The halting problem is decidable on a set of asymptotic probability one
  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. 
Joel David Hamkins (The City University of New York) and Alexei   Miasnikov (The City University of New York)