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Springer Monographs in Mathematics

Second Edition

By: Heinz-Dieter Ebbinghaus, JÃ¶rg Flum

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Hardcover | 6 October 2005 | Edition Number 2

At a Glance

Hardcover
374 Pages

Edition Type
Revised

Dimensions(cm)
23.5 x 15.88 x 1.91

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Finite model theory, the model theory of finite structures, has roots in clas- sical model theory; however, its systematic development was strongly influ- enced by research and questions of complexity theory and of database theory. Model theory or the theory of models, as it was first named by Tarski in 1954, may be considered as the part of the semantics of formalized languages that is concerned with the interplay between the syntactic structure of an axiom system on the one hand and (algebraic, settheoretic, . . . ) properties of its models on the other hand. As it turned out, first-order language (we mostly speak of first-order logic) became the most prominent language in this respect, the reason being that it obeys some fundamental principles such as the compactness theorem and the completeness theorem. These principles are valuable modeltheoretic tools and, at the same time, reflect the expressive weakness of first-order logic. This weakness is the breeding ground for the freedomwhich modeltheoretic methods rest upon. By compactness, any first-order axiom system either has only finite models of limited cardinality or has infinite models. The first case is trivial because finitely many finite structures can explicitly be described by a first-order sentence. As model theory usually considers all models of an axiom system, modeltheorists were thus led to the second case, that is, to infinite structures. In fact, classical model theory of first-order logic and its generalizations to stronger languages live in the realm of the infinite.

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Non-Fiction Mathematics Mathematical Foundation Mathematical Logic Applied Mathematics Mathematical Modelling Computing & I.T.Computer Science Mathematical Theory of Computation

Preface	p. V
Preliminaries	p. 1
The Ehrenfeucht-Fraisse Method	p. 13
Elementary Classes	p. 13
Ehrenfeucht's Theorem	p. 15
Examples and Fraisse's Theorem	p. 20
Hanf's Theorem	p. 26
Gaifman's Theorem	p. 30
More on Games	p. 37
Second-Order Logic	p. 37
Infinitary Logic: The Logics L[subscript infinity omega] and L[subscript [omega]1[omega]]	p. 40
The Logics FO[superscript s] and L[superscript s subscript infinity omega]	p. 46
Pebble Games	p. 49
The s-Invariant of a Structure	p. 54
Scott Formulas	p. 56
Logics with Counting Quantifiers	p. 58
Failure of Classical Theorems in the Finite	p. 62
0-1 Laws	p. 71
0-1 Laws for FO and L[superscript omega subscript infinity omega]	p. 71
Parametric Classes	p. 74
Unlabeled 0-1 Laws	p. 77
Appendix	p. 82
Examples and Consequences	p. 84
Probabilities of Monadic Second Order Properties	p. 88
Satisfiability in the Finite	p. 95
Finite Model Property of FO[superscript 2]	p. 95
Finite Model Property of [characters not reproducible]-Sentences	p. 99
Finite Automata and Logic: A Microcosm of Finite Model Theory	p. 105
Languages Accepted by Automata	p. 105
Word Models	p. 108
Examples and Applications	p. 111
First-Order Definability	p. 114
Descriptive Complexity Theory	p. 119
Some Extensions of First-Order Logic	p. 120
Turing Machines and Complexity Classes	p. 124
Digression: Trahtenbrot's Theorem	p. 127
Structures as Inputs	p. 129
Logical Descriptions of Computations	p. 133
The Complexity of the Satisfaction Relation	p. 147
The Main Theorem and Some Consequences	p. 151
Appendix	p. 162
Logics with Fixed-Point Operators	p. 165
Inflationary and Least Fixed-Points	p. 165
Simultaneous Induction and Transitivity	p. 177
Partial Fixed-Point Logic	p. 191
Fixed-Point Logics and L[superscript omega subscript infinity omega]	p. 198
The Logic FO(PFP[subscript PTIME])	p. 205
Fixed-Point Logic with Counting	p. 207
Fixed-Point Logics and Second-Order Logic	p. 210
Digression: Implicit Definability	p. 217
Transitive Closure Logic	p. 220
FO(DTC) < FO(TC)	p. 221
FO(posTC) and Normal Forms	p. 224
FO(TC) < FO(LFP)	p. 229
Bounded Fixed-Point Logic	p. 235
Logic Programs	p. 239
DATALOG	p. 239
I-DATALOG and P-DATALOG	p. 245
A Preservation Theorem	p. 250
Normal Forms for Fixed-Point Logics	p. 253
An Application of Negative Fixed-Point Logic	p. 263
Hierarchies of Fixed-Point Logics	p. 268
Optimization Problems	p. 275
Polynomially Bounded Optimization Problems	p. 275
Approximable Optimization Problems	p. 280
Logics for PTIME	p. 287
Logics and Invariants	p. 288
PTIME on Classes of Structures	p. 295
Quantifiers and Logical Reductions	p. 307
Lindstrom Quantifiers	p. 308
PTIME and Quantifiers	p. 314
Logical Reductions	p. 320
Quantifiers and Oracles	p. 330
References	p. 339
Index	p. 349
Table of Contents provided by Ingram. All Rights Reserved.

Springer Monographs in Mathematics

Second Edition

At a Glance

Hardcover

Shipping

How to return your order

Defective items

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