Algorithmical Definition of Finite Binary Random Sequence

Abstract

In [5] A.N. Kolmogorov has defined the complexity K_F(x) of binary sequence x with respect to program F for the description of x. He prooved the existance of a universal program F₀ i.e. of such a program that ${{\rm{K}}_{{{\rm{F}}_0}}}(x) \le {{\rm{K}}_{\rm{F}}}(x) + {{\rm{C}}_{\rm{F}}}$ C_F-const.,for each x and F. Loosely speaking,the sequence x is random if K_{F o}(x) is large enough.It is clear that,if the set J of sequences x is finite, then each program F is universal.This means that, in this case, the randomness can be defined arbitrary.For that reason this approach is more successful in case of infinite set J (see, for Instance[4].