17 found
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  1.  18
    Computably Compact Metric Spaces.Rodney G. Downey & Alexander G. Melnikov - 2023 - Bulletin of Symbolic Logic 29 (2):170-263.
    We give a systematic technical exposition of the foundations of the theory of computably compact metric spaces. We discover several new characterizations of computable compactness and apply these characterizations to prove new results in computable analysis and effective topology. We also apply the technique of computable compactness to give new and less combinatorially involved proofs of known results from the literature. Some of these results do not have computable compactness or compact spaces in their statements, and thus these applications are (...)
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  2.  4
    Computable Abelian groups.Alexander G. Melnikov - 2014 - Bulletin of Symbolic Logic 20 (3):315-356,.
    We provide an introduction to methods and recent results on infinitely generated abelian groups with decidable word problem.
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  3.  4
    Computably Isometric Spaces.Alexander G. Melnikov - 2013 - Journal of Symbolic Logic 78 (4):1055-1085.
  4.  6
    On Δ 2 0 -categoricity of equivalence relations.Rod Downey, Alexander G. Melnikov & Keng Meng Ng - 2015 - Annals of Pure and Applied Logic 166 (9):851-880.
  5.  11
    Abelian p-groups and the Halting problem.Rodney Downey, Alexander G. Melnikov & Keng Meng Ng - 2016 - Annals of Pure and Applied Logic 167 (11):1123-1138.
  6.  11
    A Friedberg enumeration of equivalence structures.Rodney G. Downey, Alexander G. Melnikov & Keng Meng Ng - 2017 - Journal of Mathematical Logic 17 (2):1750008.
    We solve a problem posed by Goncharov and Knight 639–681, 757]). More specifically, we produce an effective Friedberg enumeration of computable equivalence structures, up to isomorphism. We also prove that there exists an effective Friedberg enumeration of all isomorphism types of infinite computable equivalence structures.
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  7.  12
    Jump degrees of torsion-free abelian groups.Brooke M. Andersen, Asher M. Kach, Alexander G. Melnikov & Reed Solomon - 2012 - Journal of Symbolic Logic 77 (4):1067-1100.
    We show, for each computable ordinal α and degree $\alpha > {0^{\left( \alpha \right)}}$, the existence of a torsion-free abelian group with proper α th jump degree α.
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  8.  5
    On the complexity of classifying lebesgue spaces.Tyler A. Brown, Timothy H. Mcnicholl & Alexander G. Melnikov - 2020 - Journal of Symbolic Logic 85 (3):1254-1288.
    Computability theory is used to evaluate the complexity of classifying various kinds of Lebesgue spaces and associated isometric isomorphism problems.
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  9.  7
    Uniform procedures in uncountable structures.Noam Greenberg, Alexander G. Melnikov, Julia F. Knight & Daniel Turetsky - 2018 - Journal of Symbolic Logic 83 (2):529-550.
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  10.  18
    Decidability and Computability of Certain Torsion-Free Abelian Groups.Rodney G. Downey, Sergei S. Goncharov, Asher M. Kach, Julia F. Knight, Oleg V. Kudinov, Alexander G. Melnikov & Daniel Turetsky - 2010 - Notre Dame Journal of Formal Logic 51 (1):85-96.
    We study completely decomposable torsion-free abelian groups of the form $\mathcal{G}_S := \oplus_{n \in S} \mathbb{Q}_{p_n}$ for sets $S \subseteq \omega$. We show that $\mathcal{G}_S$has a decidable copy if and only if S is $\Sigma^0_2$and has a computable copy if and only if S is $\Sigma^0_3$.
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  11.  9
    New Degree Spectra of Abelian Groups.Alexander G. Melnikov - 2017 - Notre Dame Journal of Formal Logic 58 (4):507-525.
    We show that for every computable ordinal of the form β=δ+2n+1>1, where δ is zero or a limit ordinal and n∈ω, there exists a torsion-free abelian group having an X-computable copy if and only if X is nonlowβ.
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  12.  10
    Computable Topological Groups.K. O. H. Heer Tern, Alexander G. Melnikov & N. G. Keng Meng - forthcoming - Journal of Symbolic Logic:1-33.
    We investigate what it means for a (Hausdorff, second-countable) topological group to be computable. We compare several potential definitions based on classical notions in the literature. We relate these notions with the well-established definitions of effective presentability for discrete and profinite groups, and compare our results with similar results in computable topology.
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  13. Lempp s question for torsion free abelian groups of finite rank.Alexander G. Melnikov - 2007 - Bulletin of Symbolic Logic 13 (2):208.
  14.  11
    Torsion-free abelian groups with optimal Scott families.Alexander G. Melnikov - 2018 - Journal of Mathematical Logic 18 (1):1850002.
    We prove that for any computable successor ordinal of the form α = δ + 2k there exists computable torsion-free abelian group that is relatively Δα0 -categorical and not Δα−10 -categorical. Equivalently, for any such α there exists a computable TFAG whose initial segments are uniformly described by Σαc infinitary computable formulae up to automorphism, and there is no syntactically simpler family of formulae that would capture these orbits. As far as we know, the problem of finding such optimal examples (...)
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  15.  4
    Turing reducibility in the fine hierarchy.Alexander G. Melnikov, Victor L. Selivanov & Mars M. Yamaleev - 2020 - Annals of Pure and Applied Logic 171 (7):102766.
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  16.  3
    Reviewed Work(s): A new spectrum of recursive models using an amalgamation construction. The Journal of Symbolic Logic, vol. 73 by Uri Andrews; A computable N₀-categorical structure whose theory computes true arithmetic. The Journal of Symbolic Logic, vol. 72 by Bakhadyr Khoussainov; Antonio Montalbán. [REVIEW]Alexander G. Melnikov - forthcoming - Association for Symbolic Logic: The Bulletin of Symbolic Logic.
    Review by: Alexander G. Melnikov The Bulletin of Symbolic Logic, Volume 19, Issue 3, Page 400-401, September 2013.
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  17.  7
    Uri Andrews. A new spectrum of recursive models using an amalgamation construction. The Journal of Symbolic Logic, vol. 73 (2011), no. 3, pp. 883–896. - Bakhadyr Khoussainov and Antonio Montalbán. A computable ℵ 0 -categorical structure whose theory computes true arithmetic. The Journal of Symbolic Logic, vol. 72 (2010), no. 2, pp. 728–740. [REVIEW]Alexander G. Melnikov - 2013 - Bulletin of Symbolic Logic 19 (3):400-401.
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