when quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2009.1806
URN: urn:nbn:de:0030-drops-18061
URL: http://drops.dagstuhl.de/opus/volltexte/2009/1806/

Jez, Artur ; Okhotin, Alexander

### Equations over Sets of Natural Numbers with Addition Only

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### Abstract

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.

### BibTeX - Entry

@InProceedings{jez_et_al:LIPIcs:2009:1806,
author =	{Artur Jez and Alexander Okhotin},
title =	{{Equations over Sets of Natural Numbers with Addition Only}},
booktitle =	{26th International Symposium on Theoretical Aspects of Computer Science},
pages =	{577--588},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-09-5},
ISSN =	{1868-8969},
year =	{2009},
volume =	{3},
editor =	{Susanne Albers and Jean-Yves Marion},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},