Recursive Computational Depth

In the s 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 identi ed the classes of weakly deep and strongly deep sequences and showed that the halting problem is strongly deep Juedes Lath rop and Lutz subsequently extended this result by de ning the class of weakly useful sequences and proving that every weakly useful sequence is strongly deep The present paper investigates re nements 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 re nements naturally capture Bennett s idea that deep objects are those which contain internal evidence of a nontrivial causal history The fun damental 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 strength ened 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 This research was supported in part by National Science Foundation Grant CCR with matching funds from Rockwell Microware Systems Corporation and Amoco Foundation