Elements Of Finite Model Theory
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Elements of Finite Model Theory
Author: Leonid Libkin
language: en
Publisher: Springer Science & Business Media
Release Date: 2013-03-09
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.
Elements of Finite Model Theory
Author: Leonid Libkin
language: en
Publisher: Springer Science & Business Media
Release Date: 2004-07-02
Emphasizes the computer science aspects of the subject. Details applications in databases, complexity theory, and formal languages, as well as other branches of computer science.
Finite Model Theory
Author: Heinz-Dieter Ebbinghaus
language: en
Publisher: Springer Science & Business Media
Release Date: 2005-12-29
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.