Program Size Complexity for Possibly Infinite Computations

We define a program size complexity function H∞ as a variant of the prefix-free Kolmogorov complexity, based on Turing monotone machines performing possibly unending computations. We consider definitions of randomness and triviality for sequences in {0, 1} relative to the H∞ complexity. We prove that the classes of Martin-Löf random sequences and H∞-random sequences coincide, and that the H∞-trivial sequences are exactly the recursive ones. We also study some properties of H∞ and compare it with other complexity functions. In particular, H∞ is different from H, the prefix-free complexity of monotone machines with oracle A.