Bruno Courcelle. The expression of graph properties and graph transformations in monadic second-order logic. In G. Rozenberg, editor, Handbook of graph grammars and computing by graph transformations, vol. 1: Foundations, chapter 5, pages 313-400. World Scienti c, New-Jersey, London, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... the constraints reduce to domino systems [17] We therefore view regular graph constraints as a natural generalization of constraints on trees and grids, a generalization that is much weaker than the monadic second order logic (for which undecidability over non tree like domains is well known [7]) In this paper we consider as the class C the set of heaps. Our notion of heap (Definition 2) is motivated by the garbage collected heap in programming languages such as Java or ML. Heaps contain a root node (which models the roots of the heap such as global and local variables) and a null ....

....have the form 1 . #x#y. # where X 1 , X k denote sets of nodes, x, y denote individual nodes and # is quantifier free [22, Page 4] The isomorphism # has the following property: H G i# H #(G) where is the standard Tarskian semantics of monadic second order logic formulas [7] expressing that the closed formula #(G) is true in the model H. With the isomorphism # in mind, we introduce constraints that are propositional combinations of regular graph constraints and correspond to propositional combinations of the corresponding formulas: H G 1 and H i# H ....

Bruno Courcelle. The expression of graph properties and graph transformations in monadic second-order logic. In Handbook of graph grammars and computing by graph transformations, Vol. 1 : Foundations, chapter 5. World Scientific, 1997.

....to notice that, unlike the authors of the papers just quoted, we consider graph structures and not just tree structures. In [16] a spatial logic is introduced for reasoning about labelled directed graphs; the logic is closely related to monadic second order logic (MSO) for graphs (see e.g. [19]) and its main use is in querying structures from the semi structured data model. Both logics have a model checking problem in PSPACE, whereas satis sability for MSO is undecidable. By contrast, our logic PDL is better attuned to the reasoning tasks about path constraints and we do not ....

B. Courcelle. The expression of graph properties and graph transformations in monadic second-order logic. In G. Rozenberg, editor, Handbook of Graph Grammars and Computing by Graph Transformations - Volume 1: Foundations, pages 313-400. World Scienti c, New-Jersey, London, 1997.

.... a simpli cation of role analysis constraints generalize the notions of tree automata [27, 3] and a whole family of equivalent systems over grids [12] The remarkable fact that MSOL over trees is equivalent to tree automata inspired the question which classes of graphs have decidable MSOL theory [4]. In this paper we have introduced regular graph constraints which can be seen as an alternative to MSOL as a generalization of projections of local properties over trees and grids. Although regular graph constraints are strictly weaker than MSOL (and in fact the satis ability of regular graph ....

Bruno Courcelle. The expression of graph properties and graph transformations in monadic second-order logic. In Handbook of graph grammars and computing by graph transformations, Vol. 1 : Foundations, chapter 5. World Scienti c, 1997.

....to notice that, unlike the authors of the papers just quoted, we consider graph structures and not just tree structures. In [16] a spatial logic is introduced for reasoning about labelled directed graphs; the logic is closely related to monadic second order logic (MSO) for graphs (see e.g. [19]) and its main use is in querying structures from the semi structured data model. Both logics have a model checking problem in PSPACE, whereas satisfisability for MSO is undecidable. By contrast, our logic PDL is better attuned to the reasoning tasks 8 about path constraints and we do not ....

B. Courcelle. The expression of graph properties and graph transformations in monadic second-order logic. In G. Rozenberg, editor, Handbook of Graph Grammars and Computing by Graph Transformations - Volume 1: Foundations, pages 313--400. World Scientific, New-Jersey, London, 1997.

....x)G (G 1 j G 2 ) j G 3 G 1 j (G 2 j G 3 ) local x) G 1 j G 2 ) local x)G 1 j G 2 ; x 62 fn(G 2 ) G 1 j G 2 G 2 j G 1 (local x)nil nil (local x)G (local y)Gfy=xg; y 62 fn(G) 2. 1 Comparison with Courcelle We give a set theoretic description of graphs in the spirit of Courcelle [Cou97] which is equivalent to our graph description. We have made some di erent choices to Courcelle, which we will discuss after the de nition. We assume disjoint in nite sets of vertices V , edge identi ers E , edge labels A, and names X . The graph structure GS = hV [ E [ A; fedge E A V ....

