Unification in pretabular extensions of S4

Abstract

L.L. Maksimova and L. Esakia, V. Meskhi showed that the modal logic S4 has exactly 5 pretabular extensions PM1-PM5. In this paper, we study the problem of unification for all given logics. We showed that PM2 and PM3 have finitary, and PM1, PM4, PM5 have unitary types of unification. Complete sets of unifiers in logics are described. Keywords pretabular logic• Kripke semantic • unification • ground unifier • projective formula • unitary • finitary Unification: tasks and methodsThe unification problem, apparently, was first investigated in the works of J. Robinson [23] in developing a resolution method for of automatic proof systems. Having gone through a way of half a century development, the problem has become self-sufficient theory, actively studied in parallel in the areas of system programming [20] and non-classical logic [4]. In this paper, we consider exactly last field, where the main thesis of the theory was transformed into a statement about the possibility of turning formula into a theorem by replacement of variables.Interest in the study of unification today includes establishing the unifiability of formulas and determining the boundaries of unifiability, searching for effective algorithms for constructing unifiers (and determining their forms) for formulas, as well as determining the best such substitutions, determining the type of unification in logic, as well as a number of related tasks.One of the key modern methods is the algebraized approach proposed by S. Ghilardi [13,14] through projective approximation, which allowed describing the complete sets of formula unifiers quite efficiently. A few years beforein 1992 -S. Burris had proved the unitary unification for logics whose algebras contain a discriminatory term [8]. As was shown later [10], actually S. Burris proved a projective unification in such logics, studied in detail by S. Ghilardi [13] and actively developed in many subsequent works [11,15] (including the papers of the author, [5][6][7]).The projectivity of unification in logic allowed to prove the "good" types of unification in many logics and by many authors. One of the alternative approaches to describe complete sets of unifiers was proposed by V. Rybakov using adaptation of his method of n-characteristic models [27], successfully applied to solve the admissibility problem in a number of non-classical logics, starting in 1984 [24, 25]. Both of these approaches are reflected in this paper. Pretabular Extensions of S4By a Kripke scale, we will standardly understand a pair F = W, R , where W is a basic set, and R is a binary relation on W . The Kripke model M := F, υ is defined as a scale with a valuation υ : P rop → 2 W , where Prop is a