On the Computational Completeness of Equations over Sets of Natural Numbers

Systems of equations of the form φj(X1, . . . ,Xn) = ψj(X1, . . . ,Xn) with 1 6 j 6 m are considered, in which the unknowns Xi are sets of natural numbers, while the expressions φj , ψj may contain singleton constants and the operations of union and pairwise addition S + T = {m + n | m ∈ S, n ∈ T}. It is shown that the family of sets representable by unique (least, greatest) solutions of such systems is exactly the family of recursive (r.e., co-r.e., respectively) sets of numbers. Basic decision problems for these systems are located in the arithmetical hierarchy. The same results are established for equations with addition and intersection.