Abstract
We adjust methods of computable model theory to effective analysis. We use index sets and infinitary logic to obtain classification-type results for compact computable metric spaces. We show that every compact computable metric space can be uniquely described, up to isometry, by a computable Π3 formula, and that orbits of elements are uniformly given by computable Π2 formulas. We show that deciding if two compact computable metric spaces are isometric is a \(\Pi^0_2\) complete problem within the class of compact computable spaces, which in itself is \(\Pi^0_3\). On the other hand, if there is an isometry, then ∅ ′′ can compute one. In fact, there is a set low relative to ∅ ′ which can compute an isometry. We show that the result can not be improved to ∅ ′. We also give further results for special classes of compact spaces, and for other related classes of Polish spaces.
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Melnikov, A.G., Nies, A. (2013). The Classification Problem for Compact Computable Metric Spaces. In: Bonizzoni, P., Brattka, V., Löwe, B. (eds) The Nature of Computation. Logic, Algorithms, Applications. CiE 2013. Lecture Notes in Computer Science, vol 7921. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39053-1_37
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DOI: https://doi.org/10.1007/978-3-642-39053-1_37
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