section 2 can be brought here for analyzing on SP Terms by checking on marked SP Term (a SP Term with ....

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M. Bauderon and B. Courcelle. Graph expressions and graph rewritings. Mathematical System Theory, 20:83--127, 1987.

....formal approaches and specific proposals that must be considered. As we shall see, though, none of them have all the properties needed to enable the active document processing we envision. Graph rewriting algebras: Graph rewriting algebras specify how to replace subgraphs with other subgraphs [6]. There is a structure governing these substitutions, such as formal composition, reversion (contraction) and so on. These algebras are inadequate for active document workflow because they contain no notion of the document itself controlling the rewriting. Action refinement: In concurrency ....

Michel Bauderon and Bruno Courcelle. Graph expressions and graph rewriting. Mathematical Systems Theory, 20(83--127), 1987.

....in the calculus. The fan in fan outs represent a complex way of synchronising di erent parts of a graph, whereas our dag rewrites only perform a simple use once synchronisation. The rewriting mechanism is also akin to graph substitution in hyper graph grammars (see Bauderon Courcelle [2] for a non category theory formulation) except that we allow any number of leaves to be rewritten and do not allow inner nodes to be rewritten. Strength wise, hyper graph grammars are apparently equivalent to attribute grammars. At present, we are not sure of the generative power of dag ....

Bauderon, M., and Courcelle, B. Graph expressions and graph rewritings. Mathematical Systems Theory 20, 2-3 (1987), 83-127.

....can specify the top down design, or syntactic generation, of such a model. Three main types of hypergraph grammars which are context free in the sense of [Cou87] are considered in the literature, depending on the kind of subgraph which is rewritten in one derivation step: hyperedge rewriting [BC87,HK87], separated handle rewriting [CER93] see also [KJ99] and con uent node rewriting [Kle96] With respect to their generative power, hyperedge rewriting and separated handle rewriting grammars are known to be incomparable [CER93] and the union of their generated languages is properly contained in ....

Michel Bauderon and Bruno Courcelle. Graph expressions and graph rewriting. Mathematical Systems Theory, 20:83-127, 1987.

....approaches (some of which are already present at this workshop) However, the boundary between terms and graphs is not as strict as it might seem. In fact, there are several ways of representing graphs and graph transformations as (equivalence classes of) terms and term rewrite steps (see, e.g. [6, 10, 21]) We expect that such inductive definitions of graphs provide the glue between the visual specification of diagram languages and the use of existing tools based on tree and term representations. 4 Conclusion In this paper, we have discussed the application of some of the most important ....

BAUDERON, M., AND COURCELLE, B. Graph expressions and graph rewritings. Mathematical Systems Theory 20 (1987), 83--127.

....the Technical University of Berlin and the University of Pisa. In the double pushout (DPO) approach to graph transformation [15, 13] and in most of the other approaches) the operational definition is by far more popular. Inductive definitions of DPO graph transformation have been given in [1, 12], but they do not have the same role as in the theory of term rewriting. One reason may be that, unlike for strings and terms, there is no straightforward inductive definition of graphs. Rather, each possible interpretation suggests a different choice of the basic operations. In [1] for example, ....

....given in [1, 12] but they do not have the same role as in the theory of term rewriting. One reason may be that, unlike for strings and terms, there is no straightforward inductive definition of graphs. Rather, each possible interpretation suggests a different choice of the basic operations. In [1], for example, a hyper graph is considered as a set of edges glued by means of vertices. The operations for building graphs are disjoint union, and renaming and fusion of nodes. In [12] a graph is described in a logical style by a set of edge predicates over node variables, using (partly ....

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M. Bauderon and B. Courcelle. Graph expressions and graph rewritings. Mathematical Systems Theory, 20:83--127, 1987.

....representation of term graphs) thus our definition just formalises a common way of reasoning up to isomorphism . Thirdly, our term graphs are ranked. The idea of equipping graphs with lists of distinguished nodes in order to define composition operations on them is not new (see for example [5, 25]) but for the first time, to our knowledge, it is applied here to the class of term graphs. In [4] this technique is not used simply because it is not needed for the goals of the paper, and the single root used there has a different role, being used to relate a term graph with a term unfolded ....

M. Bauderon and B. Courcelle. Graph expressions and graph rewritings. Mathematical Systems Theory, 20:83--127, 1987.

....to concatenation in the world of hypergraphs: hypergraphs have so called external nodes , their interface to the outside, and in order to attach two (or more) hypergraphs, information is needed on how these external nodes should be merged. Such a form of graph concatenation was proposed in [1] in the form of graph expressions , where three operators (disjoint sum, node fusion, rede nition of external nodes) were introduced. These operators can be used in order to gradually construct hypergraphs out of smaller hypergraphs. We propose a similar approach of graph construction where the ....

