On the computational completeness of equations over sets of natural numbers

Abstract

Systems of equations of the form φj(X1,...,X n)=ψj(X1,...,Xn) with 1 ≤ j ≤ 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 (possibly replaced by intersection) 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. © 2008 Springer-Verlag.