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### Tree-Width for First Order Formulae [chapter]

2009 Lecture Notes in Computer Science
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.  ...

### Tree-width for first order formulae

Adler Isolde, Mark Weyer, Erich Grädel
2012 Logical Methods in Computer Science
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.  ...

### Query evaluation via tree-decompositions

Jörg Flum, Markus Frick, Martin Grohe
2002 Journal of the ACM
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  ...

### Query Evaluation via Tree-Decompositions [chapter]

Jörg Flum, Markus Frick, Martin Grohe
2001 Lecture Notes in Computer Science
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  ...

### On the Complexity of Resolution-based Proof Systems

Sergi Oliva
2013 Bulletin of the European Association for Theoretical Computer Science
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ...

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

Martin Grohe, Julian Mariño
1999 Lecture Notes in Computer Science
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  ...

### Learnability and Definability in Trees and Similar Structures

Martin Grohe, Gy. Tur�n
2004 Theory of Computing Systems
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.  ...

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

Bruno Courcelle
2008 Lecture Notes in Computer Science
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.  ...

### Learnability and Definability in Trees and Similar Structures [chapter]

Martin Grohe, Gyorgy Turán
2002 Lecture Notes in Computer Science
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.  ...

### Algorithmic Meta-Theorems [article]

Stephan Kreutzer
2009 arXiv   pre-print
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  ...

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

Bruno Courcelle
2009 Lecture Notes in Computer Science
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.  ...

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

Markus Frick, Martin Grohe
2000 arXiv   pre-print
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.  ...

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

Bruno Courcelle
2011 Lecture Notes in Computer Science
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).  ...

### Lower Bounds for the Complexity of Monadic Second-Order Logic

Stephan Kreutzer, Siamak Tazari
2010 2010 25th Annual IEEE Symposium on Logic in Computer Science
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]}.  ...

### Query Preserving Watermarking Schemes for Locally Treelike Databases

Agnishom Chattopadhyay, M. Praveen, Michael Wagner
2019 Foundations of Software Technology and Theoretical Computer Science
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?  ...
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