....on how to concatenate graphs is not provided by operators, but rather by part of a colimit. The construction itself consists of completing the colimit and is related to the double pushout approach [3] The expressive power of graph rewriting in our approach is the same as for graph expressions [1] (and thus the same as in the double pushout approach [8] Compared to [1] we have additional information in the uniqueness property of the colimit and embeddings of the subgraphs into the constructed graph which are provided by the colimit. This information can be helpful when we want to reason ....

[Article contains additional citation context not shown here]

Michel Bauderon and Bruno Courcelle. Graph expressions and graph rewritings. Mathematical Systems Theory, 20:83-127, 1987.

....to concatenation in the world of hypergraphs: hypergraphs have so called external nodes , their interface to the outside, and in order to attach two (or more) hypergraphs, information is needed on how these external nodes should be merged. Such a form of graph concatenation was proposed in [1] in the form of graph expressions , where three operators (disjoint sum, node fusion, redefinition of external nodes) were introduced. These operators can be used in order to gradually construct hypergraphs out of smaller hypergraphs. We propose a similar approach of graph construction where the ....

....on how to concatenate graphs is not provided by operators, but rather by a part of a colimit. The construction itself consists of completing the colimit and is related to the doublepushout approach [3] The expressive power of graph rewriting in our approach is the same as for graph expressions [1] (and thus the same as in the double pushout approach [7] Compared to [1] we have additional information in the uniqueness property of the colimit and embeddings of the subgraphs into the constructed graph which are provided by the colimit. This information can be helpful when we want to reason ....

[Article contains additional citation context not shown here]

Michel Bauderon and Bruno Courcelle. Graph expressions and graph rewritings. Mathematical Systems Theory, 20:83--127, 1987.

....to concatenation in the world of hypergraphs: hypergraphs have so called external nodes , their interface to the outside, and in order to attach two (or more) hypergraphs, information is needed on how these external nodes should be merged. Such a form of graph concatenation was proposed in [1] in the form of graph expressions , where three operators (disjoint sum, node fusion, redefinition of external nodes) were introduced. These operators can be used in order to gradually construct hypergraphs out of smaller hypergraphs. We propose a similar approach of graph construction where the ....

....on how to concatenate graphs is not provided by operators, but rather by part of a co limit. The construction itself consists of completing the co limit and is related to the doublepushout approach [3] The expressive power of graph rewriting in our approach is the same as for graph expressions [1] (and thus the same as in the double pushout approach [6] Compared to [1] we have additional information in the uniqueness property of the co limit and embeddings of the subgraphs into the constructed graph, which are provided by the colimit. This information can be helpful when we want to ....

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Michel Bauderon and Bruno Courcelle. Graph expressions and graph rewritings. Mathematical Systems Theory, 20:83--127, 1987.

....additional labels or structures to a type represented as a graph than to a type represented by a term. This point will become clearer in section 5 where we will assign lattice elements to pairs of nodes. 2 Categorical Hypergraph Construction We work with a variant of graphs: so called hypergraphs [7, 2], where each edge has several (ordered) source nodes. There are two kinds of labels: edge sorts and and edge labels. De nition 1. Hypergraph) Let Z be a xed set of edge sorts and let L be a xed set of labels. A simple hypergraph G is a tuple G = V; E; s; z; l) where V is a set of nodes, E is ....

Michel Bauderon and Bruno Courcelle. Graph expressions and graph rewritings. Mathematical Systems Theory, 20:83-127, 1987.

....(e) this hyperedge is not present in H 1 j V 1 , but nevertheless it may be in Hj V 1 [V 2 because port H 2 (p) 2 VH . Therefore, we only get H 1 j V 1 = H 2 j V 2 Hj V 1 [V 2 in this case. Another type of hypergraph operations is provided by the well known notion of hyperedge replacement (cf. [BC87, HK87, Hab92]) For this, we consider hypergraphs whose sets of port labels are initial segments of N . Call a hypergraphs H with PH = f1; kg for some k 2 N a k hypergraph. In particular, a 0 hypergraph is a portless hypergraph. Then, a hyperedge e in a k hypergraph H can be replaced with a type H ....

