K-trivial Sets Research Articles

Martin–Löf reducibility and cost functions

Martin-Löf (ML)-reducibility compares the complexity of K-trivial sets of natural numbers by examining the Martin-Löf random sequences that compute them. One says that a K-trivial set A is ML-reducible to a K-trivial set B if every ML-random computing B also computes A. We show that every K-trivial set is computable from a c.e. set of the same ML-degree. We investigate the interplay between ML-reducibility and cost functions, which are used to both measure the number of changes in a computable approximation, and the type of null sets intended to capture ML-random sequences. We show that for every cost function there is a c.e. set that is ML-complete among the sets obeying it. We characterise the K-trivial sets computable from a fragment of the left-c.e. random real Ω given by a computable set of bit positions. This leads to a new characterisation of strong jump-traceability.

Nullifying randomness and genericity using symmetric difference

For a class C of sets, let us say that a set A is Cstabilising if A△X∈C for every X∈C. We prove that the Martin–Löf stabilising sets are exactly the K-trivial sets, as are the weakly 2-random stabilising sets. We also show that the 1-generic stabilising sets are exactly the computable sets.

COARSE REDUCIBILITY AND ALGORITHMIC RANDOMNESS

AbstractA coarse description of a set A ⊆ ω is a set D ⊆ ω such that the symmetric difference of A and D has asymptotic density 0. We study the extent to which noncomputable information can be effectively recovered from all coarse descriptions of a given set A, especially when A is effectively random in some sense. We show that if A is 1-random and B is computable from every coarse description D of A, then B is K-trivial, which implies that if A is in fact weakly 2-random then B is computable. Our main tool is a kind of compactness theorem for cone-avoiding descriptions, which also allows us to prove the same result for 1-genericity in place of weak 2-randomness. In the other direction, we show that if $A \le _{{\rm{T}}} \emptyset {\rm{'}}$ is a 1-random set, then there is a noncomputable c.e. set computable from every coarse description of A, but that not all K-trivial sets are computable from every coarse description of some 1-random set. We study both uniform and nonuniform notions of coarse reducibility. A set Y is uniformly coarsely reducible to X if there is a Turing functional Φ such that if D is a coarse description of X, then ΦD is a coarse description of Y. A set B is nonuniformly coarsely reducible to A if every coarse description of A computes a coarse description of B. We show that a certain natural embedding of the Turing degrees into the coarse degrees (both uniform and nonuniform) is not surjective. We also show that if two sets are mutually weakly 3-random, then their coarse degrees form a minimal pair, in both the uniform and nonuniform cases, but that the same is not true of every pair of relatively 2-random sets, at least in the nonuniform coarse degrees.

Comparing the Medvedev and Turing degrees of Π01 classes

Every co-c.e. closed set (Π01 class) in Cantor space is represented by a co-c.e. tree. Our aim is to clarify the interaction between the Medvedev and Muchnik degrees of co-c.e. closed subsets of Cantor space and the Turing degrees of their co-c.e. representations. Among other results, we present the following theorems: if v and w are different c.e. degrees, then the collection of the Medvedev (Muchnik) degrees of all Π01 classes represented by v and the collection represented by w are also different; the ideals generated from such collections are also different; the collections of the Medvedev and Muchnik degrees of all Π01 classes represented by incomplete co-c.e. sets are upward dense; the collection of all Π01 classes represented by K-trivial sets is Medvedev-bounded by a single Π01 class represented by an incomplete co-c.e. set; and the Π01 classes have neither nontrivial infinite suprema nor infima in the Medvedev lattice.

Denjoy, Demuth and density

We consider effective versions of two classical theorems, the Lebesgue density theorem and the Denjoy–Young–Saks theorem. For the first, we show that a Martin-Löf random real z ∈ [0, 1] is Turing incomplete if and only if every effectively closed class 𝒞 ⊆ [0, 1] containing z has positive density at z. Under the stronger assumption that z is not LR-hard, we show that every such class has density one at z. These results have since been applied to solve two open problems on the interaction between the Turing degrees of Martin-Löf random reals and K-trivial sets: the noncupping and covering problems. We say that f : [0, 1] → ℝ satisfies the Denjoy alternative at z ∈ [0, 1] if either the derivative f′(z) exists, or the upper and lower derivatives at z are +∞ and -∞, respectively. The Denjoy–Young–Saks theorem states that every function f : [0, 1] → ℝ satisfies the Denjoy alternative at almost every z ∈ [0, 1]. We answer a question posed by Kučera in 2004 by showing that a real z is computably random if and only if every computable function f satisfies the Denjoy alternative at z. For Markov computable functions, which are only defined on computable reals, we can formulate the Denjoy alternative using pseudo-derivatives. Call a real zDA-random if every Markov computable function satisfies the Denjoy alternative at z. We considerably strengthen a result of Demuth (Comment. Math. Univ. Carolin.24(3) (1983) 391–406) by showing that every Turing incomplete Martin-Löf random real is DA-random. The proof involves the notion of nonporosity, a variant of density, which is the bridge between the two themes of this paper. We finish by showing that DA-randomness is incomparable with Martin-Löf randomness.

Elementary differences between the degrees of unsolvability and degrees of compressibility

Given two infinite binary sequences A , B we say that B can compress at least as well as A if the prefix-free Kolmogorov complexity relative to B of any binary string is at most as much as the prefix-free Kolmogorov complexity relative to A , modulo a constant. This relation, introduced in Nies (2005) [14] and denoted by A ≤ L K B , is a measure of relative compressing power of oracles, in the same way that Turing reducibility is a measure of relative information. The equivalence classes induced by ≤ L K are called L K degrees (or degrees of compressibility) and there is a least degree containing the oracles which can only compress as much as a computable oracle, also called the ‘low for K ’ sets. A well-known result from Nies (2005) [14] states that these coincide with the K-trivial sets, which are the ones whose initial segments have minimal prefix-free Kolmogorov complexity. We show that with respect to ≤ L K , given any non-trivial Δ 2 0 sets X , Y there is a computably enumerable set A which is not K-trivial and it is below X , Y . This shows that the local structures of Σ 1 0 and Δ 2 0 Turing degrees are not elementarily equivalent to the corresponding local structures in the LK degrees. It also shows that there is no pair of sets computable from the halting problem which forms a minimal pair in the LK degrees; this is sharp in terms of the jump, as it is known that there are sets computable from 0 ″ which form a minimal pair in the LK degrees. We also show that the structure of LK degrees below the LK degree of the halting problem is not elementarily equivalent to the Δ 2 0 or Σ 1 0 structures of LK degrees. The proofs introduce a new technique of permitting below a Δ 2 0 set that is not K -trivial, which is likely to have wider applications.

Using random sets as oracles

AbstractLet R be a notion of algorithmic randomness for individual subsets of ℕ. A set B is a base for R randomness if there is a Z ≥ T B such that Z is R random relative to B. We show that the bases for 1-randomness are exactly the K-trivial sets, and discuss several consequences of this result. On the other hand, the bases for computable randomness include every Δ 2 0 set that is not diagonally noncomputable, but no set of PA-degree. As a consequence, an n-c.e. set is a base for computable randomness if and only if it is Turing incomplete.