Asymptotic behaviour of Turing Machines

In this paper we show that the probability that a program p of n bits halts at discrete time t (with t > 2n+1 − 2; we define the discrete time t as the integer number of steps done by an Universal Turing Machine during the execution of p, until the halt) tends to zero as the inverse law ∝ 1/t. This means that the greater is the time an n-bit program is running on an Universal Turing Machine (provided that t > 2n+1 − 2), the lesser is the probability that it will ever halt. 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 ǫ bits with ǫ ≪ n) in order to include a progressive, integer counter that starts counting soon after p starts to run on an Universal Turing Machine (UTM) and stops, printing the total number of steps done by the UTM during the execution of p, soon after p halts. Let us call this new program p. Its algorithmic size is then of n+ ǫ bits. As it has been proved by Alan Turing in 1936 [1], for an arbitrary program of n bits, no general, finite and deterministic procedure exists that allows us to know whether such program will ever halt or will keep running forever on an UTM. This is to say that even if a program will surely halt, we are not able to know in advance when it will halt [2]. Thus, in our case, we have no algorithmic procedure to decide whether and when p will print the integer number of steps done by the UTM during the execution of p, until the halt. According to the definition of algorithmic complexity of a bit string [3], we necessarily have that: