On the expressive power of univariate equations over sets of natural numbers

Equations of the form [email protected](X) are considered, where the unknown X is a set of natural numbers. The expression @f(X) may contain the operations of set addition, defined as S+T={m+n|[email protected]?S,[email protected]?T}, union, intersection, as well as ultimately periodic constants. An equation with a non-periodic solution of exponential growth rate is constructed. At the same time it is demonstrated that no sets with super-exponential growth rate can be represented. It is also shown that restricted classes of these equations cannot represent sets with super-linearly growing complements nor sets that are additive bases of order 2. The results have direct implications on the power of unary conjunctive grammars with one nonterminal symbol.

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