Elements of Finite Model Theory

Finite model theory is an area of mathematical logic that grew out of computer science applications. The main sources of motivational examples for finite model theory are found in database theory, computational complexity, and formal languages, although in recent years connections with other areas, such as formal methods and verification, and artificial intelligence, have been discovered. The birth of finite model theory is often identified with Trakhtenbrot's result from 1950 stating that validity over finite models is not recursively enumerable; in other words, completeness fails over finite models. The tech­ nique of the proof, based on encoding Turing machine computations as finite structures, was reused by Fagin almost a quarter century later to prove his cel­ ebrated result that put the equality sign between the class NP and existential second-order logic, thereby providing a machine-independent characterization of an important complexity class. In 1982, Immerman and Vardi showed that over ordered structures, a fixed point extension of first-order logic captures the complexity class PTIME of polynomial time computable propertiE~s. Shortly thereafter, logical characterizations of other important complexity classes were obtained. This line of work is often referred to as descriptive complexity. A different line of finite model theory research is associated with the de­ velopment of relational databases. By the late 1970s, the relational database model had replaced others, and all the basic query languages for it were es­ sentially first-order predicate calculus or its minor extensions.

1 Introduction.- 2 Preliminaries.- 3 Ehrenfeucht-Fraïssé Games.- 4 Locality and Winning Games.- 5 Ordered Structures.- 6 Complexity of First-Order Logic.- 7 Monadic Second-Order Logic and Automata.- 8 Logics with Counting.- 9 Turing Machines and Finite Models.- 10 Fixed Point Logics and Complexity Classes.- 11 Finite Variable Logics.- 12 Zero-One Laws.- 13 Embedded Finite Models.- 14 Other Applications of Finite Model Theory.- References.- List of Notation.- Name Index.

The author has been with the department of computer science at the University of Toronto since 2000. Prior to that, he was a researcher at Bell Laboratories, and he spent two years visiting INRIA in France. His research interests are in the areas of database theory and applications of logic in computer science.

He is coauthor/editor of:

Constraint Databases
Kuper, G., Libkin, L., Paredaens, J. (Eds.), 12.04.2000, ISBN 3-540-66151-4

Finite-Model Theory and Its Applications
Grädel, E., Kolaitis, P.G. (et al.), 07.2004, ISBN 3-540-00428-9

Semantics in Databases
Thalheim, B., Libkin, L. (Eds.), Vol. 1358, 25.02.1998, ISBN 3-540-64199-8