Equality sets for recursively enumerable languages
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 39 (2005) no. 4, pp. 661-675.

We consider shifted equality sets of the form ${E}_{G}\left(a,{g}_{1},{g}_{2}\right)=\left\{w\mid {g}_{1}\left(w\right)=a{g}_{2}\left(w\right)\right\}$, where ${g}_{1}$ and ${g}_{2}$ are nonerasing morphisms and $a$ is a letter. We are interested in the family consisting of the languages $h\left({E}_{G}\left(J\right)\right)$, where $h$ is a coding and ${E}_{G}\left(J\right)$ is a shifted equality set. We prove several closure properties for this family. Moreover, we show that every recursively enumerable language $L\subseteq {A}^{*}$ is a projection of a shifted equality set, that is, $L={\pi }_{A}\left({E}_{G}\left(a,{g}_{1},{g}_{2}\right)\right)$ for some (nonerasing) morphisms ${g}_{1}$ and ${g}_{2}$ and a letter $a$, where ${\pi }_{A}$ deletes the letters not in $A$. Then we deduce that recursively enumerable star languages coincide with the projections of equality sets.

DOI: 10.1051/ita:2005035
Classification: 03D25,  68Q45
Keywords: morphism, equality set, shifted post correspondence problem, closure properties, recursively enumerable sets
Halava, Vesa ; Harju, Tero ; Hoogeboom, Hendrik Jan 1; Latteux, Michel 2

1 Department of Computer Science, Leiden University PO Box 9512, 2300 RA Leiden, The Netherlands;
2 Université des Sciences et Technologies de Lille, Bâtiment M3, 59655 Villeneuve d’Ascq Cedex, France
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Halava, Vesa; Harju, Tero; Hoogeboom, Hendrik Jan; Latteux, Michel. Equality sets for recursively enumerable languages. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 39 (2005) no. 4, pp. 661-675. doi : 10.1051/ita:2005035. http://www.numdam.org/articles/10.1051/ita:2005035/

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