Resolution systems and their applications I

Abstract

The central method employed today for theorem-proving is the resolution method introduced by J. A. Robinson in 1965 for the classical predicate calculus. Since then many improvements of the resolution method have been made. On the other hand, treatment of automated theorem-proving techniques for non-classical logics has been started, in connection with applications of these logics in computer science.

In this paper a generalization of a notion of the resolution principle is introduced and discussed. A certain class of first order logics is considered and deductive systems of these logics with a resolution principle as an inference rule are investigated. The necessary and sufficient conditions for the so-called resolution completeness of such systems are given. A generalized Herbrand property for a logic is defined and its connections with the resolution-completeness are presented. A class of binary resolution systems is investigated and a kind of a normal form for derivations in such systems is given. On the ground of the methods developed the resolution system for the classical predicate calculus is described and the resolution systems for some non-classical logics are outlined. A method of program synthesis based on the resolution system for the classical predicate calculus is presented. A notion of a resolution-interpretability of a logic $\mathcal{L}$ in another logic ${\mathcal{L}}^{\prime }$ is introduced. The method of resolution-interpretability consists in establishing a relation between formulas of the logic $\mathcal{L}$ and some sets of formulas of the logic ${\mathcal{L}}^{\prime }$ with the intention of using the resolution system for ${\mathcal{L}}^{\prime }$ to prove theorems of $\mathcal{L}$ . It is shown how the method of resolution-interpretability can be used to prove decidability of sets of unsatisfiable formulas of a given logic.