Computational Depth and Reducibility

This paper investigates Bennett\u27s notions of strong and weak computational depth (also called logical depth) for infinite binary sequences. Roughly, an infinite binary sequence x is defined to be weakly useful if every element of a non-negligible set of decidable sequences is reducible to x in recursively bounded time. It is shown that every weakly useful sequence is strongly deep. This result (which generalizes Bennett\u27s observation that the halting problem is strongly deep) implies that every high Turing degree contains strongly deep sequences. It is also shown that, in the sense of Baire category, almost every infinite binary sequence is weakly deep, but not strongly deep