Finite Self-Information

Abstract

We present a definition, due to Levin, of mutual information I(A : B) for infinite sequences. We say that a set A has finite self-information if I(A : A) < ∞. It is easy to see that every K-trivial set has finite self-information. We answer a question of Levin by showing that the converse does not hold. Finally, we investigate the connections between having finite self-information and other notions of weakness such as jump-traceability. In particular, we show that our proof can be adapted to produce a set that is low for both effective Hausdorff dimension and effective packing dimension, but not K-trivial.

Depth, Highness and DNR Degrees

“…The result of Herbert is optimal, Csima and Montalbán [8] showed that such sets do not exist when using ∆ 0 4 orders and Baartse and Barmpalias [4] improved this non-existence to the level ∆ 0 3 . We also point to related work of Hirschfeldt and Weber [14]. Theorem 4.9 (Baartse and Barmpalias [4]) There is a ∆ 0 3 order g such that a set A is K-trivial iff K(A ↾ n) ≤ + K(n) + g(n) for all n.…”

A Perfect Set of Reals with Finite Self-Information

“…The distinction we will be interested in for different pairs of reals is whether I(A : B) is finite or infinite. Definition 1.1 is equivalent to a definition proposed by Levin in [5], and is the one used by Hirschfeldt and Weber in [3]. In the same paper, Levin proposed another defintion of mutual information that has since been called simplified mutual information, where the sum in Definition 1.1 is only over pairs with σ = τ .…”

“…Theorem 1.3. (Hirschfeldt, Weber [3] ) There is a real that has finite selfinformation that is not K-trivial.…”

On Reals With -Bounded Complexity and Compressive Power

“…Hirschfeldt and Weber, though they did not use this terminology, first showed that for any finite-to-one approximable f, LK(f) contains an r.e. set that is not in LK(0) [9]. In an earlier paper [8], the author has shown that there is a perfect set of reals in LK(f) for any finite-to-one approximable f, and in fact in LK(Δ 0 2 ).…”

“…Hirschfeldt and Weber, though they did not use this terminology, first showed that for any finite-to-one approximable f , LKpf q contains an r.e. set that is not in LKp0q [9]. In an earlier paper [8], the author has shown that there is a perfect set of reals in LKpf q for any finite-to-one approximable f , and in fact in LKp∆ 0 2 q. Additionally, the perfect set constructed in that paper has the property that for any real A there are two elements B 1 and B 2 of the set such that B 1 ' B 2 ě T A, so in general LKpf q and LKp∆ 0 2 q are not closed under effective join (i.e., except for the trivial case when LKpf q " 2 ω ).…”