[Paper Review] Independence Properties of Algorithmically Random Sequences

Steven M. Kautz|ArXiv.org|Jan 16, 2003

Computability, Logic, AI Algorithms18 references3 citations

TL;DR

This paper establishes that when an algorithmically random infinite binary sequence A is partitioned via a bounded Kolmogorov-Loveland selection rule into selected bits A₀ and non-selected bits A₁, the subsequence A₁ remains algorithmically random relative to A₀. This independence property confirms that no information about the random sequence can be extracted from its selected subsequence, even under adaptive, non-recursive selection rules.

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.

Motivation & Objective

• To investigate the independence properties of algorithmically random sequences under various subsequence selection rules.
• To determine conditions under which a subsequence selected from a random sequence remains algorithmically random.
• To establish that the non-selected subsequence remains random relative to the selected one under bounded Kolmogorov-Loveland rules.
• To provide a foundational result for complexity-theoretic applications involving random oracles.
• To formalize the intuition that random sequences should not contain exploitable information about their own parts.

Proposed method

• Defines a bounded Kolmogorov-Loveland selection rule as an adaptive, non-recursive strategy that selects bits based on past history and a function F.
• Uses a recursive construction of sequences αₖ and βₖ to simulate the selection process and track the state of selected and non-selected bits.
• Introduces auxiliary sequences σ and τ to model the interaction between the selection rule and the random sequence A.
• Employs induction on stages k to prove invariants: αₖ is a prefix of A, αₖ* is a prefix of A, and the selection rule maintains consistency with the oracle B.
• Applies the convergence of a function S(σ, τ) to verify that the selection process can be extended indefinitely without contradiction.
• Uses the concept of relative algorithmic randomness and van Lambalgen’s theorem to formalize independence between A₀ and A₁.

Experimental results

Research questions

• RQ1 Under what selection rules is the complement of a selected subsequence algorithmically random relative to the selected subsequence?
• RQ2 Can the independence of selected and non-selected parts of a random sequence be preserved under non-recursive, adaptive selection strategies?
• RQ3 Does a bounded Kolmogorov-Loveland selection rule preserve the algorithmic randomness of the non-selected subsequence?
• RQ4 Can this independence property be used to derive complexity-theoretic results about the structure of NP relative to random oracles?
• RQ5 How does this result strengthen or generalize the Mises-Church stochasticity condition for random sequences?

Key findings

• For any algorithmically random sequence A, if A₀ is selected via a bounded Kolmogorov-Loveland rule, then A₁ (the non-selected bits) is algorithmically random relative to A₀.
• The independence of A₀ and A₁ holds even when the selection rule is non-recursive, demonstrating robustness of randomness under adaptive selection.
• The proof constructs a recursive process that maintains consistency between the selection rule, the oracle B, and the target sequence A.
• The result implies that no time-bounded Turing machine can extract useful information about the non-selected bits from the selected query sequence.
• This independence property is used to show that NP does not have p-measure zero relative to a random oracle in exponential time classes.
• The result strengthens earlier results on P ≠ NP relative to random oracles by providing a more refined analysis of information content in random sequences.