Generic algorithms for halting problem and optimal machines revisited

A classical result says that the halting problem is undecidable: there is no algorithm that, given a computation, says whether it terminates or not. A related result says that for some computations the termination statement is undecidable in Gödel’s sense (neither provable nor refutable). Still in many cases the termination question is not that hard. Maybe, the difficult cases are rare exceptions and for most cases the answer can be obtained effectively (and even easily)? This question, while natural, is difficult to formulate. For the qualitative questions it does not matter which computational model or programming language we use in the formulation of the halting problem. Technically speaking, all reasonable formulations lead to m-complete computably enumerable sets, and all m-complete sets are computably isomorphic (Myhill isomorphism theorem, see, e.g., [15]). However, for quantitative questions the choice of the programming language is very important: it is easy to imagine some universal programming language for which most programs terminate (or hang) for some trivial reasons. One may try to fix some computational model or programming language. For example, we may consider Turing machines with a fixed alphabet, and then ask whether there exists an approximation algorithm for the halting problem, whose success rate among all machines with n states converges to 1 as n→ ∞. This question, however, is sensitive to the details of the definition. For example, it was shown in [8] that for Turing machines with one-sided tape the success rate may converge to 1: speaking informally, this happens because most machines fall off the tape rather quickly. This argument, however, does not work for two-sided tape machines, for which the similar question remains open. Looking for more invariant statements, one should put some restrictions on the way the computations are encoded, and these restrictions could be quite technical. For example, in [16, 9] numberings of computable functions are considered where each function occupies a Ω(1)-fraction of n-bit programs for all sufficiently large n, together with encodings for pairs that have some special property. However, already in [12] a more natural requirement for the programming language (motivated by the algorithmic information theory) was suggested. We show (Section 3) that results about approximate algorithms for halting problem from [16, 9] remain true in this simple setting, and can be easily proved using Kolmogorov complexity. Moreover, they remain true for a weaker requirement than used in [17, 12]; we discuss it (and some related questions) in Section 2. The next three sections (Sections 4–6) are devoted to different questions related to halting problem and its approximate solutions. In Section 4 we consider the fraction of terminating programs among all programs of length at most n. We prove that this fraction has no limit as n→ ∞, and limit points are Martin-Löf random reals (even relative to 0′). We prove that the limsup of this fraction is an upper semicomputable 0′-random number and every number of this