The halting problem is decidable on a set of asymptotic probability one

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  • J. D. Hamkins and A. Miasnikov, “The halting problem is decidable on a set of asymptotic probability one,” Notre Dame J.~Formal Logic, vol. 47, iss. 4, pp. 515-524, 2006.  
    Journal MR Citation arχiv 
    @ARTICLE{HamkinsMiasnikov2006:HaltingProblemDecidable,
AUTHOR = {Hamkins, Joel David and Miasnikov, Alexei},
TITLE = {The halting problem is decidable on a set of asymptotic probability one},
JOURNAL = {Notre Dame J.~Formal Logic},
FJOURNAL = {Notre Dame Journal of Formal Logic},
VOLUME = {47},
YEAR = {2006},
NUMBER = {4},
PAGES = {515--524},
ISSN = {0029-4527},
CODEN = {NDJFAM},
MRCLASS = {03D10 (68Q05)},
MRNUMBER = {2272085 (2007m:03082)},
MRREVIEWER = {Maurice Margenstern},
DOI = {10.1305/ndjfl/1168352664},
URL = {http://dx.doi.org/10.1305/ndjfl/1168352664},
eprint = {math/0504351},
archivePrefix = {arXiv},
primaryClass = {math.LO},
file = F,
}

The halting problem for Turing machines is decidable on a set of asymptotic probability one. Specifically, there is a set B of Turing machine programs such that (i) B has asymptotic probability one, so that as the number of states n increases, the proportion of all n-state programs that are in B goes to one; (ii) B is polynomial time decidable; and (iii) the halting problem H intersect B is polynomial time decidable. The proof is sensitive to the particular computational model.

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