4 research outputs found
Inherent enumerability of strong jump-traceability
We show that every strongly jump-traceable set obeys every benign cost
function. Moreover, we show that every strongly jump-traceable set is
computable from a computably enumerable strongly jump-traceable set. This
allows us to generalise properties of c.e.\ strongly jump-traceable sets to all
such sets. For example, the strongly jump-traceable sets induce an ideal in the
Turing degrees; the strongly jump-traceable sets are precisely those that are
computable from all superlow Martin-L\"{o}f random sets; the strongly
jump-traceable sets are precisely those that are a base for
-randomness; and strong jump-traceability is
equivalent to strong superlowness.Comment: 25 page
Strong jump traceability and Demuth randomness
We solve the covering problem for Demuth randomness, showing that a
computably enumerable set is computable from a Demuth random set if and only if
it is strongly jump-traceable. We show that on the other hand, the class of
sets which form a base for Demuth randomness is a proper subclass of the class
of strongly jump-traceable sets.Comment: 28 page
Demuth's path to randomness
Osvald Demuth (1936--1988) studied constructive analysis from the viewpoint
of the Russian school of constructive mathematics. In the course of his work he
introduced various notions of effective null set which, when phrased in
classical language, yield a number of major algorithmic randomness notions. In
addition, he proved several results connecting constructive analysis and
randomness that were rediscovered only much later.
In this paper, we trace the path that took Demuth from his constructivist
roots to his deep and innovative work on the interactions between constructive
analysis, algorithmic randomness, and computability theory. We will focus
specifically on (i) Demuth's work on the differentiability of Markov computable
functions and his study of constructive versions of the Denjoy alternative,
(ii) Demuth's independent discovery of the main notions of algorithmic
randomness, as well as the development of Demuth randomness, and (iii) the
interactions of truth-table reducibility, algorithmic randomness, and
semigenericity in Demuth's work
Logic Blog 2011
This year's logic blog has focussed on:
1. Demuth randomness
2. traceability
3. The connection of computable analysis and randomness
4. -triviality in metric spaces