Sophisticated Infinite Sequences

In this paper we revisit the notion of sophistication for infinite sequences. Koppel defined sophistication of an object as the length of the shortest (finite) total program (p) that with some (finite or infinite) data (d) produce it and |p|+ |d| is smaller than the shortest description of the object plus a constant. However the notion of “description of infinite sequences” is not appropriately defined. In this work, we propose a new definition of sophistication for infinite sequences as the limit of the ratio of sophistication of the initial segments and its length. As the main result we prove that highly sophisticated sequences are dense when the sophistication is defined with lim sup and the set of sequences with lower sophistication equal to zero is also dense. We also prove that, similarly to what happens for finite strings, sophistication and depth, for infinite sequences are distinct complexity measures.