In this paper we describe a simple semi-decision algorithm applicable to a wide class of quantified formulas. The formulas we consider are built using the propositional connectives from prenex formulas in a language for which a decision algorithm for the corresponding quantifier-free theory T is available. In general the algorithm to be presented is not complete, but for certain formulas whose matrix belongs to one of several special sublanguages of set theory the algorithm is complete. Proper choice of the quantifier-free theory one starts from will allow many significant set theoretic constructs to be expressed by a quantified formula of the class for which our algorithm is complete. In particular, we shall show that various elementary but useful statements concerning simple set theoretic operations and relations, ordinals, and maps can be verified automatically by the technique to be described.
A Naive Unsatisfibility TestLet T be a quantifier-free theory (cf. [ 131) for which a satisfiability algorithm Aprenex formula is a formula of the form is available.where n is an integer greater than or equal to 0, Q, is either (Vx,) or (ax,), and p is a quantifier-free formula of T. We shall describe a semi-decision algorithm for formulas which have been built from prenex formulas by means of the usual logical connectives 1, &, V , -+, t) .1. Let cp be a formula of this type. Our algorithm is as follows: Bring cp into disjunctive normal form, i.e., rewrite it as a disjunction ' P O V C P I .
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