Verificação automática de lógicas finitas multivalentes

Abstract

In recent years, there is a growing interest in many-valued logics in many areas such as computer sciences automated theorem proving, approximate reasoning, multi-agent systems, program verication electrical engineering and digital circuits, linguistics, mathematics and algebra, philosophy, etc. In this context, the general problem of nding an axiomatization for nite many-valued logics has not yet been solved satisfactorily. For this, we must demonstrate the correction and completeness theorems of these logic systems, both mathematically rigorous. Soundness theorem is straightforward, however, it is known that the demonstration of the "completeness theorem"is much more sosticated and unique for each logic system. Completeness is one of the most important notions in logic and the foundations of mathematics. Completeness means the possibility of getting all correct and reliable schemata of inference by use of logical methods. When one wishes to build or design a many-valued logic system, he or she is implicitly looking for a system with a minimal set of axioms and rules. The purpose of this thesis is to establish a general algorithmic completeness proof procedure for nite many-valued logics. It is shown that a matrix is characteristic (sound and complete) for a many-valued nite logic system when, all successive correct extensions of this matrix have the same set of tautologies. It is also shown that in order to determine if a matrix is characteristic, all one has to do is to fetch from all the unit extensions those which are correct and then verify if they are repetitions of the matrix being considered. How to implement a computational procedure to demonstrate the completeness of a many-valued nite logic system is shown from these results. The new approach is a simpler and more uniform solution for the problem of nite many-valued logics axiomatization.