Recursive Computational Depth

In the 1980'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, Lathrop, 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 fundamental properties of recursive computational depth are developed, and it is shown that the recursively weakly (respectivel...

• Type: Conference
• Year: 1997
• Where: ICALP
• Authors: James I. Lathrop, Jack H. Lutz
• Keywords: Computational Depth, Deep Sequence, ICALP 1997, Programming Languages, Useful Sequence