WHAT IS. . . a Halting Probability?

Notices of the AMS 57:236-237 (2010)
  Copy   BIBTEX

Abstract

Turing’s famous 1936 paper “On computable numbers, with an application to the Entscheidungsproblem” defines a computable real number and uses Cantor’s diagonal argument to exhibit an uncomputable real. Roughly speaking, a computable real is one that one can calculate digit by digit, that there is an algorithm for approximating as closely as one may wish. All the reals one normally encounters in analysis are computable, like π, √2 and e. But they are much scarcer than the uncomputable reals because, as Turing points out, the computable reals are countable, whilst the uncomputable reals have the power of the continuum. Furthermore, any countable set of reals has measure zero, so the computable reals have measure zero. In other words, if one picks a real at random in the unit interval with uniform probability distribution, the probability of obtaining an uncomputable real is unity. One may obtain a computable real, but that is in- finitely improbable. But how about individual examples of uncomputable reals? We will show two: H and the halting probability Ω, both contained in the unit interval. Their construction was anticipated in..

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 90,616

External links

  • This entry has no external links. Add one.
Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Relative Randomness and Real Closed Fields.Alexander Raichev - 2005 - Journal of Symbolic Logic 70 (1):319 - 330.
Schnorr Randomness.Rodney G. Downey & Evan J. Griffiths - 2004 - Journal of Symbolic Logic 69 (2):533 - 554.
Mapping a set of reals onto the reals.Arnold W. Miller - 1983 - Journal of Symbolic Logic 48 (3):575-584.
Subclasses of the Weakly Random Reals.Johanna N. Y. Franklin - 2010 - Notre Dame Journal of Formal Logic 51 (4):417-426.
Alan Turing and the foundations of computable analysis.Guido Gherardi - 2011 - Bulletin of Symbolic Logic 17 (3):394-430.
Cohen reals from small forcings.Janusz Pawlikowski - 2001 - Journal of Symbolic Logic 66 (1):318-324.
Church's Thesis and the Conceptual Analysis of Computability.Michael Rescorla - 2007 - Notre Dame Journal of Formal Logic 48 (2):253-280.
Degrees of Monotone Complexity.William C. Calhoun - 2006 - Journal of Symbolic Logic 71 (4):1327 - 1341.
The Block Relation in Computable Linear Orders.Michael Moses - 2011 - Notre Dame Journal of Formal Logic 52 (3):289-305.
Degrees of Unsolvability of Continuous Functions.Joseph S. Miller - 2004 - Journal of Symbolic Logic 69 (2):555 - 584.

Analytics

Added to PP
2010-12-22

Downloads
69 (#214,873)

6 months
1 (#1,040,386)

Historical graph of downloads
How can I increase my downloads?

Author's Profile

Cristian S. Calude
University of Auckland

Citations of this work

No citations found.

Add more citations

References found in this work

No references found.

Add more references