We study Tukey types of ultrafilters on ω, focusing on the question of when Tukey reducibility is equivalent to Rudin-Keisler reducibility. We give several conditions under which this equivalence holds. We show that there are only c many ultrafilters that are Tukey below any basically generated ultrafilter. The class of basically generated ultrafilters includes all known ultrafilters that are n...

We introduce a notion of width-bounded reducibility. Width-bounded reducibility provides a circuit-based realization of RuzzoSimon-Tompa reducibility [RS-84], and allows us to generalize that notion of reducibility. We show that reductions of simultaneously restricted width and depth provide a characterization of binary search over complexity classes, as introduced by Wagner [Wa-89] and Buss an...

Preface In this course we survey some recent work on the strongly bounded Turing re-ducibilities on the computably enumerable sets. Bounded Turing reducibilities are obtained from classical Turing reducibility by imposing upper bounds on the use functions (i.e., on the size of the oracle queries) of the reductions. The most popular bounded Turing reducibility which has been intensively studied ...

A class L is called mitotic if it admits a splitting L0,L1 such that L,L0,L1 are all equivalent with respect to a certain reducibility. Such a splitting might be called a symmetric splitting. In this paper we investigate the possibility of constructing a class which has a splitting and where any splitting of the class is a symmetric splitting. We call such a class a symmetric class. In particul...

We study the notion of polynomial-time relation reducibility among computable equivalence relations. We identify some benchmark equivalence relations and show that the reducibility hierarchy has a rich structure. Specifically, we embed the partial order of all polynomial-time computable sets into the polynomial-time relation reducibility hierarchy between two benchmark equivalence relations Eλ ...

Preface This survey deals with some aspects of the problem of reducibility for linear equations with quasi-periodic coefficients. It is a compilation of results on this problem, some already classical and some other more recent. Our motivation comes from the study of stability of quasi-periodic motions and preservation of invariant tori in Hamiltonian mechanics (where the reducibility of linear...

Properties of topological spaces are not absolute, as they may depend on the ambient mathematical universe. For instance, the answers to some questions in classical point-set topology hinge on non-standard set-theoretic axioms, while in intuitionistic mathematics even such basic properties as the Heine-Borel compactness of the closed interval depend on the ambient constructive variety. We show ...

We describe a new method to bound certain higher-dimensional exponential sums which are associated with tori in symplectic groups over finite fields. Our method is based on the self-reducibility property of the Weil representation. As a result, we obtain a sharp form of the Hecke quantum unique ergodicity theorem for generic linear symplectomorphisms of the 2Ndimensional torus.

We introduce polynomial time enumeration reducibility (≤pe) and we retrace Selman’s analysis of this reducibility and its relationship with non deterministic polynomial time conjunctive reducibility. We discuss the basic properties of the degree structure induced by ≤pe over the computable sets and we show how to construct meets and joins. We are thus able to prove that this degree structure is...