Computational completeness of equations over sets of natural numbers ?
Systems of finitely many equations of the form ?(X1,,Xn)=?(X1,,Xn) ? ( X 1 ,  , X n ) = ? ( X 1 ,  , X n ) are considered, in which the unknowns Xi X i are sets of natural numbers, while the expressions ?,? ? , ? 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.
computability language equations unary languages