DEGREES OF RANDOMIZED COMPUTABILITY

Degrees of Computability(1)

Introduction. In the theory of recursive functions, several decision problems have been proved unsolvable. A decision problem arises when we are confronted with a set TV of mathematical objects and a subset S of 7Y. We desire a mechanical procedure which will operate on elements of TV, determining if they are elements of 5. Now, we can operate on a mathematical object using such an effective pr...

Randomness and the linear degrees of computability

We show that there exists a real α such that, for all reals β, if α is linear reducible to β (α ≤` β, previously denoted α ≤sw β) then β ≤T α. In fact, every random real satisfies this quasi-maximality property. As a corollary we may conclude that there exists no `-complete ∆2 real. Upon realizing that quasi-maximality does not characterize the random reals—there exist reals which are not rando...

Generic computability, Turing degrees, and asymptotic density

Generic decidability has been extensively studied in group theory, and we now study it in the context of classical computability theory. A set A of natural numbers is called generically computable if there is a partial computable function which agrees with the characteristic function of A on its domain D, and furthermore D has density 1, i.e. limn→∞ |{k < n : k ∈ D}|/n = 1. A set A is called co...

Weihrauch Degrees, Omniscience Principles and Weak Computability

In this paper we study a reducibility that has been introduced by Klaus Weihrauch or, more precisely, a natural extension of this reducibility for multi-valued functions on represented spaces. We call the corresponding equivalence classes Weihrauch degrees and we show that the corresponding partial order induces a lower semi-lattice with the disjoint union of multi-valued functions as greatest ...

CSE200: Computability and complexity Randomized algorithms