We prove that there exist sets of natural numbers A and B such that A and B form a minimal pair with respect to Turing reducibility, enumeration reducibility, hyperarithmetical reducibility and hyperenumer-ation reducibility. Relativized versions of this result are presented as well. 1. Introduction In the present paper we consider four kinds of reducibilities among sets of natural numbers: Tur...

Secure Function Evaluation (SFE) protocols are very hard to design, and reducibility has been recognized as a highly desirable property of SFE protocols. Informally speaking, reducibility (sometimes called modular composition) is the automatic ability to break up the design of complex SFE protocols into several simpler, individually secure components. Despite much effort, only the most basic ty...

Wadge reducibility provides a rich and nice analysis of Borel sets in Polish zero dimensional spaces. However, outside this framework, reducibility by continuous functions was shown to be ill behaved in many important cases. We define a notion of reducibility for subsets of a second countable T0 topological space based on the notions of admissible representations and relatively continuous relat...

Let [Formula: see text] be a commutative Noetherian ring. For finitely generated text]-module text], Northcott introduced the reducibility index of which is number submodules appearing in an irredundant irreducible decomposition submodule text]. On other hand, for Artinian Macdonald proved that sum-irreducible representation does not depend on choice representation. This called sum-reducibility...

We investigate structural properties of interactive perfect zero-knowledge (PZK) proofs. Specifically, we look into the closure properties of PZK languages under monotone boolean formula composition. This gives rise to new protocol techniques. We show that interactive PZK for random self-reducible languages (RSR) (and for co-RSR) is closed under monotone boolean formula composition. Namely, we ...

Q-reducibility is a natural generalization of many-one reducibility (m-reducibility): if A ≤m B via a computable function f(x), then A ≤Q B via the computable function g(x) such that Wg(x) = {f(x)}. Also this reducibility is connected with enumeration reducibility (e-reducibility) as follows: if A ≤Q B via a computable function g(x), then ω −A ≤e ω −B via the c. e. set W = {〈x, 2y〉 | x ∈ ω, y ∈...

We investigate the relationship between two di erent notions of reducibility among prediction (learning) problems within the distribution-free learning model of Valiant (PAC learning model [7, 2].) The notions of reducibility we consider are the analogues for prediction problems of the many-one reducibility and of the Turing reducibility. The former is the notion of prediction preserving reduci...

Reducibility is a powerful proof method which applies to various properties of typed terms in different type systems. For strong normalization, different variants are known, such as Girard’s reducibility candidates, Tait’s saturated sets and biorthogonals. They differ by the closure conditions imposed to types interpretations, called here reducibility families. This paper is about the computati...

Reducibility without qualification means in this lecture reducibility over the rational field. Questions on such reducibility occupy an intermediate place between questions on reducibility of polynomials over an algebraically closed field and those on primality. I shall refer to these two cases as to the algebraic and the arithmetic one and I shall try to exhibit some of the analogies consideri...