Abstract
If \(\mathcal{L}\) is a finite relational language then all computable \(\mathcal{L}\)-structures can be effectively enumerated in a sequence \(\mathcal{L}\)-structure \(\mathcal{B}\) an index n of its isomorphic copy \(\mathcal{L}\)-structures closed under isomorphism we denote by K c the set of all computable members of K. We measure the complexity of a description of K c or of an equivalence relation on K c via the complexity of the corresponding sets of indices. If the index set of K c is hyperarithmetical then (the index sets of) such natural equivalence relations as the isomorphism or bi-embeddability relation are \(\Sigma^1_1\). In the present paper we study the status of these \(\Sigma^1_1\) equivalence relations (on classes of computable structures with hyperarithmetical index set) within the class of \(\Sigma^1_1\) equivalence relations as a whole, using a natural notion of hyperarithmetic reducibility.
This work was partially supported by FWF Grant number P 19375 - N18.
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Fokina, E.B., Friedman, SD. (2009). Equivalence Relations on Classes of Computable Structures. In: Ambos-Spies, K., Löwe, B., Merkle, W. (eds) Mathematical Theory and Computational Practice. CiE 2009. Lecture Notes in Computer Science, vol 5635. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03073-4_21
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DOI: https://doi.org/10.1007/978-3-642-03073-4_21
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