I gave a talk at the CUNY MAMLS conference March 9-10, 2012 at the City University of New York.

This talk will be about a generalization of the concept of Turing degrees to the hierarchy of equivalence relations on $\mathbb{N}$ under computable reducibility. The idea is to develop a computable analogue of the enormously successful theory of equivalence relations on $\mathbb{R}$ under Borel reducibility, a theory which has led to deep insights on the complexity hierarchy of classification problems arising throughout mathematics. In the computable analogue, we consider the corresponding reduction notion in the context of Turing computability for relations on $\mathbb{N}$. Specifically, one relation $E$ is computably reducible to another, $F$, if there is a computable function $f$ such that $x\mathrel{E} y$ if and only if $f(x)\mathrel{F} f(y)$. This is a very different concept from mere Turing reducibility of $E$ to $F$, for it sheds light on the comparative difficulty of the classification problems corresponding to $E$ and $F$, rather than on the difficulty of computing the relations themselves. In particular, the theory appears well suited for an analysis of equivalence relations on classes of c.e. structures, a rich context with many natural examples, such as the isomorphism relation on c.e. graphs or on computably presented groups. In this regard, our exposition extends earlier work in the literature concerning the classification of computable structures. An abundance of open questions remain. This is joint work with Sam Coskey and Russell Miller.

article | slides | abstract on conference web page | related talk Florida MAMLS 2012 | Sam’s post on this topic