Independence Properties of Algorithmically Random Sequences

TL;DR

This paper proves that for an algorithmically random sequence, the subsequence selected by a bounded Kolmogorov-Loveland rule is independent of the nonselected bits, showing a form of randomness preservation.

Contribution

It establishes that the nonselected bits of a random sequence remain algorithmically random relative to the selected subsequence, extending understanding of randomness properties under adaptive selection rules.

Findings

Selected subsequence is independent of nonselected bits.

Nonselected bits are random relative to the selected subsequence.

Supports applications in complexity theory and randomness in oracles.

Abstract

A bounded Kolmogorov-Loveland selection rule is an adaptive strategy for recursively selecting a subsequence of an infinite binary sequence; such a subsequence may be interpreted as the query sequence of a time-bounded Turing machine. In this paper we show that if A is an algorithmically random sequence, A_0 is selected from A via a bounded Kolmogorov-Loveland selection rule, and A_1 denotes the sequence of nonselected bits of A, then A_1 is independent of A_0; that is, A_1 is algorithmically random relative to A_0. This result has been used by Kautz and Miltersen [1] to show that relative to a random oracle, NP does not have p-measure zero (in the sense of Lutz [2]). [1] S. M. Kautz and P. B. Miltersen. Relative to a random oracle, NP is not small. Journal of Computer and System Sciences, 53:235-250, 1996. [2] J. H. Lutz. Almost everywhere high nonuniform complexity. Journal of Computer and System Sciences, 44:220-258, 1992.