ISBN: 3540657584
TITLE: Finite Model Theory
AUTHOR: Ebbinghaus, Heinz-Dieter; Flum, Jrg
TOC:

Preface.VII
1. Preliminaries 1
2. The Ehrenfeucht-Frass Method 13
2.1 Elementary Classes 13
2.2 Ehrenfeucht's Theorem 15
2.3 Examples and Frass's Theorem 20
2.4 Hanf's Theorem 26
2.5 Gaifman's Theorem 30
3. More on Games 37
3.1 Second-Order Logic 37
3.2 Infinitary Logic: The Logics L_infinity omega and L_omega 1 omega 40
3.3 The Logics FO^s and L^s_infinity omega 46
3.3.1 Pebble Games 49
3.3.2 The s-Invariant of a Structure 54
3.3.3 Scott Formulas 56
3.4 Logics with Counting Quantifiers 58
3.5 Failure of Classical Theorems in the Finite 62
4. 0-l Laws 71
4.1 0-l Laws for FO and L^omega _infinity omega 71
4.2 Parametric Classes 74
4.3 Unlabeled 0-l Laws 77
4.3.1 Appendix 82
4.4 Examples and Consequences 84
4.5 Probabilities of Monadic Second Order Properties 88
5. Satisfiability in the Finite 95
5.1 Finite Model Property of FO^2 95
5.2 Finite Model Property of forall^2 exists* Sentences 99
6. Finite Automata and Logic: A Microcosm of Finite Model Theory 105
6.1 Languages Accepted by Automata 105
6.2 Word Models 108
6.3 Examples and Applications 111
6.4 First-Order Definability 114
7. Descriptive Complexity Theory 119
7.1 Some Extensions of First-Order Logic 120
7.2 Turing Machines and Complexity Classes 124
7.2.1 Digression: Trahtenbrot's Theorem 127
7.2.2 Structures as Inputs 129
7.3 Logical Descriptions of Computations 133
7.4 The Complexity of the Satisfaction Relation 147
7.5 The Main Theorem and Some Consequences 151
7.5.1 Appendix 162
8. Logics with Fixed-Point Operators 165
8.1 Inflationary and Least Fixed-Points 165
8.2 Simultaneous Induction and Transitivity 177
8.3 Partial Fixed-Point Logic 191
8.4 Fixed-Point Logics and L^omega_infinity omega 198
8.4.1 The Logic FO(PFP_PTIME) 205
8.4.2 Fixed-Point Logic with Counting 207
8.5 Fixed-Point Logics and Second-Order Logic 210
8.5.1 Digression: Implicit Definability 217
8.6 Transitive Closure Logic 220
8.6.1 FO(DTC) < FO(TC) 221
8.6.2 FO(posTC) and Normal Forms 224
8.6.3 FO(TC) < FO(LFP) 229
8.7 Bounded Fixed-Point Logic 235
9. Logic Programs 239
9.1 DATALOG 239
9.2 I-DATALOG and P-DATALOG 245
9.3 A Preservation Theorem 250
9.4 Normal Forms for Fixed-Point Logics 253
9.5 An Application of Negative Fixed-Point Logic 263
9.6 Hierarchies of Fixed-Point Logics 268
10. Optimization Problems 275
10.1 Polynomially Bounded Optimization Problems 275
10.2 Approximable Optimization Problems 280
11. Logics for PTIME 287
11.1 Logics and Invariants 288
11.2 PTIME on Classes of Structures 295
12. Quantifiers and Logical Reductions 307
12.1 Lindstrm Quantifiers 308
12.2 PTIME and Quantifiers 314
12.3 Logical Reductions 320
12.4 Quantifiers and Oracles 330
References 339
Index 349
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