Random numbers as probabilities of machine behavior

Abstract

2017-08-25T16:27:31

The Probability of a Computable Output from a Random Oracle

“…A summary of basic facts about the representation of open Σ 0 n classes of streams as Σ 0 n prefixfree sets of strings and the characterization of their measures as the 0 (n) -left c.e. reals can be found in [BCP17]. If Q is a set of strings, we let Q be the set of streams which have a prefix in Q.…”

“…almost everywhere totality of computed function) often produce 3-random probabilities (e.g. [BC02,BCP17]) and some Σ 0 4 properties (e.g. universality) produce 4-random probabilities [BD12].…”

“…By the Kraft-Chaitin theorem it follows (see [BCP17] for a detailed proof) that given any e ∈ N, any string ρ of length e and any 0 ′ -left-c.e. real α ∈ (0, 1 − 2 −e ) there exists a Σ 0 2 prefix-free set S of strings which are incompatible with ρ and µ( S ) = α.…”

“…1 Chaitin [Cha75] originally used self-delimiting machines in this thought experiment, but our version can easily be seen to be equivalent. 2 for example, work by Becher and her collaborators [BDC01, BC02,BG05], and the more recent work by the authors in [BCP17]; see Section 2.1 for a discussion. 3 Alternatively we could consider as input the pairs (X, n) of oracle binary streams and integers, and the output as partial function n → M(X, n) that is computed by M on oracle X and input n. The point here is that the eventual content of the output tape of U is a representation (i.e.…”

“…the characteristic sequence) of the partial function n → U(X, n). The restriction that the output tape is written in a contiguous manner does not make any difference to our discussion, since we do not consider events that involve partial functions (something which is no longer true for many of the results in [BCP17]). Remark 1.3 (Significance of binary output alphabet).…”