Jez, Artur ;
Okhotin, Alexander
Equations over Sets of Natural Numbers with Addition Only
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},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2009/1806},
URN = {urn:nbn:de:0030-drops-18061},
doi = {http://dx.doi.org/10.4230/LIPIcs.STACS.2009.1806},
annote = {Keywords: }
}
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26th International Symposium on Theoretical Aspects of Computer Science
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2009 |
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2009 |
2009
