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. The proof is sensitive to the particular computational model. The halting problem for Turing machines is perhaps the canonical undecidable set. Nevertheless, we prove that there is an algorithm deciding almost all instances of it. The halting problem is therefore among the growing collection of those exhibiting the “black hole” phenomenon of complexity theory, by which the difficulty of an unfeasible or undecidable problem is confined to a very small region, a black hole, outside of which the problem is easy. We use the most natural method for measuring the size of a set of Turing machine programs, namely, that of asymptotic density. The asymptotic density or probability of a set B of Turing machine programs is the limit of MSC: 03D10; 68Q17.