On the Monadic Second-Order Transduction Hierarchy

Blumensath, Achim; Courcelle, Bruno

doi:10.2168/lmcs-6(2:2)2010

Cited by 23 publications

(40 citation statements)

References 26 publications

(42 reference statements)

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“…A graph M is a minor of a graph G if there exists a minor map from M to G, which is a map µ defined on M where When G is a graph, we use minorspGq to denote the class of all minors of G, and we extend this notation to a class G setting minorspGq " Ť GPG minorspGq. The following theorem is known; the first two parts are due to Robertson and Seymour's graph minor series (see [6]) and the third is due to Blumensath and Courcelle [5]. PROPOSITION 2.2.…”

Section: Graphsmentioning

confidence: 99%

One hierarchy spawns another

Chen

Müller

2014

Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Nin

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We study the problem of conjunctive query evaluation relative to a class of queries; this problem is formulated here as the relational homomorphism problem relative to a class of structures A, wherein each instance must be a pair of structures such that the first structure is an element of A. We present a comprehensive complexity classification of these problems, which strongly links graph-theoretic properties of A to the complexity of the corresponding homomorphism problem. In particular, we define a binary relation on graph classes and completely describe the resulting hierarchy given by this relation. This binary relation is defined in terms of a notion which we call graph deconstruction and which is a variant of the well-known notion of tree decomposition. We then use this hierarchy of graph classes to infer a complexity hierarchy of homomorphism problems which is comprehensive up to a computationally very weak notion of reduction, namely, a parameterized version of quantifier-free reductions. In doing so, we obtain a significantly refined complexity classification of homomorphism problems, as well as a unifying, modular, and conceptually clean treatment of existing complexity classifications. We then present and develop the theory of Ehrenfeucht-Fraïssé-style pebble games which solve the homomorphism problems where the cores of the structures in A have bounded tree depth. Finally, we use our framework to classify the complexity of model checking existential sentences having bounded quantifier rank.

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Section: Graphsmentioning

confidence: 99%

One hierarchy spawns another

Chen

Müller

2014

Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Nin

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“…Here we provide a brief definition based on [1], simplified to target only the FO graph case. A basic FO-transduction τ 0 is a triple (χ, ν, µ) of FO formulas with 0, 1 and 2 free variables, respectively, such that τ 0 maps a graph G into a graph on the vertex set {v | G |= ν(v)} and the edge set {{u, v} | G |= µ(u, v)} (an induced subgraph of I µ (G)), or τ 0 (G) is undefined if G |= χ.…”

Section: Fo Logicmentioning

confidence: 99%

“…Note that the model checking problem for first-order logic -given a graph G and an FO formula φ we want to decide whether G satisfies φ (written as G |= φ) -is trivially solvable in time |V (G)| O(|φ|) . "Efficient solvability" hence in this context often means fixed-parameter tractability (FPT); that is, solvability in time f (|φ|) · |V (G)| O (1) for some computable function f .…”

Section: Introductionmentioning

confidence: 99%

A New Perspective on FO Model Checking of Dense Graph Classes

Gajarský

Hliněný

Óbdržálek

et al. 2016

Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science

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We study the first-order (FO) model checking problem of dense graphs, namely those which have FO interpretations in (or are FO transductions of) some sparse graph classes. We give a structural characterization of the graph classes which are FO interpretable in graphs of bounded degree. This characterization allows us to efficiently compute such an FO interpretation for an input graph. As a consequence, we obtain an FPT algorithm for successor-invariant FO model checking of any graph class which is FO interpretable in (or an FO transduction of) a graph class of bounded degree. The approach we use to obtain these results may also be of independent interest.

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“…As a second example, consider the across-connect operation which takes two copies of a graph G and connects corresponding nodes across. To implement this operation, we first construct the 2-copy of G [9]. Specifically, we take isomorphic copies G 1 and G 2 of G, where the universe of G i is…”

Section: Closure Of L-ebsp(· ·) Under Operations Implemented Using Tmentioning

confidence: 99%

A generalization of the Łoś–Tarski preservation theorem

Sankaran

Adsul

Chakraborty

2016

Annals of Pure and Applied Logic

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Chairman Date:Place: DeclarationI declare that this written submission represents my ideas in my own words and where others' ideas or words have been included, I have adequately cited and referenced the original sources.I also declare that I have adhered to all principles of academic honesty and integrity and have not misrepresented or fabricated or falsified any idea/data/fact/source in my submission. I understand that any violation of the above will be cause for disciplinary action by the Institute and can also evoke penal action from the sources which have thus not been properly cited or from whom proper permission has not been taken when needed.(Signature of student) and preservation under extensions, when k equals 0. As a consequence, we get a parameterized generalization of the Łoś-Tarski theorem for sentences, in both its substructural and extensional forms. We call our characterizations collectively the generalized Łoś-Tarski theorem for sentences at level k, abbreviated GLT(k). To the best of our knowledge, GLT(k) is the first to relate counts of quantifiers appearing in the sentences of the Σ In summary, the properties introduced in this thesis are interesting in both the classical and finite model theory contexts, and yield in both these contexts, a new parameterized generalization of the Łoś-Tarski preservation theorem.viii

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On the Monadic Second-Order Transduction Hierarchy

Cited by 23 publications

References 26 publications

One hierarchy spawns another

One hierarchy spawns another

A New Perspective on FO Model Checking of Dense Graph Classes

A generalization of the Łoś–Tarski preservation theorem

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