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