Recursive Computational Depth

In the 1980s, Bennett introduced computational depth as a formal measure of the amount of computational history that is evident in an object's structure. In particular, Bennett identified the classes of weakly deep and strongly deep sequences and showed that the halting problem is strongly deep. Juedes, Lathrop, and Lutz subsequently extended this result by defining the class of weakly useful sequences and proving that every weakly useful sequence is strongly deep. The present paper investigates refinements of Bennett's notions of weak and strong depth, called recursively weak depth (introduced by Fenner, Lutz, and Mayordomo) and recursively strong depth (introduced here). It is argued that these refinements naturally capture Bennett's idea that deep objects are those which “contain internal evidence of a nontrivial causal history.” The fundamental properties of recursive computational depth are developed, and it is shown that the recursively weakly (respectively, strongly) deep sequences form a proper subclass of the class of weakly (respectively, strongly) deep sequences. The above-mentioned theorem of Juedes, Lathrop, and Lutz is then strengthened by proving that every weakly useful sequence is recursively strongly deep. It follows from these results that not every strongly deep sequence is weakly useful, thereby answering a question posed by Juedes.