....graphs. Courcelle considers two systems, one where src is injective and one where it is not. The graphs presented here correspond to the injective case; the non injective case corresponds to adding name fusions x = y to our graphical description, as introduced by Gardner and Wischik [GW00] In [Cou97] Courcelle studies a graph grammar which is similar to ours. Courcelle s motivation is to explore the expressive power of MSOL. In contrast, our motivation is to use our graphs to model semi structured data, and to introduce a spatial logic for locally reasoning about such data. 3 The Graph ....

Bruno Courcelle. The expression of graph properties and graph transformations in monadic second-order logic. Graph grammars and computing by graph transformations, 1:313-400, 1997.

....have been developed for proving, constructively, the existence of such algorithms. Among them, we mention the work of Courcelle in [Cou92] where the existence of polynomial algorithms is guarantied by the expressability of the corresponding problem by Monadic Second Order formulas (see also [Cou97]) Similar results have been developed in [CMR] for solving counting problems for partial k trees. As a consequence of these results, it is possible to contruct a polynomial time algorithm solving the #H coloring problem for partial k trees, when k and the size of H are xed constants. ....

B. Courcelle. The Expression of Graph Properties and Graph Transformations in Monadic Second-Order Logic. In Chap 5 of Handbook of graph grammars and computing by graph transformation. Vol. 1 Foundations, Edited by Grzegorz Rozenberg (Leiden University, The Netherlands), World Sci. Publishing, River Edge, NJ, 1997.

....summaries concise. Our system achieves this goal by using e ects as nodeterministic procedure speci cations that enable compositional interprocedural analysis. Graph types and the pointer assertion logic [54, 52, 64] are heap invariant descrip82 tion languages based on monadic second order logic [85, 17, 55]. In these systems, each graph type data structure must be represented as a spanning tree with additional pointer elds [64] constrained to denote exactly one target node. If a data structure is expressible in this way, the system [64] can verify strong properties about it, an example is ....

Bruno Courcelle. The expression of graph properties and graph transformations in monadic second-order logic. In Handbook of graph grammars and computing by graph transformations, Vol. 1 : Foundations, chapter 5. World Scientic, 1997.

....say that K is L polynomial if for every L property Phi there is a polynomial time Turing machine A Phi such that, for every G 2 K, A Phi accepts (the string code of) G iff A 2 Phi. Our starting point is the following, by now well known, generalization of Slissenko s theorem cf. Bod88,ALS91,Cou97] Theorem 10 Let K be a class of graphs of bounded treewidth. Then K is MSOL polynomial. In other words, the monadic quantifier hierarchy K collapses to the lowest level of the polynomial hierarchy, i.e. P. The same holds for classes K of cliquewidth at most 2, which includes the cliques, and ....

....conditions MSOL polynomial classes are of bounded treewidth. We have shown that closure under topological minors is sufficient, but closure under substructures is not. MSOL as a logic is very sensitive to the choice of representations of graphs and the choice of vocabulary in general, cf. Cou97] A more stable version is Bounded Second Order Logic BSLO, introduced in [Mak99] which also appears under the name of Guarded Second Order Logic GSLO in [GHO00] Here we can quantify over subsets of the basic relations as well. By changing the vocabulary such that tuples of the old relations ....

B. Courcelle. The expression of graph properties and graph transformations in monadic second-order logic. In G. Rozenberg, editor, Handbook of graph grammars and computing by graph transformations, Vol. 1 : Foundations, chapter 5, pages 313--400. World Scientific, 1997.

.... one may compare the expressive power of recognizers over graphs with that of monadic second order logic (or natural fragments of it) Much research on nite state recognizability of graph properties focused on the comparison with monadic second order de nability (see for example [Co90a] GRST96] [Co96]) However, the set up of monadic second order logic involves quite a lot of syntactical detail and thus may be regarded as natural only at second sight. The purpose of the present paper is to suggest a simpler algebraic framework in which deterministic and nondeterministic recognizability can ....

B. Courcelle, The expression of graph properties and graph transformations in monadic second-order logic, in: Handbook of Graph Transformations, Vol. I: Foundations (G. Rozenberg, ed.), World Scienc, Singapore 1996.

