Abstract
Equations of the form X=φ(X) are considered, where the unknown X is a set of natural numbers. The expression φ(X) may contain the operations of set addition, defined as S+T= {m+n ∣ m ∈ S, n ∈ T}, union and intersection, as well as ultimately periodic constants. An equation with a non-periodic solution of exponential growth is constructed. At the same time it is demonstrated that no sets with super-exponential growth can be represented. It is also shown that a restricted class of these equations cannot represent sets with super-linearly growing complements. The results have direct implications on the power of conjunctive grammars with one nonterminal symbol.
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© 2008 IFIP International Federation for Information Processing
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Okhotin, A., Rondogiannis, P. (2008). On the expressive power of univariate equations over sets of natural numbers. In: Ausiello, G., Karhumäki, J., Mauri, G., Ong, L. (eds) Fifth Ifip International Conference On Theoretical Computer Science – Tcs 2008. IFIP International Federation for Information Processing, vol 273. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-09680-3_15
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DOI: https://doi.org/10.1007/978-0-387-09680-3_15
Publisher Name: Springer, Boston, MA
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