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1 1 Introduction A Database Example An Example from Complexity Theory An Example from Formal Language Theory An Overview of the Book Exercises Preliminaries Background from Mathematical Logic Background from Automata and Computability Theory Background from Complexity Theory Bibliographic Notes Ehrenfeucht-Fra ssé Games First Inexpressibility Proofs Definition and Examples of Ehrenfeucht-Fra ssé Games Games and the Expressive Power of FO Rank-k Types Proof of the Ehrenfeucht-Fra ssé Theorem More Inexpressibility Results Bibliographic Notes Exercises Locality and Winning Games Neighborhoods, Hanf-locality, and Gaifman-locality Combinatorics of Neighborhoods LocalityofFO Structures of Small Degree Locality offo Revisited Bibliographic Notes Exercises

2 XII 5 Ordered Structures Invariant Queries The Power of Order-invariant FO Locality of Order-invariant FO Bibliographic Notes Exercises Complexity of First-Order Logic Data, Expression, and Combined Complexity Circuits and FO Queries Expressive Power with Arbitrary Predicates Uniformity and AC Combined Complexity offo Parametric Complexity and Locality Conjunctive Queries Bibliographic Notes Exercises Monadic Second-Order Logic and Automata Second-Order Logic and Its Fragments MSO Games and Types Existential and Universal MSO on Graphs MSO on Strings and Regular Languages FO on Strings and Star-Free Languages Tree Automata Complexity of MSO Bibliographic Notes Exercises Logics with Counting Counting and Unary Quantifiers An Infinitary Counting Logic Games for L ω(cnt) Counting and Locality Complexity of Counting Quantifiers Aggregate Operators Bibliographic Notes Exercises Turing Machines and Finite Models Trakhtenbrot's Theorem and Failure of Completeness Fagin's Theorem and NP Bibliographic Notes Exercises

3 XIII 10 Fixed Point Logics and Complexity Classes Fixed Points of Operators on Sets Fixed Point Logics Properties of LFP and IFP LFP, PFP, and Polynomial Time and Space Datalog and LFP Transitive Closure Logic A Logic for Ptime? Bibliographic Notes Exercises Finite Variable Logics Logics with Finitely Many Variables Pebble Games Definability of Types Ordering of Types Canonical Structures and the Abiteboul-Vianu Theorem Bibliographic Notes Exercises Zero-One Laws Asymptotic Probabilities and Zero-One Laws Extension Axioms The Random Graph Zero-One Law and Second-Order Logic Almost Everywhere Equivalence of Logics Bibliographic Notes Exercises Embedded Finite Models Embedded Finite Models: the Setting Analyzing Embedded Finite Models Active-Generic Collapse Restricted Quantifier Collapse The Random Graph and Collapse to MSO An Application: Constraint Databases Bibliographic Notes Exercises Other Applications of Finite Model Theory Finite Model Property and Decision Problems Temporal and Modal Logics Constraint Satisfaction and Homomorphisms of Finite Models Bibliographic Notes

4 XIV References List of Notation Index Name Index...313

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