Equations over Sets of Natural Numbers with Addition Only

Authors Artur Jez, Alexander Okhotin

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Artur Jez
Alexander Okhotin

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Artur Jez and Alexander Okhotin. Equations over Sets of Natural Numbers with Addition Only. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 577-588, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)


Systems of equations of the form $X=YZ$ and $X=C$ are considered, in which the unknowns are sets of natural numbers, ``$+$'' denotes pairwise sum of sets $S+T=\ensuremath{ \{ m+n \: | \: m \in S, \; n \in T \} }$, and $C$ is an ultimately periodic constant. It is shown that such systems are computationally universal, in the sense that for every recursive (r.e., co-r.e.) set $S \subseteq \mathbb{N}$ there exists a system with a unique (least, greatest) solution containing a component $T$ with $S=\ensuremath{ \{ n \: | \: 16n+13 \in T \} }$. This implies undecidability of basic properties of these equations. All results also apply to language equations over a one-letter alphabet with concatenation and regular constants.


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