On the polynomial depth of various sets of random strings

Abstract

This paper proposes new notions of polynomial depth (called monotone poly depth), based on a polynomial version of monotone Kolmogorov complexity. We show that monotone poly depth satisfies all desirable properties of depth notions i.e., both trivial and random sequences are not monotone poly deep, monotone poly depth satisfies the slow growth law i.e., no simple process can transform a non deep sequence into a deep one, and monotone poly deep sequences exist (unconditionally).We give two natural examples of deep sets, by showing that both the set of Levinrandom strings and the set of Kolmogorov random strings are monotone poly deep. 1 complexity, nonuniform circuit complexity, and efficient search for satisfying assignments to Boolean formulas. In [9], both a notion of finite-state and polynomial depth were investigated, and the depth of polynomial weakly useful languages was shown.

Depth, Highness and DNR Degrees

“…Several variants of logical depth have been studied in the past [2,9,16,18,21]. As shown in [21], all depth notions proposed so far can be interpreted in the compression framework which says a sequence is deep if given (arbitrarily) more than t(n) time steps, a compressor can compress the sequence r(n) more bits than if given at most t(n) time steps only. By considering different time bound families for t(n) (e.g.…”

“…and the magnitude of compression improvement r(n) -for short: the depth magnitude -(e.g. O(1), O(log n)) one can capture all existing depth notions [2,9,16,18,21] in the compression framework [21]. E.g.…”

“…Bennett's notion is obtained by considering all recursive time bounds t and a constant depth magnitude, i.e., r(n) = O (1). Several authors studied variants of Bennett's notion, by considering different time bounds and/or different depth magnitude from Bennett's original notion [2,3,9,16,21].…”

Polylog depth, highness and lowness for E

“…⊓ ⊔ As noticed in [15], for most depth notions it can be shown that easy and random sequences are not deep. The following two results show that this is also the case for deep (P,EXP) .…”

“…In this paper, we revisit their results [16] within the context of computational complexity theory. Adapting Bennett's logical depth to the computational complexity setting is an elusive task, and several authors have proposed polynomial versions of logical depth [2,7,15]; see [15] for a summary of most notions. Here we study a polynomial version of depth as close as possible to the original notion by Bennett, namely the difference of two Kolmogorov complexities with different time bounds.…”