Tree-width for first order formulae

We introduce tree-width for first order formulae ϕ, fotw(ϕ). We show that computing fotw is fixed-parameter tractable with parameter fotw. ... In this way, Chen and Dalmau obtain a notion of elimination-width 1 , which 1 Actually, the notion is called tree-width for quantified constraint formulae in [9] , But since the notion is defined via elimination ... TREE-WIDTH FOR FIRST ORDER FORMULAE 5 3. First order tree-width 3.1. Stratified tree-width. We start with defining stratified tree-width. Then, first order tree-width is defined as a special case. ...

doi:10.1007/978-3-642-04027-6_8 fatcat:b2ayc5nucvb3ljf2znqii3ghrm

We introduce tree-width for first order formulae \phi, fotw(\phi). We show that computing fotw is fixed-parameter tractable with parameter fotw. ... Our notion of tree-width generalises tree-width of conjunctive queries to arbitrary formulae of first order logic by taking into account the quantifier interaction in a formula. ... TREE-WIDTH FOR FIRST ORDER FORMULAE 5 3. First order tree-width 3.1. Stratified tree-width. We start with defining stratified tree-width. Then, first order tree-width is defined as a special case. ...

doi:10.2168/lmcs-8(1:32)2012 fatcat:faq5y4oa6rddxfdkpjdjnc6eb4

DOAJ Multiple Versions

We generalize a theorem of Courcelle [2] by showing that on structures of bounded tree-width a monadic second-order formula (with free first-and second-order variables) can be evaluated in time linear ... We generalize the notions of acyclicity and bounded tree-width from conjunctive queries to arbitrary first-order formulas in a straightforward way and analyze the complexity of evaluating formulas of these ... We generalize a theorem of Courcelle [2] by showing that on structures of bounded tree-width a monadic second-order formula (with free first-and second-order variables) can be evaluated in time linear ...

doi:10.1145/602220.602222 fatcat:t3zwdhdqdjesndualpsi3cixou

We generalize a theorem of Courcelle [2] by showing that on structures of bounded tree-width a monadic second-order formula (with free first-and second-order variables) can be evaluated in time linear ... We generalize the notions of acyclicity and bounded tree-width from conjunctive queries to arbitrary first-order formulas in a straightforward way and analyze the complexity of evaluating formulas of these ... We generalize a theorem of Courcelle [2] by showing that on structures of bounded tree-width a monadic second-order formula (with free first-and second-order variables) can be evaluated in time linear ...

doi:10.1007/3-540-44503-x_2 fatcat:ozgbjfyezrdmfmbblvofhfhb2y

The first result of this thesis is that TQBF, the problem of determining if a fully-quantified propositional CNF-formula is true, is PSPACE-complete even when restricted to instances of bounded tree-width ... On the negative side, our first result implies that, unless NP = PSPACE, the class of fully-quantified CNF-formulas of bounded tree-width does not have short proofs in any proof system (and in particular ... 3.1 Tree-width and path-width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 Quantified boolean formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

dblp:journals/eatcs/Oliva13 fatcat:tegco4ztnvfe7jjjcx63n2pyx4

Furthermore, for each ½ the class of databases of tree-width at most is definable in fixed-point logic. ... Our first theorem says that fixed-point logic with counting captures polynomial time on classes of databases of bounded tree-width. ... The set of IFP-formulas is obtained adding the following formula-formation rule to the usual rules to form first-order formulas: Given a formula ³ over the schema , where is a -ary relation symbol for ...

doi:10.1007/3-540-49257-7_6 fatcat:4yje2r6bmbfbjmjpa7d7nt3xge

We show that monadic second-order (MSO) formulas with parameters have bounded Vapnik-Chervonenkis dimension over structures of bounded clique-width, and first-order formulas with parameters have bounded ... We also show that MSO formulas of a fixed size have bounded strong consistency dimension over MSO formulas of a fixed larger size, for labeled trees. ... We also show that first-order formulas with parameters have bounded VCdimension for classes of structures of bounded local clique-width. ...

doi:10.1007/s00224-003-1112-8 fatcat:myaxkm645zg5zorfkla24llnhu

Hierarchical decompositions yielding the notions of tree-width and clique-width, expressed by terms written with appropriate graph operations and associated with Monadic Second-order Logic are important ... tools for the construction of Fixed-Parameter Tractable algorithms and also for the extension of methods and results of Formal Language Theory to the description of sets of finite graphs. ... Walukiewicz for many useful comments on a first draft of this overview. The first 11 references are books and survey articles. ...

