Computational Depth of Infinite Strings Revisited

Bennett introduced the notion of logical depth of an object as the amount of time required for an algorithm to derive the object from a shorter description. He also defined logical depth for infinite strings, in particular strongly and weakly deep sequences. Later Lutz, Juedes and Mayordomo have further studied and related these measures. Recently Antunes et al. noted that logical depth, as introduced by Bennett is connected to Kolmogorov and Levin’s notion of “randomness deficiency”. Based on this connection, we revisit the notion of computational depth for infinite strings, introducing the notion of super deep sequences and relate it with other approaches. Classification: Kolmogorov Complexity, Computational Depth.