Decision procedures for elementary sublanguages of set theory. II. Formulas involving restricted quantifiers, together with ordinal, integer, map, and domain notions

Abstract

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 .

Decision procedures for elementary sublanguages of set theory VII. Validity in set theory when a choice operator is present

“…Choose a representative x' in each equivalence class { y : y -x } , replace every variable by its respresentative in Q and let 0 be the resulting formula. The following theorem contained in [2]. There is an ordering y,, y2 , a *, y,,, of the variables of Q such that y, -0 , and such that yi E y) in Q implies i < j .…”

“…Moreover we assume that only restrictions R,-R, must hold and that variables can range over arbitrary sets (not necessarily finite). We describe a satisfiability algorithm for Q originally given in [2]. Let -be the smallest equivalence relation on the set of all the variables of Q such that x = y in Q implies x -y .…”

Decision procedures for elementary sublanguages of set theory IX. Unsolvability of the decision problem for a restricted subclass of the Δ0‐formulas in set theory

“…This problem has been investigated in several papers (see [2], [3], [4], [5], [6], [8], [lo]), and various classes of formulas for which the decision problem has a positive solution (i.e., is solvable) have been found.…”

Decision procedures for elementary sublanguages of set theory. IV. Formulae involving a rank operator or one occurrence of Σ(x)={{y}| y ϵ x}

“…This implies the decidability of MLS extended with a singleton operator { } (see [l], [2], [3]). Notice that this I: operator is somehow related both to the Cartesian product and to the powerset operator.…”

“…For X = 0 the values of V;., wi are given by step (3) and ( Proof: Let V;. be as in algorithm a t ' at the end of execution.…”