....operation on hypergraphs. The result of its application to n argument hypergraphs is obtained by replacing the n hyperedges by the arguments. To the author s knowledge the first to take this view of hyperedge replacement was Courcelle in [Cou91] 3. 4 Definition (hyperedge replacement, cf. [BC87, HK87, Cou91, Hab92]) Let H be a k hypergraph for some k 2 N and let e 1 ; e n 2 EH be pairwise distinct, so called virtual hyperedges. For i = 1; n let H i be a type H (e i ) hypergraph. 1. Let H 0 = HtF (H 1 )t Delta Delta Delta tF (H n ) The (simultaneous) replacement of H 1 ; H n ....

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M. Bauderon, B. Courcelle. Graph expressions and graph rewriting. Mathematical Systems Theory 20, 83--127, 1987.

....formalisms generate the same class of hypergraph languages. twentysix Hypergraphs, hypergraph languages and hypergraph expressions 4.1 Hypergraphs A variety of different definitions of hypergraphs and hypergraph grammars is in use. Examples can be found in [EngHey91] HabKre87] Hab92] and [BauCou87]. We chose for the definitions in [Hab92] since this book was also the source for the predicates and numerical functions on hypergraphs and hypergraph languages. All definitions and theorems in this section and the sections 4.2 and 4.3 are from [Hab92] sometimes with a minor alteration. A ....

....called the substitution operators. Then we use regular tree grammars over these substitution operators, to generate expressions over the thirtysix Hypergraphs, hypergraph languages and hypergraph expressions operators. This method for the generation of hypergraphs was first presented in [BauCou87]. For the definition of the substitution operators, recall the definition of Y as a set of variables fy 1 ; y 2 ; y 3 ; g Omega . Definition 4.29 (substitution operator) Let k 0. A completely repetition free hypergraph H 2 HG m;n is called a substitution operator of rank k, if there ....

M. Bauderon, B. Courcelle, Graph Expressions and Graph Rewritings, Mathematical Systems Theory, Vol. 20, 1987, pp. 83--127.

....First, we consider the notion of hyperedge replacement grammar, introduced by Habel and Kreowski. We only give an informal description here: for a good introduction to this topic, see e.g. 45, 46] The framework of context free graph grammars of Bauderon and Courcelle is essentially similar [8]. See also [66] Hyperedge replacement grammars work with hypergraphs, where each hyperedge is represented as a sequence of vertices. A hyperedge also has a label, which is either a terminal label, or a non terminal label. More or less similar to context free string grammars, a hyperedge ....

M. Bauderon and B. Courcelle, Graph expressions and graph rewritings, Mathematical Systems Theory, 20 (1987), pp. 83--127.

....is recalled. Hyperedge replacement is a simple generation mechanism for hypergraphs that was introduced in the early seventies (under other names) by Feder and Pavlidis [Fed71, Pav72] Its systematic study began more than 15 years later, initiated through papers by Bauderon and Courcelle [BC87] and Habel and Kreowski [HK87] This work was continued in diverse directions. For references see, e.g. Hab92] Eng97] and [Roz97, Chapters 1, 2, and 5] Results that establish in different ways connections between tree transductions and hyperedge replacement can be found in, e.g. EH91, ....

Michel Bauderon and Bruno Courcelle. Graph expressions and graph rewriting. Mathematical Systems Theory, 20:83--127, 1987.

.... of subderivations to be synchronized in a given grammar, we obtain a subfamily of parallel rewriting systems that includes the so called finite copying top down tree to string transducers (yT fc ) of [ En2 gelfriet et al. 1980 ] the string generating context free hypergraph grammars (CFHG) of [ Bauderon and Courcelle, 1987 ] the multiple context free grammars (MCFG) of [ Kasami et al. 1987 ] and [ Seki et al. 1991 ] and the stringbased linear context free rewriting systems (LCFRS) of [ Vijay Shanker et al. 1987; Joshi et al. 1991 ] All these rewriting systems are weakly equivalent, as shown in [ Engelfriet ....

....Engelfriet for drawing our attention to the relevance of our result to this issue. Context free hypergraph grammars (CFHG) are rewriting systems that derive sets of edge labeled hypergraphs; these systems were introduced as a generalization of edge rewriting graph grammars (see for instance [ Bauderon and Courcelle, 1987 ] In a CFHG, each production specifies some replacement of a labeled hyperedge with a hypergraph, along with particular conditions that allow the replacing hypergraph to be embedded within the host hypergraph. In this way a derivation proceeds sequentially, by replacing hyperedges in a ....

