Skolem reduction classes

Abstract

Among the earliest and best-known theorems on the decision problem is Skolem's result [7] that the class of all closed formulas with prefixes of the form ∀···∀∃···∃ is a reduction class for satisfiability for the whole of quantification theory. This result can be refined in various ways. If the Skolem prefix alone is considered, the best result [8] is that the ∀∀∀∃ class is a reduction class, for Gödel [3], Kalmár [4], and Schütte [6] showed the ∀∀∃···∃ class to be solvable. The purpose of this paper is to describe the more complex situation that arises when (Skolem) formulas are restricted with respect to the arguments of their atomic subformulas. Before stating our theorem, we must introduce some notation.Let x, y1, y2, be distinct variables (we shall use v1, v2, ··· and w1, w2, ··· as metavariables ranging over these variables), and for each i ≥ 1 let Y(i) be the set {y1, ···, yi}. An atomic formula Pv1 ··· vk will be said to be {v1, ···, vk}-based. For any n ≥ 1, p ≥ 1, and any subsets Y1, ··· Yp of Y(n), let C(n, Y1, ···, Yp) be the class of all those closed formulas with prefix ∀y1 ··· ∀yn∃x such that each atomic subformula not containing the variable x is Yi-based for some i, 1 ≤ i ≤ p.