Complexity of equivalence relations and preorders from computability theory

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\le S \iff \ex f \, \forall x, y \, [xRy \lra f(x) Sf(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 $\Pi_1$-complete equivalence relation, but no $\Pi k$-complete for $k \ge 2$. We show that $\Sigma k$ preorders arising naturally in the above-mentioned areas are $\Sigma k$-complete. This includes polynomial time $m$-reducibility on exponential time sets, which is $\Sigma 2$, almost inclusion on r.e.\ sets, which is $\Sigma 3$, and Turing reducibility on r.e.\ sets, which is $\Sigma 4$.Comment: To appear in J. Symb. Logi

The Complexity of Orbits of Computably Enumerable Sets

The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, \E, such that the question of membership in this orbit is $\Sigma^1_1$-complete. This result and proof have a number of nice corollaries: the Scott rank of \E is \wock +1; not all orbits are elementarily definable; there is no arithmetic description of all orbits of \E; for all finite $\alpha \geq 9$, there is a properly $\Delta^0_\alpha$ orbit (from the proof). A few small corrections made in this versionComment: To appear in the Bulletion of Symbolic Logi

The Bolzano-Weierstrass Theorem is the Jump of Weak K\"onig's Lemma

We classify the computational content of the Bolzano-Weierstrass Theorem and variants thereof in the Weihrauch lattice. For this purpose we first introduce the concept of a derivative or jump in this lattice and we show that it has some properties similar to the Turing jump. Using this concept we prove that the derivative of closed choice of a computable metric space is the cluster point problem of that space. By specialization to sequences with a relatively compact range we obtain a characterization of the Bolzano-Weierstrass Theorem as the derivative of compact choice. In particular, this shows that the Bolzano-Weierstrass Theorem on real numbers is the jump of Weak K\"onig's Lemma. Likewise, the Bolzano-Weierstrass Theorem on the binary space is the jump of the lesser limited principle of omniscience LLPO and the Bolzano-Weierstrass Theorem on natural numbers can be characterized as the jump of the idempotent closure of LLPO. We also introduce the compositional product of two Weihrauch degrees f and g as the supremum of the composition of any two functions below f and g, respectively. We can express the main result such that the Bolzano-Weierstrass Theorem is the compositional product of Weak K\"onig's Lemma and the Monotone Convergence Theorem. We also study the class of weakly limit computable functions, which are functions that can be obtained by composition of weakly computable functions with limit computable functions. We prove that the Bolzano-Weierstrass Theorem on real numbers is complete for this class. Likewise, the unique cluster point problem on real numbers is complete for the class of functions that are limit computable with finitely many mind changes. We also prove that the Bolzano-Weierstrass Theorem on real numbers and, more generally, the unbounded cluster point problem on real numbers is uniformly low limit computable. Finally, we also discuss separation techniques.Comment: This version includes an addendum by Andrea Cettolo, Matthias Schr\"oder, and the authors of the original paper. The addendum closes a gap in the proof of Theorem 11.2, which characterizes the computational content of the Bolzano-Weierstra\ss{} Theorem for arbitrary computable metric space