Abstract
We study the algorithmic complexity of embeddings between bi-embeddable equivalence structures. We define the notions of computable bi-embeddable categoricity, (relative) \(\Delta ^0_\alpha \) bi-embeddable categoricity, and degrees of bi-embeddable categoricity. These notions mirror the classical notions used to study the complexity of isomorphisms between structures. We show that the notions of \(\Delta ^0_\alpha \) bi-embeddable categoricity and relative \(\Delta ^0_\alpha \) bi-embeddable categoricity coincide for equivalence structures for \(\alpha =1,2,3\). We also prove that computable equivalence structures have degree of bi-embeddable categoricity \(\mathbf {0},\mathbf {0}'\), or \(\mathbf {0}''\). We furthermore obtain results on the index set complexity of computable equivalence structure with respect to bi-embeddability.
References
Fröhlich, A., Shepherdson, J.C.: Effective procedures in field theory. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 248(950), 407–432 (1956)
Maltsev, A.I.: On recursive Abelian groups. In: Soviet Mathematics, vol. 3. Doklady (1962)
Fokina, E., Harizanov, V., Melnikov, A.G.: Computable model theory. Turing’s legacy: developments from Turing’s ideas in logic 42, 124–191 (2014)
Montalbán, A.: Up to equimorphism, hyperarithmetic is recursive. J. Symb. Log. 70, 360–378 (2005)
Montalbán, A.: On the equimorphism types of linear orderings. Bull. Symb. Log. 13, 71–99 (2007)
Greenberg, N., Montalbán, A.: Ranked structures and arithmetic transfinite recursion. Trans. Am. Math. Soc. 360(3), 1265–1307 (2008)
Fokina, E., Rossegger, D., Mauro, L.S.: Bi-embeddability spectra and bases of spectra. arXiv preprint arXiv:1808.05451 (2018)
Robert, I.S.: Turing Computability. Theory and Applications of Computability. Springer, Berlin Heidelberg (2016)
Calvert, W., Cenzer, D., Harizanov, V., Morozov, A.: Effective categoricity of equivalence structures. Ann. Pure Appl. Log. 141, 61–78 (2006)
Fokina, E., Kalimullin, I.S., Miller, R.: Degrees of categoricity of computable structures. Arch. Math. Log. 49(1), 51–67 (2010)
Russell, M.: D-computable categoricity for algebraic fields. J. Symb. Log. 74(12), 1325–1351 (2009)
Fokina, E., Frolov, A., Kalimullin, I.S.: Categoricity spectra for rigid structures. Notre Dame J. Form. Log. 57(1), 45–57 (2016)
Csima, B.F., Franklin, J.N.Y., Shore, R.A.: Degrees of categoricity and the hyperarithmetic hierarchy. Notre Dame J. Form. Log. 54(2), 215–231 (2013)
Csima, B.F., Stephenson, J.: Finite computable dimension and degrees of categoricity. Annals of Pure and Applied Logic (2018)
Bazhenov, N.A., Kalimullin, I.S., Yamaleev, M.M.: Degrees of categoricity vs. strong degrees of categoricity. Algebra Log. 55(2), 173–177 (2016)
Bazhenov, N.A., Kalimullin, I.S., Yamaleev, M.M.: Degrees of categoricity and spectral dimension. J. Symb. Log. 83(1), 103–116 (2018)
Kach, A.M., Turetsky, D.: \({\Delta ^{0}}_2\)-categoricity of equivalence structures. N. Z. J. Math. 39, 143–149 (2009)
Downey, R.G., Melnikov, A.G., Meng Ng, K.: On \({\Delta ^{0}}_2\)-categoricity of equivalence relations. Ann. Pure Appl. Log. 166(9), 851–880 (2015)
Rodney, G.D., Asher, M.K., Turetsky, D.: Limitwise monotonic functions and their applications. In: Proceedings Of The 11Th Asian Logic Conference, pp. 59–85 (2011)
Acknowledgements
Open access funding provided by Austrian Science Fund (FWF). The first author was supported by the Russian Foundation for Basic Research, according to the research Project No. 16-31-60058 mol_a_dk. The second, third and fourth author were supported by the Austrian Science Fund FWF through project P 27527.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Bazhenov, N., Fokina, E., Rossegger, D. et al. Degrees of bi-embeddable categoricity of equivalence structures. Arch. Math. Logic 58, 543–563 (2019). https://doi.org/10.1007/s00153-018-0650-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-018-0650-3
Keywords
- Computable categoricity
- Bi-embeddability
- Degrees of categoricity
- Degrees of bi-embeddable categoricity
Mathematics Subject Classification
- 03C57