The Monotonous Elimination of Predicate Variables

Abstract

I describe an algorithm which, when given a sentence Γ involving universally quantified predicate variables, first reduces this sentence to a normal form Γ* and then, if Γ* satisfies certain extra restrictions, eliminates the predicate variables of Γ* to produce a predicate-free equivalent of Γ. This process may involve the introduction of skolem functions but in many cases does not (and so produces a more elementary version of Γ). The algorithm eliminates the predicate variables by witnessing them with certain definable predicates. The choice of these predicates is guided by the shape of the normal form Γ*. The algorithm was initially designed to deal with modal correspondence problems. In this respect it is an extension of Sahlqvist's algorithm which produces first-order equivalents of a wide class of modal formulas. The algorithm is illustrated by several examples from modal logic.