M. GROHE AND J. MARINO, Definability and descriptive complexity on databases of bounded tree-width, in Proceedings of ICDT 99, vol. 1540 of Lecture Notes in Computer Science, Springer, 1999, pp. 70-82.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... class, and let b L( be a formula defining ]C inside Z) Further, let I be an L[ U t) interpretation that defines a canonization on 7) By the interpretation lemma Hence L captures PTIME on ) This result is important, because it has been shown, in particular in the work of Grohe [53, 54, 55], that a number of interesting domains admit canonization via fixed point logic with counting, IFP C) Among these are (1) The domain of finite (labelled) trees (see Proposition 5.12) 2) The class of planar graphs [53] and, more generally, any domain of structures, whose Gaifman graphs are ....

....(IFP C) Among these are (1) The domain of finite (labelled) trees (see Proposition 5. 12) 2) The class of planar graphs [53] and, more generally, any domain of structures, whose Gaifman graphs are embeddable in a fixed surface [54] 3) Any domain of structures of bounded tree width [55]. Corollary 5.14. IFP C) captures PTIME on any of these domains. Further, the results extend to domains that can be reduced to any of the mentioned ones by simple definable operations like adding or deleting a vertex or edge. An example are the so called nearly planar (or apex) graphs which ....

M. GROHE AND J. MARINO, Definability and descriptive complexity on databases of bounded tree-width, in Proceedings of ICDT 99, vol. 1540 of Lecture Notes in Computer Science, Springer, 1999, pp. 70-82.

....CONN (connected) is Pi 1 MSOL but not Sigma 1 MSOL. ii) However, on graphs of treewidth at most 1 (undirected forests) monadic PH collapses to Sigma 1 MSOL. i) is proven by Fagin [Fag75] for graphs of treewidth at most 2. ii) follows from the padding lemma for MSOL of Grohe and Mari no, GM99] A predecessor to theorem 11 may be found in [See91] who showed a similar theorem, were 9MSOL polynomial is replaced by decidability of the MSOL theory of K, and no closure condition is needed. Not so similar but related to theorem 11 are the characterization of classes of formulas for which ....

M. Grohe and J. Mari~no. Definability and descriptive complexity on databases of bounded tree--width. In C. Beeri and P. Bunemann, editors, Database Theory--ICDT'99, volume 1540 of Lecture Notes in Computer Science, pages 70--82. Springer, 1999.

....the are also computable from the Kauffman brackets, as shown in [ABS] Problem 2. Are the interlace polynomials computable in polynomial time on graphs of bounded tree width Finally, our methods may have some bearing on the search of complete invariant for graphs of bounded tree width. In [GM99], it is shown that there is a graph canonization, and hence a complete graph invariant, for graphs of bounded tree width which is computable in polynomial time. However, this canonozation is not given as a graph polynomial. Problem 3. Is there a natural family Inv(k) of polynomials which is a ....

M. Grohe and J. Mari~no. Definability and descriptive complexity on databases of bounded tree--width. In C. Beeri and P. Bunemann, editors, Database Theory--ICDT'99, volume 1540 of Lecture Notes in Computer Science, pages 70--82. Springer, 1999.

....to the recognition and construction problems. We have presented here our results for graphs (and problems expressible as graph problems) Tree width has been generalized to arbitrary relational structures by Feder and Vardi in [36] Interesting applications may be found in [56] Grohe and Mari no [44] have shown the following remarkable connection between MS( and the formulas of the Fixed Point Logic LFP. 29 Theorem 48 (Grohe and Mari no 1999) On classes of structures of bounded tree width every MS( sentence (formula) is equivalent a LFP( formula. Similarly, clique width can also ....

M. Grohe and J. Mari~no. Definability and descriptive complexity on databases of bounded tree--width. In C. Beeri and P. Bunemann, editors, Database Theory--ICDT'99, volume 1540 of Lecture Notes in Computer Science, pages 70--82. Springer, 1999.

....inflationary fixed point logic. In particular, Immerman and Lander [8] proved that inflationary fixed point logic with counting (that is, where there is an additional universe of numbers and a total ordering on this universe) captures P on the class of trees, and Grohe [6] and Grohe and Mari no [7] proved that this same logic does likewise on the class of planar graphs and the class of graphs of bounded treewidth, respectively. Grohe [6] also proved that inflationary fixed point logic (without counting) captures P on the class of 3 connected planar graphs. In this paper, we show that a ....

