Notre Dame Journal of Formal Logic

Classifications of Computable Structures

Karen Lange, Russell Miller, and Rebecca M. Steiner

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Let K be a family of structures, closed under isomorphism, in a fixed computable language. We consider effective lists of structures from K such that every structure in K is isomorphic to exactly one structure on the list. Such a list is called a computable classification of 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 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

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

Received: 27 June 2014
Accepted: 31 March 2015
First available in Project Euclid: 30 June 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03C57: Effective and recursion-theoretic model theory [See also 03D45]
Secondary: 03D45: Theory of numerations, effectively presented structures [See also 03C57; for intuitionistic and similar approaches see 03F55]

Friedberg enumeration computable structure theory equivalence structure effective classification


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.

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