Geodesic
On $\Sigma^0_1$-Classification of Relations on Computable Structures
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 8 (2008) no. 4, pp. 23-32
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In the paper the example of computable structure in the language with infinite number of binary predicate symbols without $\Sigma^0_1$-classification of all unary relations, which are definable by $\Sigma^0_1$-formulas on the structure, is constructed. 
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S. S. Goncharov; N. T. Kogabaev. On $\Sigma^0_1$-Classification of Relations on Computable Structures. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 8 (2008) no. 4, pp. 23-32. http://geodesic.mathdoc.fr/item/VNGU_2008_8_4_a2/

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