doi:10.1007/978-3-540-70575-8_1 fatcat:ftpcqszcb5fs5nswbqnbnmy5ja

We show that monadic second-order (MSO) formulas with parameters have bounded Vapnik-Chervonenkis dimension over structures of bounded clique-width, and first-order formulas with parameters have bounded ... We also show that MSO formulas of a fixed size have bounded strong consistency dimension over MSO formulas of a fixed larger size, for labeled trees. ... We also show that first-order formulas with parameters have bounded VCdimension for classes of structures of bounded local clique-width. ...

doi:10.1007/3-540-45841-7_53 fatcat:ypcer5wsh5eo7kh7pxkbbvubc4

As many meta-theorems use results from graph minor theory, we give a brief introduction to the theory developed by Robertson and Seymour for their proof of the graph minor theorem and state the main algorithmic ... Many thanks to Javier Esparza for proofreading this manuscript and for his patience as an editor. ... Acknowledgements I want to take this opportunity to express my gratitude towards Martin Grohe for the long discussions we had on graph minors and for his advise, patience and inspiration during the time ...

arXiv:0902.3616v1 fatcat:wsuguljgrvfndhxlsps2ibes2i

This tutorial will present an overview of the use of Monadic Second-Order Logic to describe sets of finite graphs and graph transformations, in relation with the notions of tree-width and clique-width. ... First-order formulas are rather weak in this respect because they can only express local properties, like having degree or diameter at most k for fixed k. ... Tree-width and clique-width are important for language theoretical issues as well as for the construction of polynomial algorithms. ...

doi:10.1007/978-3-642-00982-2_2 fatcat:ryjvoaa7dnefrcruygfnpqso2i

We show that for each property P of structures that is definable in first-order logic and for each locally tree-decomposable class C of graphs, there is a linear time algorithm deciding whether a given ... For classes C of bounded local tree-width, we show that for every k> 1 there is an algorithm that solves the same problem in time O(n^1+(1/k)) (where n is the cardinality of the input structure). ... Let C be a class of relational structures of bounded local tree-width and ϕ a first-order definable property. ...

arXiv:cs/0004007v1 fatcat:yhr7ooestza45ayjc7be4f567u

Engelfriet : Graph structure and monadic second-order logic, book to be published by Cambridge University Press (740 p.). B.C. : Tutorial in 3 parts presented in Bangalore in 2010. ... Monadic second-order logic = First-order logic on power-set structures = First-order logic extended with (quantified) set variables denoting subsets of the domains. ... (But this is possible if the graph is linearly ordered or if some linear order is definable by an MSO formula). ...

doi:10.1007/978-3-642-24288-5_3 fatcat:yngx6xn555bhjchbvt27wyfgoe

Courcelle's famous theorem from 1990 states that any property of graphs definable in monadic second-order logic (MSO2) can be decided in linear time on any class of graphs of bounded tree-width, or in ... In particular, we show that if C is any class of graphs which is closed under taking sub-graphs and whose tree-width is not bounded by a poly-logarithmic function (in fact, log c n for some small c suffices ... If ϕ(x) is a formula with a free first-order variable x and G is a structure, we write ϕ(G) for the set {v ∈ U (G) : [9] for more on MSO 2 . G |= ϕ[v]}. ...

doi:10.1109/lics.2010.39 dblp:conf/lics/KreutzerT10 fatcat:f6m5vckbgjfotl44spf5rygt3e

We prove that for relational structures whose Gaifman graphs belong to a class of graphs that have locally bounded tree-width and is closed under minors, watermarking schemes exist that preserve first-order ... For relational structures whose Gaifman graphs have bounded degree, watermarking can be done preserving first-order queries. ... Why first-order logic? ...

doi:10.4230/lipics.fsttcs.2019.36 dblp:conf/fsttcs/ChattopadhyayP19 fatcat:vyecore2onayvoml34pdaf4xi4

Tree-Width for First Order Formulae [chapter]

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Query evaluation via tree-decompositions

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Query Evaluation via Tree-Decompositions [chapter]

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On the Complexity of Resolution-based Proof Systems

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Definability and Descriptive Complexity on Databases of Bounded Tree-Width [chapter]

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Learnability and Definability in Trees and Similar Structures

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Graph Structure and Monadic Second-Order Logic: Language Theoretical Aspects [chapter]

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Learnability and Definability in Trees and Similar Structures [chapter]

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Algorithmic Meta-Theorems [article]

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Monadic Second-Order Logic for Graphs: Algorithmic and Language Theoretical Applications [chapter]

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Deciding first-order properties of locally tree-decomposable structures [article]

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Automata for Monadic Second-Order Model-Checking [chapter]

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Lower Bounds for the Complexity of Monadic Second-Order Logic

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Query Preserving Watermarking Schemes for Locally Treelike Databases

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