A Quadratic Reduction of Constraints over Nested Sets to Purely Boolean Formulae in CNF

A translation is proposed of conjunctions of literals of the forms x = y z, x /= y z, and x ∈ y, 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 field 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 z, x /= y 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.