Finite Model Theory and Its Applications

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1 Erich Grädel Phokion G. Kolaitis Leonid Libkin Maarten Marx Joel Spencer Moshe Y. Vardi Yde Venema Scott Weinstein Finite Model Theory and Its Applications With 35 Figures and 2 Tables Springer

Contents 1 Unifying Themes in Finite Model Theory 1 1.1 Definability Theory 2 1.1.1 Classification of Concepts in Terms of Definitional Complexity 2 1.1.2 What More Do We Know When We Know a Concept Is L-Definable?

2 Contents 1 Unifying Themes in Finite Model Theory Definability Theory Classification of Concepts in Terms of Definitional Complexity What More Do We Know When We Know a Concept Is L-Definable? Logics with Finitely Many Variables Distinguishing Struktures: L-Equivalence and Comparison Games Random Graphs and 0-1 Laws Constraint Satisfaction Problems Descriptive Complexity Satisfaction What Is a Logic for PTIME? Finite Model Theory and Infinite Structures Tarne Fragments and Tarne Classes 20 References 22 2 On the Expressive Power of Logics on Finite Models Introduction Basic Concepts Ehrenfeucht-Frai'sse Games for First-Order Logic Computational Complexity Complexity Classes The Complexity of Logic Ehrenfeucht-Frai'sse Games for Existential Second-Order Logic Logics with Fixed-Point Operators Operators and Fixed Points Least Fixed-Point Logic Datalog and Datalog(^) 74

VIII Contents 2.6.4 The Complementation Problem for LFPi and a Normal Form for LFP 79 2.6.5 Partial Fixed-Point Logic 82 2.7 Infmitary Logics with Finitely Mariy Variables 86 2.7.1 The Infmitary Logic L^ 87 2.

3 VIII Contents The Complementation Problem for LFPi and a Normal Form for LFP Partial Fixed-Point Logic Infmitary Logics with Finitely Mariy Variables The Infmitary Logic L^ Pebble Games and L^-Definability Laws for L^ Definability and Complexity of L^-Equivalence Least Fixed-Point Logic vs. Partial Fixed-Point Logic on Finite Structures Existential Infmitary Logics with Finitely Many Variables The Infmitary Logics 3L^ and 3L*, W (^) Existential Pebble Games Descriptive Complexity of Fixed Subgraph Homeomorphism Queries 116 References Finite Model Theory and Descriptive Complexity Definability and Complexity Complexity Issues in Logic Model Checking for First-Order Logic The Strategy Problem for Finite Games Complexity of First-Order Model Checking Encoding Finite Structures by Words Capturing Complexity Classes Capturing NP: Fagin's Theorem Logics That Capture Complexity Classes Capturing Polynomial Time on Ordered Structures Capturing Logarithmic Space Complexity Transitive Closure Logics Fixed-Point Logics Some Fixed-Point Theory Least Fixed-Point Logic The Modal ^-Calculus Parity Games Model-Checking Games for Least Fixed-Point Logic Definability of Winning Regions in Parity Games Simultaneous Fixed-Point Inductions Infiationary Fixed-Point Logic Partial Fixed-Point Logic Datalog and Stratified Datalog Logics with Counting Logics with Counting Terms Fixed-Point Logic with Counting Datalog with Counting 190

Contents IX 3.5 Capturing PTIME via Canonization 193 3.5.1 Definable Linear Orders 193 3.5.2 Canonizations and Interpretations 194 3.5.3 Capturing PTIME up to Bisimulation 198 3.5.4 Is There a Logic for PTIME?

4 Contents IX 3.5 Capturing PTIME via Canonization Definable Linear Orders Canonizations and Interpretations Capturing PTIME up to Bisimulation Is There a Logic for PTIME? Algorithmic Model Theory Beyond Finite Struktures Finitely Presentable Structures Metafinite Structures Metafinite Spectra Descriptive Complexity over the Real Numbers Appendix: Alternating Complexity Classes 222 References Logic and Random Structures An Instructive Example Random Graphs Extension Statements Closure The Almost Sure Theory The Case p Constant Countable Models A Dynamic View Infinite Spectra via Almost Sure Encoding The Jump Condition The Complexity Condition Nonconvergence via Almost Sure Encoding No Almost Sure Representation of Evenness The Ehrenfeucht Game About the References 254 References Embedded Finite Models and Constraint Databases Introduction Organization Relational Databases and Embedded Finite Models Constraint Databases Collapse and Genericity: An Overview Approaches to Proving Expressivity Bounds Active-Generic Collapse The Ramsey Property Collapse Results Natural-Active Collapse Collapse: Failure and Success Good Structures vs. Bad Structures: O-minimality 276

X Contents 5.6.3 Collapse Theorem and Corollaries 277 5.6.4 Collapse Algorithm: the Linear Case 278 5.6.5 Collapse Algorithm: the General Case 280 5.6.6 Collapse Without O-minimality 285 5.6.7 Natural-Generic Collapse 287 5.

5 X Contents Collapse Theorem and Corollaries Collapse Algorithm: the Linear Case Collapse Algorithm: the General Case Collapse Without O-minimality Natural-Generic Collapse Model Theory and Collapse Results Pseudo-finite Homogeneity Finite Cover Property and Collapse Isolation and Collapse The VC Dimension and Collapse Results Random Graph and Collapse to MSO Complexity Bounds for Generic Queries Expressiveness of Constraint Query Languages Reductions to the Finite Case Topological Properties Linear vs. Polynomial Constraints Query Safety Finite and Infinite Query Safety Safe Translations Finite Query Safety: Characterization Infinite Query Safety: Reduction Deciding Safety Dichotomy Theorem for Embedded Finite Models Database Considerations Aggregate Operators Higher-Order Features Bibliographie Notes 330 References A Logical Approach to Constraint Satisfaction Introduction Preliminaries The Computational Complexity of Constraint Satisfaction Nonuniform Constraint Satisfaction Monotone Monadic SNP and Nonuniform Constraint Satisfaction Datalog and Nonuniform Constraint Satisfaction Datalog, Games, and Constraint Satisfaction Games and Consistency Uniform Constraint Satisfaction and Bounded Treewidth 362 References 367

Contents XI 7 Local Variations on a Loose Theme: Modal Logic and Decidability 371 7.1 Introduction 371 7.2 Modal Systems and Bisimulations 373 7.3 Basic Modal Logic 378 7.3.1 Notes 388 7.

6 Contents XI 7 Local Variations on a Loose Theme: Modal Logic and Decidability Introduction Modal Systems and Bisimulations Basic Modal Logic Notes Some Variations Neither Locality nor Looseness: Grid Logics Universal Access: K* Generalizing Looseness: the Until Operator Modal Complexity NP and the Polysize Model Property PSPACE and Polynomially Deep Paths EXPTIME and Exponentially Deep Paths NEXPTIME Notes Modal Logic and First-Order Logic Guarded Fragments Decidability and Complexity Notes 425 References 426 Index 431

A Retrospective on Datalog 1.0

A Retrospective on Datalog 1.0 Phokion G. Kolaitis UC Santa Cruz and IBM Research - Almaden Datalog 2.0 Vienna, September 2012 2 / 79 A Brief History of Datalog In the beginning of time, there was E.F.