M. Grohe and J. Mari~no, Definability and descriptive complexity on databases of bounded tree-width, manuscript (1998).

.... [Rot98b] has given a positive answer both to the recognition and construction problem We have presented here our results for graphs (and problems expressible as graph problems) Treewidth has been generalized to arbitrary relational structures by Feder and Vardi in [FV98] Grohe and Mari no [Gn98] have shown the following remarkable connection betwee MS( and the formulas of the Fixed Point Logic LFP . Theorem 15 ( Grohe and Mari no 1998) On classes of structures of bounded treewidth every MS( sentence (formula) is equivalent a LFP ( formula. Open problem 16 Is theorem 15 ....

M. Grohe and J. Mari no. Definability and descriptive complexity on databases of bounded tree-width. Preprint, 1998.

....and the notion of the Gaifman graph of a structure will notice that a structure B is a k tree if and only the Gaifman graph of B has tree width at most k. Definition 3.5 gives a natural notion of tree width for arbitrary structures. The very same definition has been introduced independently in [16]. Let B be a k tree, with T and F as in Definition 3.5. We call a tuple b = b 1 ; b r ) in B local if fb 1 ; b r g F (v) for some node v of T . It is immediate from the definition that every tuple b that satisfies an atomic formula in B is local. A simple graph theoretic ....

M. Grohe and J. Mari~ no, Definability and descriptive complexity on databases of bounded tree-width, submitted for publication.

....is not expressible in first order logic with the addition of the fixed point operator and counting quantifiers. Some beautiful work by Martin Grohe and his students has shown that there are large and important classes of graphs on which the language FO(wo) LFP; C) does capture order independent P [11, 8, 10]. For these classes of graphs, isomorphism is testable in polynomial time using the natural algorithm for equivalence in a sublanguage of FO(wo) LFP; C) These results include natural classes of graphs for which no polynomial time graph isomorphism algorithm had been previously known. Most ....

M Grohe and J. Mari~no, "Definability and Descriptive Complexity on Databases of Bounded Tree-Width," to appear in Intl. Conf. on Database Theory(1999)

.... [Rot98b] has given a positive answer both to the recognition and construction problem We have presented here our results for graphs (and problems expressible as graph problems) Treewidth has been generalized to arbitrary relational structures by Feder and Vardi in [FV98] Grohe and Mari no [Gn98] have shown the following remarkable connection betwee MS( and the formulas of the Fixed Point Logic LFP . Theorem41 (Grohe and Mari no 1998) On classes of structures of bounded treewidth every MS( sentence (formula) is equivalent a LFP ( formula. Problem 42. Is theorem 41 also ....

M. Grohe and J. Mari no. Definability and descriptive complexity on databases of bounded tree-width. Preprint, 1998.

....goes on like this, and since the level decreases each round the spoiler eventually wins. # We state three further results without proof. Theorem 6.24. i) Grohe [32] There is a k # 1 such that on the class of planar graphs, isomorphism and C k equivalence coincide. ii) Grohe, Mari no [33]) Let k # 1. On graphs of tree width at most k, isomorphism and C k 3 equivalence coincide. iii) Immerman, Lander [49] On graphs of color class size at most 3, isomorphism and L 3 equivalence coincide. The previous results can all be extended to colored graphs of the respective ....

....canonization. But this is only an empirical observation, it is not known whether ( # = # C ) # C k # C ) always implies that C admits IFP C definable canonization. Theorem 7.7. i) Grohe [32] The class of planar graphs admits IFP C definable canonization. ii) Grohe, Mari no [33]) For each w # 1, the class of graphs of tree width at most w admits IFP C definable canonization. The following result is implicit in Otto s [64] Proposition 7.8. Let C be a class of # structures and k # N. i) If I k is PTIME invertible on C then # L k on C admits ....

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M. Grohe and J. Mari no, Definability and descriptive complexity on databases of bounded tree-width, Proceedings of the 7th international conference on database theory (C. Beeri, editor), Lecture Notes in Computer Science, Springer-Verlag, 1999, to appear.

