The Hierarchy of Equivalence Relations on the Natural Numbers Under Computable Reducibility

The notion of computable reducibility between equivalence relations on the natural numbers provides a natural computable analogue of Borel reducibility. We investigate the computable reducibility hierarchy, comparing and contrasting it with the Borel reducibility hierarchy from descriptive set theory. Meanwhile, the notion of computable reducibility appears well suited for an analysis of equivalence relations on the c.e. sets, and more specifically, on various classes of c.e. structures. This is a rich context with many natural examples, such as the isomorphism relation on c.e. graphs or on computably presented groups. Here, our exposition extends earlier work in the literature concerning the classification of computable structures. An abundance of open questions remains.

[1]  W. Calvert,et al.  Comparing Classes of Finite Structures , 2004, 0803.3291.

[2]  Sy-David Friedman,et al.  Equivalence Relations on Classes of Computable Structures , 2009, CiE.

[3]  Andrea Sorbi,et al.  Universal computably Enumerable Equivalence Relations , 2014, J. Symb. Log..

[4]  Julia F. Knight,et al.  Turing computable embeddings , 2007, J. Symb. Log..

[5]  Alan H. Mekler.  Stability of Nilpotent Groups of Class 2 and Prime Exponent , 1981, J. Symb. Log..

[6]  R. Soare.  Recursively enumerable sets and degrees , 1987 .

[7]  Andrea Sorbi,et al.  Classifying Positive Equivalence Relations , 1983, J. Symb. Log..

[8]  Julia F. Knight,et al.  Classification from a Computable Viewpoint , 2006, Bulletin of Symbolic Logic.

[9]  Su Gao.  Invariant Descriptive Set Theory , 2008 .

[10]  V. Harizanov,et al.  Isomorphism and Bi-Embeddability Relations on Computable Structures ∗ , 2010 .

[11]  Sy-David Friedman,et al.  On Σ11 equivalence relations over the natural numbers , 2012, Math. Log. Q..

[12]  Yijia Chen,et al.  Strong isomorphism reductions in complexity theory , 2011, The Journal of Symbolic Logic.

[13]  Robert I. Soare,et al.  Recursively enumerable sets and degrees - a study of computable functions and computability generated sets , 1987, Perspectives in mathematical logic.

[14]  Sy-David Friedman,et al.  The effective theory of Borel equivalence relations , 2009, Ann. Pure Appl. Log..

[15]  Su Gao,et al.  Computably Enumerable Equivalence Relations , 2001, Stud Logica.

[16]  Harvey M. Friedman,et al.  A Borel reductibility theory for classes of countable structures , 1989, Journal of Symbolic Logic.