No context found.

B. Courcelle. The expression of graph properties and graph transformations in monadic second-order logic in Handbook of graph grammars and computing by graph transformations, chapter 5, pages 313--400. World Scientific. G. Rozenberg ed., 1997.

No context found.

B. Courcelle. The expression of graph properties and graph transformations in monadic second-order logic in Handbook of graph grammars and computing by graph transformations, chapter 5, pages 313-400. World Scienti c. G. Rozenberg ed., 1997.

....interactions [1] and the theory of optimal reductions [2] The present paper is independent of these calculus related theories. We develop several tools for the MS logic of higher level trees. Our rst tool is a signature of graph operations studied by Courcelle, Olariu, Engelfriet and others [10,16,6,9,13]. We use operations which allow to add edges to a vertex labelled graph, from every vertex labelled by, say, a to every vertex labelled by, say, b. Other operations rename labels and a binary operation builds the disjoint union of two labelled graphs. Using these operations, graphs are generated ....

....The graph represented by the term is the image of the term under the value morphism. Our rst result shows that this value morphism is a Monadic Second order transduction (an MS transduction in short) MS transductions are transformations of logical structures speci ed in MS logic ; see [7,9]. This implies that the value morphism preserves the decidability of the MS theory of the structures to which it is applied. Hence the MS theory of a graph de ned by an in nite regular term is decidable. Our second main tool is an explicit operation of rst order substitution of terms. This ....

[Article contains additional citation context not shown here]

B. Courcelle. The expression of graph properties and graph transformations in monadic second{order logic. In G. Rozenberg, editor, Handbook of Graph Grammars and Computing by Graph Transformation, volume 1, pages 313-400. World Scienti c, 1997.

....aG (e) holds iff e 2 E a (or e = #G ) and incG = f(e; x; y) j e 2 EG ; e links x to yg. Hence, sets of graphs can be defined by formulas in Monadic Second Order logic or in Counting MS logic, a refinement of MS logic using special predicate expressing cardinality of sets modulo fixed integers; see [2 4]. Such sets are said MS definable (resp. CMS definable) for short. MS logic is the extension of First Order logic with set variables. For words and binary trees, MS definability equals recognizability. Definition 6 (Tree width) A tree decomposition is a pair (T ; g) where T is a tree and where ....

....bd(L 0 ) bd(val(t) ff(K) hence is a context free language by Proposition 10. ut Our aim is now to extend Proposition 15 to sets of graphs having special tree decompositions of bounded width (as opposed to compact ones) We need some preliminary results from Courcelle (see [2] or the survey [4]) Notation 16 (Parallel composition) Let n 2 IN, let H and K be two edgedisjoint graphs having the same sequence of n sources. If every common vertex of H and K is a source, then we denote by HutnK the graph with no source obtained from the union H [ K by forgetting the sources. We denote by ....

[Article contains additional citation context not shown here]

B. Courcelle. The expression of graph properties and graph transformations in monadic second-order logic. In Handbook of Graph Grammars and Computing by Graph Transformations. Vol. I: Foundations, chapter 5, pages 313--400. World Scientific, 1997.

....aG (e) holds iff e 2 E a (or e = #G ) and incG = f(e; x; y) j e 2 EG ; e links x to yg. Hence, sets of graphs can be defined by formulas in Monadic Second Order logic or in Counting MS logic, a refinement of MS logic using special predicate expressing cardinality of sets modulo fixed integers; see [2 4]. Such sets are said MS definable (resp. CMS definable) for short. MS logic is the extension of First Order logic with set variables. For words and binary trees, MS definability equals recognizability. Definition 6 (Tree width) A tree decomposition is a pair (T ; g) where T is a tree and where ....

....bd(L 0 ) bd(val(t) ff(K) hence is a context free language by Proposition 10. ut Our aim is now to extend Proposition 15 to sets of graphs having special tree decompositions of bounded width (as opposed to compact ones) We need some preliminary results from Courcelle (see [2] or the survey [4]) Notation 16 (Parallel composition) Let n 2 IN, let H and K be two edgedisjoint graphs having the same sequence of n sources. If every common vertex of H and K is a source, then we denote by HutnK the graph with no source obtained from the union H [ K by forgetting the sources. We denote by ....

