We invistigate what is the minimal length of
a program mapping A to B and at the same time
mapping C to D (where A,B,C,D are binary strings). We prove that
it cannot be expressed
in terms of Kolmogorv complexity of A,B,C,D, their pairs (A,B), (A,C),
..., their ... more >>>

We define Kolmogorov complexity of a set of strings as the minimal
Kolmogorov complexity of its element. Consider three logical
operations on sets going back to Kolmogorov
and Kleene:
(A OR B) is the direct sum of A,B,
(A AND B) is the cartesian product of A,B,
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We study different notions of descriptive complexity of
computable sequences. Our main result states that if for almost all
n the Kolmogorov complexity of the n-prefix of an infinite
binary sequence x conditional to n
is less than m then there is a program of length
m^2+O(1) that for ... more >>>