Asymptotic behavior and halting probability of Turing Machines

Through a straightforward Bayesian approach we show that under some general conditions a maximum running time, namely the number of discrete steps done by a computer program during its execution, can be defined such that the probability that such program will halt after that time is smaller than any arbitrary fixed value. Consistency with known results and consequences are also discussed. 1 Introductory remarks As it has been proved by Alan M. Turing in 1936 [1], if we have a program p running on an Universal Turing Machine (UTM), then we have no general, finite and deterministic algorithm which allows us to know whether and when it will halt (this is the well known halting problem). This is to say that the halting behavior of a program, with the trivial exception of the simplest ones, is not computable and predictable by a unique, general procedure. In this paper we show that, for what concerns the probability of its halt, every program running on an UTM is characterized by a peculiar asymptotic behavior in time. 2 Probabilistic approach Given a program p of n bits, it is always possible to slightly change its code (increasing its size by a small, fixed amount of bits, let’s say of s bits with