The Friedman–Stanley jump, extensively studied by descriptive set theorists, is a fundamental tool for gauging the complexity of Borel isomorphism relations. This paper focuses on a natural computable analog of this jump operator for equivalence relations on $\omega $, written ${\dotplus }$, recently introduced by Clemens, Coskey, and Krakoff. We offer a thorough analysis of the computable Friedman–Stanley jump and its connections with the hierarchy of countable equivalence relations under the computable reducibility $\leq _c$. In particular, we show that this jump gives benchmark equivalence relations going up the hyperarithmetic hierarchy and we unveil the complicated highness hierarchy that arises from ${\dotplus }$.

We introduce the computable FS-jump, an analog of the classical Friedman–Stanley jump in the context of equivalence relations on the natural numbers. We prove that the computable FS-jump is proper with respect to computable reducibility. We then study the effect of the computable FS-jump on computably enumerable equivalence relations (ceers).

We examine the degree structure $\operatorname {\mathrm {\mathbf {ER}}}$ of equivalence relations on $\omega $ under computable reducibility. We examine when pairs of degrees have a least upper bound. In particular, we show that sufficiently incomparable pairs of degrees do not have a least upper bound but that some incomparable degrees do, and we characterize the degrees which have a least upper bound with every finite equivalence relation. We show that the natural classes of finite, light, and dark degrees are definable in $\operatorname {\mathrm {\mathbf {ER}}}$. We show that every equivalence relation has continuum many self-full strong minimal covers, and that $\mathbf {d}\oplus \mathbf {\operatorname {\mathrm {\mathbf {Id}}}_1}$ needn’t be a strong minimal cover of a self-full degree $\mathbf {d}$. Finally, we show that the theory of the degree structure $\operatorname {\mathrm {\mathbf {ER}}}$ as well as the theories of the substructures of light degrees and of dark degrees are each computably isomorphic with second-order arithmetic.

This paper is a contribution to the growing investigation of strong reducibilities between ${\rm{\Pi }}_2^1$ statements of second-order arithmetic, viewed as an extension of the traditional analysis of reverse mathematics. We answer several questions of Hirschfeldt and Jockusch [13] about Weihrauch (uniform) and strong computable reductions between various combinatorial principles related to Ramsey’s theorem for pairs. Among other results, we establish that the principle $SRT_2^2$ is not Weihrauch or strongly computably reducible to $D_{ < \infty }^2$, and that COH is not Weihrauch reducible to $SRT_{ < \infty }^2$, or strongly computably reducible to $SRT_2^2$. The last result also extends a prior result of Dzhafarov [9]. We introduce a number of new techniques for controlling the combinatorial and computability-theoretic properties of the problems and solutions we construct in our arguments.

We introduce the notion of finitary computable reducibility on equivalence relations on the domain ω. This is a weakening of the usual notion of computable reducibility, and we show it to be distinct in several ways. In particular, whereas no equivalence relation can be ${\rm{\Pi }}_{n + 2}^0$ -complete under computable reducibility, we show that, for every n, there does exist a natural equivalence relation which is ${\rm{\Pi }}_{n + 2}^0$ -complete under finitary reducibility. We also show that our hierarchy of finitary reducibilities does not collapse, and illustrate how it sharpens certain known results. Along the way, we present several new results which use computable reducibility to establish the complexity of various naturally defined equivalence relations in the arithmetical hierarchy.

We study the relative complexity of equivalence relations and preorders from computability theory and complexity theory. Given binary relations R, S, a componentwise reducibility is defined by

R ≤ S ⇔ ∃f ∀x, y [x R y ↔ f (x) S f (y)].

Here, f is taken from a suitable class of effective functions. For us the relations will be on natural numbers, and f must be computable. We show that there is a ${\rm{\Pi }}_1^0$-complete equivalence relation, but no ${\rm{\Pi }}_k^0$-complete for k ≥ 2. We show that ${\rm{\Sigma }}_k^0$ preorders arising naturally in the above-mentioned areas are ${\rm{\Sigma }}_k^0$-complete. This includes polynomial time m-reducibility on exponential time sets, which is ${\rm{\Sigma }}_2^0$, almost inclusion on r.e. sets, which is ${\rm{\Sigma }}_3^0$, and Turing reducibility on r.e. sets, which is ${\rm{\Sigma }}_4^0$.