On the polynomial depth of various sets of random strings

Abstract

We introduce a general framework for defining the depth of an infinite binary sequence with respect to a class of observers. We show that our general framework captures all depth notions introduced in computability/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. We use our framework to define 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. We give two natural examples of deep sets, by showing that both the set of Levin random strings and the set of Kolmogorov random strings are monotone poly deep.