## Notre Dame Journal of Formal Logic

### Classifications of Computable Structures

#### Abstract

Let $\mathcal{K}$ be a family of structures, closed under isomorphism, in a fixed computable language. We consider effective lists of structures from $\mathcal{K}$ such that every structure in $\mathcal{K}$ is isomorphic to exactly one structure on the list. Such a list is called a computable classification of $\mathcal{K}$, up to isomorphism. Using the technique of Friedberg enumeration, we show that there is a computable classification of the family of computable algebraic fields and that with a $\mathbf{0'}$-oracle, we can obtain similar classifications of the families of computable equivalence structures and of computable finite-branching trees. However, there is no computable classification of the latter, nor of the family of computable torsion-free abelian groups of rank $1$, even though these families are both closely allied with computable algebraic fields.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 59, Number 1 (2018), 35-59.

Dates
Accepted: 31 March 2015
First available in Project Euclid: 30 June 2017

https://projecteuclid.org/euclid.ndjfl/1498788255

Digital Object Identifier
doi:10.1215/00294527-2017-0015

Mathematical Reviews number (MathSciNet)
MR3744350

Zentralblatt MATH identifier
06848190

#### Citation

Lange, Karen; Miller, Russell; Steiner, Rebecca M. Classifications of Computable Structures. Notre Dame J. Formal Logic 59 (2018), no. 1, 35--59. doi:10.1215/00294527-2017-0015. https://projecteuclid.org/euclid.ndjfl/1498788255

#### References

• [1] Ash, C., and J. F. Knight, Computable Structures and the Hyperarithmetical Hierarchy, Amsterdam, North-Holland, 2000.
• [2] Calvert, W., “Algebraic structure and computable structure,” Ph.D. dissertation, University of Notre Dame, Notre Dame, Indiana, 2005.
• [3] Calvert, W., D. Cenzer, V. Harizanov, and A. Morozov, “Effective categoricity of equivalence structures,” Annals of Pure and Applied Logic, vol. 141 (2006), pp. 306–25.
• [4] Calvert, W., V. Harizanov, J. F. Knight, and S. Miller, “Index sets for computable structures,” Algebra and Logic, vol. 45 (2006), pp. 61–78.
• [5] Calvert, W., and J. F. Knight, “Classification from a computable viewpoint,” Bulletin of Symbolic Logic, 12 (2006), pp. 191–218.
• [6] Coles, R. J., R. G. Downey, and T. A. Slaman, “Every set has a least jump enumeration,” Journal of the London Mathematical Society, vol. 62 (2000), pp. 641–49.
• [7] Ershov, Yu. L., “Theorie der Numerierungen,” Zeitschrift für Mathematische Logik und Grundlagen Mathematik, vol. 23 (1977), pp. 289–371.
• [8] Friedberg, R. M., “Three theorems on recursive enumeration, I, Decomposition, II, Maximal set, III, Enumeration without duplication,” Journal of Symbolic Logic, vol. 23 (1958), pp. 309–16.
• [9] Frolov, A., I. Kalimullin, and R. Miller, “Spectra of algebraic fields and subfields,” pp. 232–41, in Mathematical Theory and Computational Practice: Fifth Conference on Computability in Europe, CiE 2009, edited by K. Ambos-Spies, B. Löwe, and W. Merkle, vol. 5635 of Lecture Notes in Computer Science, Berlin, Springer, 2009. (For Appendix A, see qcpages.qc.cuny.edu/~rmiller/research.html).
• [10] Goncharov, S. S., and J. F. Knight, “Computable structure and non-structure theorems,” Algebra and Logic, vol. 41 (2002), pp. 351–73.
• [11] Goncharov, S. S., S. Lempp, and D. R. Solomon, “Friedberg numberings of families of of $n$-computably enumerable sets,” Algebra and Logic vol. 41 (2002), pp. 81–6.
• [12] Hirschfeldt, D. R., B. Khoussainov, R. A. Shore, and A. M. Slinko, “Degree spectra and computable dimensions in algebraic structures,” Annals of Pure and Applied Logic, vol. 115 (2002), pp. 71–113.
• [13] Kronecker, L., Grundzüge einer arithmetischen Theorie der algebraischen Größen, Journal für die Reine und Angewandte Mathematik, vol. 92 (1882), pp. 1–122.
• [14] Miller, R., “$\mathbf{d}$-Computable categoricity for algebraic fields,” Journal of Symbolic Logic, vol. 74 (2009), pp. 1325–51.
• [15] Miller, R., and A. Shlapentokh, “Computable categoricity for algebraic fields with splitting algorithms,” Transactions of the American Mathematical Society, vol. 367 (2015), pp. 3955–80.
• [16] Rabin, M., “Computable algebra, general theory, and theory of computable fields,” Transactions of the American Mathematical Society, vol. 95 (1960), pp. 341–60.
• [17] Soare, R. I., Recursively Enumerable Sets and Degrees, Berlin, Springer, 1987.
• [18] Steiner, R. M., “Effective algebraicity,” Archive for Mathematical Logic, vol. 52 (2013), pp. 91–112.