Weakly useful sequences

Abstract

An infinite binary sequence x is defined to be (i) strongly useful if there is a computable time bound within which every decidable sequence is Turing reducible to x; and (ii) weakly useful if there is a computable time bound within which all the sequences in a non-measure 0 subset of the set of decidable sequences are Turing reducible to x.Juedes, Lathrop, and Lutz (1994) proved that every weakly useful sequence is strongly deep in the sense of Bennett (1988) and asked whether there are sequences that are weakly useful but not strongly useful.The present paper answers this question affirmatively. The proof is a direct construction that combines the martingale diagonalization technique of Lutz (1994) with a new technique, namely, the construction of a sequence that is "computably deep" with respect to an arbitrary, given uniform reducibility. The abundance of such computably deep sequences is also proven and used to show that every weakly useful sequence is computably deep with respect to every uniform reducibility.

Depth as Randomness Deficiency

“…The Hausdorff constructive dimension has a close connection with the information theories for infinite strings studied before, see for example [FLMR05], [Lut00], [Lut02] and [May02]. Therefore, in this section we define the dimensional computational depth of a sequence in order to study the nonrandom information on a infinite sequence.…”

“…This result generalizes Bennett's remark that the diagonal halting problem is strobgly deep, strengthening the relation between depth and usefulness. Latter Fenner et al [FLMR05] proved the existence of sequences that are weakly useful but not strongly useful.…”

“…Lathrop, and Lutz [JLL94] proved that every weakly useful sequence is strongly deep in the sense of Bennett. Later, Fenner et al [FLMR05] proved that there exist sequences that are weakly useful but not strongly useful. Lathrop and Lutz [LL99] introduced refinements (named recursive weak depth and recursive strong depth) of Bennett's notion of weak and strong depth, and studied its fundamental properties, showing that recursively weakly (resp.…”

Computing and evolving variants of computational depth

“…Juedes, Lathrop, and Lutz [38] defined the class of weakly useful sequences and proved that every weakly useful sequence is strongly deep. Fenner, Lutz, and Mayordomo [19] subsequently proved that every weakly useful sequence is rec-weakly deep. In this section, we strengthen both these results by proving that every weakly useful sequence is rec-strongly deep.…”

“…Using recursive computational depth, we investigate refinements of Bennett's notions of weak and strong depth, called recursively weak depth (introduced by Fenner, Lutz and Mayordomo [19]) and recursively strong depth (introduced here).…”