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.
@InProceedings{jez_et_al:LIPIcs.STACS.2009.1806, author = {Jez, Artur and Okhotin, Alexander}, 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 = {Albers, Susanne and Marion, Jean-Yves}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1806}, URN = {urn:nbn:de:0030-drops-18061}, doi = {10.4230/LIPIcs.STACS.2009.1806}, annote = {Keywords: } }
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