Complexity assessments for decidable fragments of Set Theory. IV: A quadratic reduction of constraints over nested sets to Boolean formulae

TL;DR

This paper presents a quadratic-time translation of certain set-theoretic constraints into Boolean formulae, enabling efficient satisfiability checking while preserving semantics, thus bridging set theory and Boolean logic.

Contribution

It introduces a quadratic reduction method translating specific set constraints into Boolean formulae, facilitating analysis within NP-complete frameworks.

Findings

Translation preserves satisfiability.

Algorithm operates in quadratic time.

Bridges set theory constraints with Boolean logic.

Abstract

As a contribution to quantitative set-theoretic inferencing, a translation is proposed of conjunctions of literals of the forms $x=y\setminus z$, $x \neq y\setminus z$, and $z =\{x\}$, where $x,y,z$ stand for variables ranging over the von Neumann universe of sets, into unquantified Boolean formulae of a rather simple conjunctive normal form. The formulae in the target language involve variables ranging over a Boolean ring of sets, along with a difference operator and relators designating equality, non-disjointness and inclusion. Moreover, the result of each translation is a conjunction of literals of the forms $x=y\setminus z$, $x\neq y\setminus z$ and of implications whose antecedents are isolated literals and whose consequents are either inclusions (strict or non-strict) between variables, or equalities between variables. Besides reflecting a simple and natural semantics, which ensures satisfiability-preservation, the proposed translation has quadratic algorithmic time-complexity, and bridges two languages both of which are known to have an NP-complete satisfiability problem.