Diagonally non-computable functions and fireworks

Abstract

A set C of reals is said to be negligible if there is no probabilistic algorithm which generates a member of C with positive probability. Various classes have been proven to be negligible, for example the Turing upper-cone of a non-computable real, the class of coherent completions of Peano Arithmetic or the class of reals of minimal Turing degree. One class of particular interest in the study of negligibility is the class of diagonally noncomputable (DNC) functions, proven by Kučera to be non-negligible in a strong sense: every Martin-Löf random real computes a DNC function. Ambos-Spies et al. showed that the converse does not hold: there are DNC functions which compute no Martin-Löf random real. In this paper, we show that the set of such DNC functions is in fact non-negligible. More precisely, we prove that for every sufficiently fast-growing computable h, every 2random real computes an h-bounded DNC function which computes no Martin-Löf random real. Further, we show that the same holds for the set of reals which compute a DNC function but no bounded DNC function. The proofs of these results use a combination of a technique due to Kautz (which, following a metaphor of Shen, we like to call a 'fireworks argument') and bushy tree forcing, which is the canonical forcing notion used in the study of DNC functions.1 Very similar ideas were used by V'yugin in [22,23] showed that the following are non-negligible: the class of hyperimmune sets, the class of 1generic reals, and the class of reals of CEA degrees. All three results are variations of the same technique. However, the way Kautz presents his technique is quite abstract and in some sense hides its true spirit, namely that what is being used is a probabilistic algorithm. In the paper [19], Rumyantsev and Shen give a more explicit and intuitive explanation of the underlying algorithm, using the metaphor of a fireworks shop in which a customer is trying to either buy a box of good fireworks, or expose the vendor by opening a flawed one, and uses a probabilistic algorithm to maximize his chances of success. 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. On the other hand, as showed by Kučera [15], the class of DNC functions is non-negligible. If we take the point of view of probabilistic algorithms, th...

Forcing With Bushy Trees

“…Dorais, Hirst, and Shafer [5], building on the aforementioned work of Ambos-Spies, et al [1], have shown that the reverse mathematics principle "there exists a k such that for every function f there is a k-bounded function that is DNC relative f " does not imply the existence of a {0, 1}-valued DNC function in the absence of Σ 0 2 induction, answering a question of Simpson. Bienvenu and Patey [3], by combining bushy tree arguments with probabilistic ones, have shown that there is a computable function h such that every 2-random real computes an h-bounded DNC function that computes no Martin-Löf random real.…”

On the Uniform Computational Content of the Baire Category Theorem

“…We prove that the jump of A − (X) can be described with the cluster point representation. This enables us to prove in Section 4 that (for perfect Polish spaces) 3 . In other words, the change of the input space from A − (X) to A + (X) can equivalently be expressed by an application of the jump.…”

“…On the other hand, BCT 0 and BCT 2 are both ω-indiscriminative and hence also indiscriminative: since BCT 0 and BCT 2 are each densely realized, 3 by the Baire Category Theorem (1.2) itself, this follows from [14,Proposition 4.3]. Moreover, 3 A notion introduced in [14], which roughly speaking, says that the image of BCT 0 and BCT 2 is densely covered over all realizers. every jump of BCT 0 or BCT 2 is also ω-indiscriminative, since it is merely a property of the image.…”

“…Upon input of q, r and k the probabilistic algorithm works in steps s = 0, 1, 2, ... and computes a sequence p by producing longer and longer prefixes v s of p. Initially the prefix v 0 is the empty sequence. We also use two sequences 5 The fireworks technique is due to Andrei Rumyantsev and Alexander Shen [31]; the fact that it can be used to prove that every 2-random degree bounds a 1-generic has been communicated to us by Laurent Bienvenu; see also [3]. of natural number programming variables (c i ) i and (l i ) i , which are initially all set to zero.…”

On the Interplay Between Effective Notions of Randomness and Genericity

“…A more intuitive proof can be given using a fireworks argument (which takes its name from the presentation by Rumyantsev and Shen [RS14] with an analogy about purchasing fireworks from a purportedly corrupt fireworks salesman), an approach that is more suitable for our purposes. The mechanics of fireworks arguments are already thoroughly explained in [RS14] and [BP17], but we shall review them for the sake of completeness.…”