Deep Classes

Abstract

A set of infinite binary sequences C ⊆ 2 N is negligible if there is no partial probabilistic algorithm that produces an element of this set with positive probability. The study of negligibility is of particular interest in the context of Π 0 1 classes. In this paper, we introduce the notion of depth for Π 0 1 classes, which is a stronger form of negligibility. Whereas a negligible Π 0 1 class C has the property that one cannot probabilistically compute a member of C with positive probability, a deep Π 0 1 class C has the property that one cannot probabilistically compute an initial segment of a member of C with high probability. That is, the probability of computing a length n initial segment of a deep Π 0 1 class converges to 0 effectively in n.We prove a number of basic results about depth, negligibility, and a variant of negligibility that we call tt-negligibility. We provide a number of examples of deep Π 0 1 classes that occur naturally in computability theory and algorithmic randomness. We also study deep classes in the context of mass problems, examine the relationship between deep classes and certain lowness notions in algorithmic randomness, and establish a relationship between members of deep classes and the amount of mutual information with Chaitin's Ω.1991 Mathematics Subject Classification. 03D32, 68Q30, 03D80. Key words and phrases. computability theory, algorithmic randomness, Π 0 1 classes, probabilistic computation.Both authors acknowledge the support of the John Templeton Foundation through the grant "Structure and Randomness in the Theory of Computation."Depth can be viewed as a strong form of negligibility. A deep Π 0 1 class C has the property that producing an initial segment of some member of C is in some sense maximally difficult: the probability of obtaining such an initial segment not only converges to 0 as we consider longer initial segments of members of C, but this convergence is fast, as we can effectively bound the rate of convergence. That initial segments of members of deep classes are so difficult to produce via probabilistic computation reflects the fact that members of deep classes are highly structured: initial segments of these sequences cannot be successfully produced by any combination of Turing machine and random oracle.Although the notion of depth is isolated for the first time in the present study, it is implicitly used in the work of both Levin [22] and Stephan [32]. In fact, in each of these papers one can extract a proof that the consistent completions of Peano arithmetic form a deep class, a result that we reprove in Section 7.The primary goals of this paper are (1) to prove a number of basic results about deep Π 0 1 classes, (2) to determine the exact relationship between negligibility and depth (and a related notion we call tt-negligibility), and (3) to provide a number of examples of deep Π 0 1 classes that occur naturally in computability theory and algorithmic randomness.The outline of this paper is as follows: In Section 2, we provide the technical background...

Highness properties close to PA-completeness

“…Take a D-computable subtree S Ď T with no dead ends such that λprSsq ą 0. Of course, every infinite path through S in Martin-Löf random, and since S has no dead ends, there are paths computable from S ď T D. We have proved: The fact that a Martin-Löf random sequence need not compute a slow growing DNC function was mentioned above; it follows from Bienvenu and Porter's [4] work on deep Π 0 1 classes. Greenberg and Miller [11] proved that slow growing DNC functions do not always compute a Martin-Löf random.…”

“…N has a DNC 2 atom. 4 A martingale computes all of its atoms, so in this case, N has PA degree. 3 In other words, for τ ď σ, we let M s`1 pτ q ´Mspτ q " 2 |τ |´n ; to preserve the martingale property, for τ ą σ, we let M s`1 pτ q ´Mspτ q " 2 |σ|´n .…”

“…3 In other words, for τ ď σ, we let M s`1 pτ q ´Mspτ q " 2 |τ |´n ; to preserve the martingale property, for τ ą σ, we let M s`1 pτ q ´Mspτ q " 2 |σ|´n . 4 That is, the associated measure µpσq " 2 ´|σ| M pσq has a DNC 2 atom.…”

Diagonally non-computable functions and fireworks

“…The faster h grows, the easier it is to obtain an element of DNC h . And indeed, depending on the growth rate of h the class DNC h can be negligible or non-negligible (more specifically, for h computable, DNC h is negligible if and only if n 1/h(n) = ∞, this is an unpublished result due to J. Miller, see [3] for a proof).This notion relativizes to an arbitrary oracle X: a DNC X function is a function f such that f (n) = ϕ X n (n) for all n. Likewise, we setOf course, the stronger the oracle X, the harder it is to compute a DNC X function.In this paper, we study the role of DNC functions in the setting of the Levin-V'yugin algebra, which is the algebra of Turing invariant Borel sets 'modulo negligibility'. That is, for two Turing-invariant sets A and B we write A ⊑ B if A B is negligible.…”

“…Following the tradition of colourful terminology in computability theory (Lerman's pinball machine, Nies' decanter argument...), we propose to refer to this method as a fireworks argument, the precise template of which will be recalled in the next section.The dichotomy between negligibility and non-negligibility has received quite a lot of attention in recent years. We refer the reader to [2,3,21] for a panorama of the existing results in this direction.One class of particular interest in the study of negligibility is the class of diagonally non-computable functions, or 'DNC functions' for short. It consists of the total functions f : N → N such that f (n) = ϕ n (n) for all n, where (ϕ n ) n is a standard enumeration of partial computable functions from N to N. By definition, there is no computable DNC function.…”