Complexity Assessments for Decidable Fragments of Set Theory. I: A Taxonomy for the Boolean Case*

Abstract

We report on an investigation aimed at identifying small fragments of set theory (typically, sublanguages of Multi-Level Syllogistic) endowed with polynomial-time satisfiability decision tests, potentially useful for automated proof verification. Leaving out of consideration the membership relator ∈ for the time being, in this paper we provide a complete taxonomy of the polynomial and the NP-complete fragments involving, besides variables intended to range over the von Neumann set-universe, the Boolean operators ∪ ∩ \, the Boolean relators ⊆, ⊈,=, ≠, and the predicates ‘• = Ø’ and ‘Disj(•, •)’, meaning ‘the argument set is empty’ and ‘the arguments are disjoint sets’, along with their opposites ‘• ≠ Ø and ‘¬Disj(•, •)’. We also examine in detail how to test for satisfiability the formulae of six sample fragments: three sample problems are shown to be NP-complete, two to admit quadratic-time decision algorithms, and one to be solvable in linear time.

On the Convexity of a Fragment of Pure Set Theory with Applications within a Nelson-Oppen Framework

“…In addition, for the purposes of simplifying some proofs, we can further restrict ourselves to MLSconjunctions involving a minimal number of literal types. As shown in [8], all the atoms in (1) and their negations can be rewritten in terms of atoms of type x ∈ y and x = y \ z only by repeatedly applying the following equivalences much as rewrite rules (the existential quantifiers are then just dropped while the quantified variables are replaced by fresh ones):…”

“…Thus, if V k = / 0, by picking any v k ∈ V k and by repeatedly applying (8), it follows that there exist…”

“…Convexity of MLS is plainly inherited by all of its fragments. In [8] and [10], we recently investigated them with the goal of spotting the polynomial ones, namely the fragments of MLS endowed with polynomial-time satisfiability tests. Specifically, we examined all the sublanguages of the theories BST := BST(∪, ∩, \, =∅, =∅, Disj, ¬Disj, ⊆, ⊆, =, =) and MST := MST(∪, ∩, \, ∈, / ∈),…”

“…In this paper, we start an investigation for combining decidable fragments of pure Zermelo-Fraenkel set theory (in which sets are recursively built up from other sets) with other theories within the Nelson-Oppen framework. More specifically, our main result is that the theory Multi-Level Syllogistic (the basic language of computable set theory-MLS for short) is convex and therefore its decision procedure (and those of its several polynomial fragments [8,10]) can be efficiently combined with the decision procedures of other basic decidable theories, such as for instance the theory of lists and the theory of linear rational arithmetic, since set theory is plainly stably infinite.…”