Given a simplicial pair (X, A), complex Y, and map $$f:A \rightarrow Y$$ , does f have an extension to X? We show that for fixed this question is algorithmically decidable all X, A, if Y has the rational homotopy type of H-space. As corollary, many questions related bundle structures over finite are likely decidable. Conversely, other at least as hard certain special cases Hilbert’s tenth probl...

In this article, we study a notion of the extraction rate Turing functionals that translate between notions randomness with respect to different underlying probability measures. We analyze several classes procedures: (1) class generalizes von Neumann’s trick for extracting unbiased from tosses biased coin, (2) based on work by Knuth and Yao (which more properly can be characterized as randomnes...

The computability of reals was introduced by Alan Turing [20] by means of decimal representations. But the equivalent notion can also be introduced accordingly if the binary expansion, Dedekind cut or Cauchy sequence representations are considered instead. In other words, the computability of reals is independent of their representations. However, as it is shown by Specker [19] and Ko [9], the ...

We examine some recent work which has made significant progress in out understanding of algorithmic randomness, relative algorithmic randomness and their relationship with algorithmic computability and relative algorithmic computability. Mathematics Subject Classification Primary 68Q30, 68Q15, 03D15, 03D25, 03D28, 03D30.

These are notes for a short introductory course on Computability Theory (or recursive function theory). The basic notion of computability is defined in terms of a simple imperative programming language.