Upper Limitation of Kolmogorov Complexity and Universal P. Martin-Löf Tests

Abstract

In this paper we study the Kolmogorov complexity of initial strings in infinite sequences (being inspired by [9]), and we relate it with the theory of P. Martin-Lof random sequences. Working with partial recursive functions instead of recursive functions we can localize on an apriori given recursive set the points where the complexity is small. The relation between Kolmogorov's complexity and P. Martin-Lof's universal tests enables us to show that the randomness theories for finite strings and infinite sequences are not compatible (e.g.because no universal test is sequential).
We lay stress upon the fact that we work within the general framework of a non-necessarily binary alphabet.