This book gives a comprehensive overview of central themes of finite model theory â expressive power, descriptive complexity, and zero-one laws â together with selected applications relating to database theory and artificial intelligence, especially constraint databases and constraint satisfaction problems. The final chapter provides a concise modern introduction to modal logic, emphasizing the continuity in spirit and technique with finite model theory. This underlying spirit involves the use of various fragments of and hierarchies within first-order, second-order, fixed-point, and infinitary logics (...) to gain insight into phenomena in complexity theory and combinatorics. The book emphasizes the use of combinatorial games, such as extensions and refinements of the Ehrenfeucht-Fraissé pebble game, as a powerful way to analyze the expressive power of such logics, and illustrates how deep notions from model theory and combinatorics, such as o-minimality and treewidth, arise naturally in the application of finite model theory to database theory and AI. Students of logic and computer science will find here the tools necessary to embark on research into finite model theory, and all readers will experience the excitement of a vibrant area of the application of logic to computer science. (shrink)
A criterion of adequacy is proposed for theories of relevant consequence. According to the criterion, scientists whose deductive reasoning is limited to some proposed subset of the standard consequence relation must not thereby suffer a reduction in scientific competence. A simple theory of relevant consequence is introduced and shown to satisfy the criterion with respect to a formally defined paradigm of empirical inquiry.
A paradigm of scientific discovery is defined within a first-order logical framework. It is shown that within this paradigm there exists a formal scientist that is Turing computable and universal in the sense that it solves every problem that any scientist can solve. It is also shown that universal scientists exist for no regular logics that extend first-order logic and satisfy the Löwenheim-Skolem condition.
A model of idealized scientific inquiry is presented in which scientists are required to infer the nature of the structure that makes true the data they examine. A necessary and sufficient condition is presented for scientific success within this paradigm.
Alternative models of idealized scientific inquiry are investigated and compared. Particular attention is devoted to paradigms in which a scientist is required to determine the truth of a given sentence in the structure giving rise to his data.
This paper provides a mathematical model of scientific discovery. It is shown in the context of this model that any discovery problem that can be solved by a computable scientist can be solved by a computable scientist all of whose conjectures are finitely axiomatizable theories.