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

As a contribution to quantitative set-theoretic inferencing, a translation is proposed of conjunctions of literals of the forms x=y∖ z, x ≠ y∖ 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∖ 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.

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