On the Computational Completeness of Equations over Sets of Natural Numbers

Systems of equations of the form φ j (X 1 , ..., X n ) = i¾? j (X 1 , ..., X n ) with $1 \leqslant j \leqslant m$ are considered, in which the unknowns X i are sets of natural numbers, while the expressions φ j ,i¾? j may contain singleton constants and the operations of union (possibly replaced by intersection) and pairwise addition . 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.

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