Computational Depth and Reducibility

This paper reviews and investigates Bennett s notions of strong and weak computational depth also called logical depth for in nite binary sequences Roughly an in nite binary sequence x is de ned 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 s 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 in nite binary sequence is weakly deep but not strongly deep