....game goes on like this, and since the level decreases each round the spoiler eventually wins. 2 We state three further results without proof. Theorem 6.24. i) Grohe [32] There is a k 1 such that on the class of planar graphs, isomorphism and C k equivalence coincide. ii) Grohe, Mari no [33]) Let k 1. On graphs of tree width at most k, isomorphism and C k 3 equivalence coincide. iii) Immerman, Lander [49] On graphs of color class size at most 3, isomorphism and L 3 equivalence coincide. The previous results can all be extended to colored graphs of the respective classes. ....

....But this is only an empirical observation, it is not known whether = C = C k C always implies that C admits IFP C definable canonization. 34 MARTIN GROHE Theorem 7.7. i) Grohe [32] The class of planar graphs admits IFP C definable canonization. ii) Grohe, Mari no [33]) For each w 1, the class of graphs of tree width at most w admits IFP C definable canonization. The following result is implicit in Otto s [64] Proposition 7.8. Let C be a class of structures and k 2 N. i) If I k is PTIME invertible on C then L k on C admits IFP definable ....

[Article contains additional citation context not shown here]

M. Grohe and J. Marino. Definability and descriptive complexity on databases of bounded tree-width. In C. Beeri, editor, Proceedings of the 7th International Conference on Database Theory, Lecture Notes in Computer Science. Springer-Verlag, 1999. To appear.

.... are at least m elements x such that : For many important classes C of graphs there is a k 1 such that two graphs in C are isomorphic if, and only if, they are equivalent in L k (or C k ) Examples are the class of planar graphs [12] and all classes of graphs of bounded tree width [13]. Immerman and Lander proved that for each fixed k there are polynomial time algorithms deciding whether two given graphs are equivalent in L k or C k , respectively. If equivalence in the respective logic coincides with isomorphism, this actually gives rise to a polynomial time canonization ....

M. Grohe and J. Marino. Definability and descriptive complexity on databases of bounded tree-width. In C. Beeri and P. Buneman, editors, Proceedings of the 7th International Conference on Database Theory, volume 1540 of Lecture Notes in Computer Science, pages 70--82. Springer-Verlag, 1999.

....game goes on like this, and since the level decreases each round the spoiler eventually wins. 2 We state three further results without proof. Theorem 6.24. i) Grohe [32] There is a k 1 such that on the class of planar graphs, isomorphism and C k equivalence coincide. ii) Grohe, Mari no [33]) Let k 1. On graphs of tree width at most k, isomorphism and C k 3 equivalence coincide. iii) Immerman, Lander [49] On graphs of color class size at most 3, isomorphism and L 3 equivalence coincide. The previous results can all be extended to colored graphs of the respective classes. ....

....empirical observation, it is not known whether Gamma = C Delta = Gamma j C k C Delta always implies that C admits IFP C definable canonization. 34 MARTIN GROHE Theorem 7.7. i) Grohe [32] The class of planar graphs admits IFP C definable canonization. ii) Grohe, Mari no [33]) For each w 1, the class of graphs of tree width at most w admits IFP C definable canonization. The following result is implicit in Otto s [64] Proposition 7.8. Let C be a class of structures and k 2 N. i) If I k is PTIME invertible on C then j L k on C admits IFP definable ....

[Article contains additional citation context not shown here]

M. Grohe and J. Marino. Definability and descriptive complexity on databases of bounded tree-width. In C. Beeri, editor, Proceedings of the 7th International Conference on Database Theory, Lecture Notes in Computer Science. Springer-Verlag, 1999. To appear.

.... has been proved by Cai, Furer, and Immerman [1] that the resulting fixed point logic with counting, denoted by IFP C, still does not capture all of polynomial time, it does capture polynomial time on several important classes of structures (on trees, planar graphs, structures of bounded tree width [14, 8, 9]) The main motivation for such capturing results is that they may give a better understanding of polynomial time. But of course this requires that the logical side is well understood. We hope that our analysis of IFP C formulas will help to understand better the expressive power of IFP C; in ....

M. Grohe and J. Marino. Definability and descriptive complexity on databases of bounded tree-width. 1998.

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