Computational completeness of equations over sets of natural numbers

Systems of finitely many equations of the form @f(X"1,...,X"n)[email protected](X"1,...,X"n) are considered, in which the unknowns X"i are sets of natural numbers, while the expressions @f,@j may contain singleton constants and the operations of union and pairwise addition S+T={m+n|[email protected]?S,[email protected]?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.

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