A General Notion of Useful Information

Abstract

In this paper we introduce a general framework for defining the depth of a sequence with respect to a class of observers. We show that our general framework captures all depth notions introduced in complexity theory so far. We review most such notions, show how they are particular cases of our general depth framework, and review some classical results about the different depth notions.

On the polynomial depth of various sets of random strings

“…As noticed in [15], the notion of depth is a relative notion, that depends on the power of the observers. Our goal is to study polynomial versions of Bennett's original depth notion [7], and its recursive version called recursive depth [11].…”

“…As noticed in [15], depth is not an absolute concept, but depends on the power of two competing group of observers ∆ and ∆ ′ . Informally a sequence is (∆, ∆ ′ )-deep if for any observer O from ∆ there is an observer O ′ in ∆ ′ such that O ′ performs (e.g.…”

“…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.…”

Unpredictability and Computational Irreducibility

“…22 Actually, for technical reasons, this definition is not totally satisfying and the correct one is: the depth at significance level l of a string s is the least time required by a universal Turing machine to generate s by a program that is not compressible itself by more than l bits. We'll ignore this subtlety in the following.23 See for example[Moser, 2008].…”