M. Bauderon and B. Courcelle. Graph expressions and graph rewritings. Mathematical Systems Theory, 20:83--127, 1987.

....a rewriting is then a pushout in a suitable category. Though the categorical apparatus leads to apparently complicated definitions, many proofs, e.g. the Critical Pair Lemma, become nothing more than commutativity of diagrams. A completely different approach to graph rewriting is taken in [ Bauderon Courcelle, 1987 ] where finite graphs are treated as algebraic expressions. Finitely oriented labeled hypergraphs are considered as a set of hyperedges glued together by means of vertices. This generalizes the situation of words, with hyperedge labels as the constants, gluing for concatenation, and a set S of ....

M. Bauderon and B. Courcelle, Graph expressions and graph rewritings, Mathematical Systems Theory 20, pp. 83-127 (1987).

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Michel Bauderon and Bruno Courcelle. Graph expressions and graph rewritings. Mathematical Systems Theory, 20:83--127, 1987. 214 215

....of graphs has to be made explicit, i.e. the composition (and construction) operations for graphs as representatives of distributed states have to be de ned. Many papers in the literature propose some (essentially) algebraic presentation of suitable classes of graphs (see for example [BC87,EV97,GH98]) Clearly, the choice of the right algebraic presentation of graphs is fundamental for our goals, because the STS method will lift via a free construction this structure to the transitions and then to the computations of a system. We do not discuss here what would be the result of applying the ....

M. Bauderon and B. Courcelle. Graph expressions and graph rewritings. Mathematical Systems Theory, 20:83-127, 1987.

....with C; C 0 K; p; p 1 ; pn 2 K etc. We obtain thus an FHR magma HS.The terms in T (FHR ) are called the HR(hypergraph) expressions. Each of them, say t, denotes a hypergraph val(t) in HS called its value. Hypergraph expressions based on di erent operations were rst introduced in [3]) Proposition 5.10 : Every nite hypergraph in HS(C) is the value of some HR expression in T (FHR ) C . Proof : Quite similar to that of Proposition 5.4. One takes a source label p(v) for each v 2 VG . One de nes G by an HR expression of the form ren h ( a(p 1 ; pn ) ....

: BAUDERON M., COURCELLE B., Graph expressions and graph rewritings, Mathematical Systems Theory 20 (1987) 83-127.

....but they are not as well accepted as, e.g. in the theory of term rewriting. One reason may be that, unlike for strings and terms, there is no straightforward inductive definition of graphs. Rather, each possible interpretation suggests a different choice of the basic operations, see for example [1,10,16,19] for different formulations of the DPO approach. Another reason might be that, except for the last and most recent one, none of the above formulations models faithfully the concurrent semantics of DPO graph rewriting based on the so called shift equivalence of derivations which captures the ....

....none of the above formulations models faithfully the concurrent semantics of DPO graph rewriting based on the so called shift equivalence of derivations which captures the abstraction from the execution order of independent steps [11,23] The reason for this failure is two fold. Some approaches [1,10,16] define rewriting on (partly) abstract graphs without providing appropriate means for the composition of isomorphism classes of arrows. As a consequence, many derivations which are not shift equivalent are identified. From the operational point of view, this problem was recognised and solved in ....

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M. Bauderon and B. Courcelle. Graph expressions and graph rewritings. Mathematical Systems Theory, 20:83--127, 1987.

....Therefore we demand above that the morphism from D into the colimit be a strong morphism and thereby determine the string of external nodes of the result. Rewriting based on our graph construction operator has the same expressive power as the graph expressions by Bauderon and Courcelle [BC87] or the double pushout approach (with injective production spans) of Ehrig [Ehr79] see also [K on00b] 1 [n] stands for the set f1; ng. 2 3 A Name based Notation for Hypergraphs We now introduce a name based notation for hypergraphs which will help us make the transition from the ....

.... systems of concurrent or interaction combinators (which are related to calculus in the same way combinator logic is related to the calculus) are given in [Yos94, Laf97] The concurrent combinators in [Yos94] are closer to the calculus and the graph representation is related to the one in [BC87]. Because of the simple nature of combinators, the resulting graphs are not hierarchical. Remark: an abridged version of this paper has appeared as [K on00a] ....

Michel Bauderon and Bruno Courcelle. Graph expressions and graph rewritings. Mathematical Systems Theory, 20:83-127, 1987.

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BAUDERON, M., AND COURCELLE, B. Graph expressions and graph rewritings. Mathematical Systems Theory 20 (1987), 83--127.

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M. Bauderon and B. Courcelle, \Graph Expressions and Graph Rewritings", Mathematical System Theory, 20 (1987) 83-127.

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