[Article contains additional citation context not shown here]

B. Courcelle. The expression of graph properties and graph transformations in monadic second-order logic. In Handbook of Graph Grammars and Computing by Graph Transformations. Vol. I: Foundations, chapter 5, pages 313--400. World Scientific, 1997.

....the theory of optimal reductions [AG98] The present paper is independent of these calculus related theories. We develop several tools for the MS logic of higher level trees. Our tools are based on graph operations extensively studied during the past decade by Courcelle, Engelfriet and others [CER93,Eng94,Cou92,Cou97]. We use operations which allow to add edges from the vertices of a given label to the vertices of another given label, to rename labels of vertices and to make disjoint union of two graphs. Using these operations, graphs are generated from constants, each of which represents a single labeled ....

.... (r) k i is called a de nition scheme. An MS de nable transduction is k copying, if k 1 (and noncopying if k = 1) In all cases, f is deterministic: L S(R 0 ) where L = fS 2 S(R) j S j= g and its domain is of course MS de nable. A parameterless MS de nable transduction f is MS compatible [Cou94,Cou97]. One can even de ne a backwards translation f # ( for MS formulas over R 0 with free variables. If has p free variables, then f # ( has kp free variables. We refer the reader to [Cou94,Cou97] for details. 4 MS transductions with parameters We now extend the previous de nitions ....

[Article contains additional citation context not shown here]

Bruno Courcelle. The expression of graph properties and graph transformations in monadic second{ order logic. In Grzegorz Rozenberg, editor, Handbook of Graph Grammars and Computing by Graph Transformation, volume 1, pages 313-400. World Scientic, 1997. 2 In fact it is even in ALG. 26

No context found.

Bruno Courcelle. The expression of graph properties and graph transformations in monadic second-order logic. In G. Rozenberg, editor, Handbook of graph grammars and computing by graph transformations, vol. 1: Foundations, chapter 5, pages 313-400. World Scienti c, New-Jersey, London, 1997.

No context found.

B. Courcelle. The expression of graph properties and graph transformations in monadic second-order logic. In G. Rozenberg, editor, Handbook of Graph Grammars and Graph Transformations, chapter 5, pages 313--400. World Scientific, 1997.

No context found.

Bruno Courcelle. The expression of graph properties and graph transformations in monadic second-order logic. In Handbook of graph grammars and computing by graph transformations, Vol. 1 : Foundations, chapter 5. World Scienti c, 1997.

No context found.

B. Courcelle. The expression of graph properties and graph transformations in monadic second-order logic. In G. Rozenberg, editor, Handbook of Graph Grammars and Computing by Graph Transformation, Vol.1: Foundations, chapter 5. World Scienti c, 1997.

No context found.

B. Courcelle. The expression of graph properties and graph transformations in monadic second-order logic. In G. Rozenberg, editor, Handbook of Graph Grammars and Computing by Graph Transformation, Vol.1: Foundations, chapter 5. World Scienti c, 1997.

No context found.

B. Courcelle. The expression of graph properties and graph transformations in monadic second-order logic. In G. Rozenberg, editor, Handbook of Graph Grammars and Computing by Graph Transformation, Vol.1: Foundations. World Scienti c, 1997.

No context found.

B. Courcelle. The expression of graph properties and graph transformations in monadic second-order logic. In G. Rozenberg, editor, Handbook of Graph Grammars and Computing by Graph Transformation, Vol.1: Foundations, chapter 5. World Scienti c, 1997.

No context found.

Bruno Courcelle. The expression of graph properties and graph transformations in monadic second-order logic. In G. Rozenberg, editor, Handbook of graph grammars and computing by graph transformations, Vol. 1: Foundations, pages 313--400. World Scientific, 1997.

No context found.

Courcelle, B., The expression of graph properties and graph transformations in monadic second-order logic, in: G. Rozenberg, editor, Handbook of Graph Grammars and Computing by Graph Transformation. Vol. I: Foundations, World Scientific, 1997 pp. 313--400.

No context found.

B. Courcelle, The expression of graph properties and graph transformations in monadic second-order logic, in: G. Rozenberg, editor, Handbook of Graph Grammars and Computing by Graph Transformations, vol. 1, (World Scienti c Publishing, Singapore, 1997) 313-400.

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