Feasible Depth

Abstract

This paper introduces two complexity-theoretic formulations of Bennett's logical depth: finite-state depth and polynomial-time depth. It is shown that for both formulations, trivial and random infinite sequences are shallow, and a slow growth law holds, implying that deep sequences cannot be created easily from shallow sequences. Furthermore, the E analogue of the halting language is shown to be polynomial-time deep, by proving a more general result: every language to which a nonnegligible subset of E can be reduced in uniform exponential time is polynomial-time deep.

A General Notion of Useful Information

“…However subtle the influence of those small changes on the corresponding depth notion is, they are not relevant to the high level understanding of the notion of depth we aim at in the present paper, and can all be seen as variations of a same common theme, which is captured by our general depth framework, and that in essence says that a sequence is deep with respect to (G, G ′ ) if given any algorithm in G there is an algorithm in G ′ that performs better (as measured by M) on the sequence. This general definition actually captures all depth notions introduced in complexity theory so far [2,7,1,3,8], as we shall see in the next section, where we will review some of these notions together with some of the results that were obtained for each of them.…”

“…As we show in this paper, our general framework actually captures all depth notions introduced in complexity theory so far [2,7,1,3,8], which can all be seen as particular instances of our general depth framework. Most of these notions, are based on some class of compression algorithm, which as seen by our general depth framework, is only one -among many others-particular way to define the depth of a sequence, therefore leaving the door open to the study of new depth notions, not necessarily based on the compression paradigm.…”

On the polynomial depth of various sets of random strings

“…In [5] Antunes, Fortnow, van Melkebeek, and Vinodchandran studied several polynomial-time formulations of depth, with connections to average-case Unfortunately, the feasible notions proposed so far suffer some limitations, e.g. a notion in [5] requires a complexity assumptions to prove the existence of deep sequences; and the polynomial depth of [9] is based on polynomial time predictors that cannot read their input (predictors must predict the nth bit of a sequence without access to the history, i.e. bits 1, 2, .…”

“…In this paper, we use the idea of competing observers from [15] to construct new notions of polynomial depth (called monotone-polynomial depth), aiming at notions that satisfy the slow growth law, and for which deep objects can be proved to exist unconditionally. The classes of observers (the classes ∆ and ∆ ′ ) we consider are based on the notion of monotone polynomial time compression [8], which is a polynomial version of monotone Kolmogorov complexity, with the advantage that unlike polynomial predictors [9], monotone polynomial compressors can read their input. We show that our notions of monotone polynomial depth have all the desired properties of a depth notion, i.e.…”

“…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.…”