I&C Journal 2005 Journal Article
Weakly useful sequences
- Stephen A. Fenner
- Jack H. Lutz
- Elvira Mayordomo
- Patrick Reardon
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 [Theorectical Computer Science 132 (1994) 37] proved that every weakly useful sequence is strongly deep in the sense of Bennett [The Universal Turing Machine: A Half-Century Survey, 1988, 227] 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 [SIAM Journal on Computing 24 (1995) 1170] 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.