Muchnik Degrees Research Articles

Mass problems and density

Recall that [Formula: see text] is the lattice of Muchnik degrees of nonempty effectively compact sets in Euclidean space. We solve a long-standing open problem by proving that [Formula: see text] is dense, i.e. satisfies [Formula: see text]. Our proof combines an oracle construction with hyperarithmetical theory.

Mass problems and intuitionistic higher-order logic

In this paper we study a model of intuitionistic higher-order logic which we call the Muchnik topos. The Muchnik topos may be defined briefly as the category of sheaves of sets over the topological space consisting of the Turing degrees, where the Turing cones form a base for the topology. We note that our Muchnik topos interpretation of intuitionistic mathematics is an extension of the well known Kolmogorov/Muchnik interpretation of intuitionistic propositional calculus via Muchnik degrees, i.e., mass problems under weak reducibility. We introduce a new sheaf representation of the intuitionistic real numbers, the Muchnik reals, which are different from the Cauchy reals and the Dedekind reals. Within the Muchnik topos we obtain a choice principle (8x9y A(x,y)) ) 9w 8xA(x,wx) and a bound

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.

Inside the Muchnik degrees II: The degree structures induced by the arithmetical hierarchy of countably continuous functions

It is known that infinitely many Medvedev degrees exist inside the Muchnik degree of any nontrivial Π10 subset of Cantor space. We shed light on the fine structures inside these Muchnik degrees related to learnability and piecewise computability. As for nonempty Π10 subsets of Cantor space, we show the existence of a finite-Δ20-piecewise degree containing infinitely many finite-(Π10)2-piecewise degrees, and a finite-(Π20)2-piecewise degree containing infinitely many finite-Δ20-piecewise degrees (where (Πn0)2 denotes the difference of two Πn0 sets), whereas the greatest degrees in these three “finite-Γ-piecewise” degree structures coincide. Moreover, as for nonempty Π10 subsets of Cantor space, we also show that every nonzero finite-(Π10)2-piecewise degree includes infinitely many Medvedev (i.e., one-piecewise) degrees, every nonzero countable-Δ20-piecewise degree includes infinitely many finite-piecewise degrees, every nonzero finite-(Π20)2-countable-Δ20-piecewise degree includes infinitely many countable-Δ20-piecewise degrees, and every nonzero Muchnik (i.e., countable-Π20-piecewise) degree includes infinitely many finite-(Π20)2-countable-Δ20-piecewise degrees. Indeed, we show that any nonzero Medvedev degree and nonzero countable-Δ20-piecewise degree of a nonempty Π10 subset of Cantor space have the strong anticupping properties. Finally, we obtain an elementary difference between the Medvedev (Muchnik) degree structure and the finite-Γ-piecewise degree structure of all subsets of Baire space by showing that none of the finite-Γ-piecewise structures is Brouwerian, where Γ is any of the Wadge classes mentioned above.

Inside the Muchnik degrees I: Discontinuity, learnability and constructivism

Every computable function has to be continuous. To develop computability theory of discontinuous functions, we study low levels of the arithmetical hierarchy of nonuniformly computable functions on Baire space. First, we classify nonuniformly computable functions on Baire space from the viewpoint of learning theory and piecewise computability. For instance, we show that mind-change-bounded learnability is equivalent to finite (Π10)2-piecewise computability (where (Π10)2 denotes the difference of two Π10 sets), error-bounded learnability is equivalent to finite Δ20-piecewise computability, and learnability is equivalent to countable Π10-piecewise computability (equivalently, countable Σ20-piecewise computability). Second, we introduce disjunction-like operations such as the coproduct based on BHK-like interpretations, and then, we see that these operations induce Galois connections between the Medvedev degree structure and associated Medvedev/ Muchnik-like degree structures. Finally, we interpret these results in the context of the Weihrauch degrees and Wadge-like games.

Medvedev degrees of two-dimensional subshifts of finite type

AbstractIn this paper, we apply some fundamental concepts and results from recursion theory in order to obtain an apparently new example in symbolic dynamics. Two setsXandYare said to beMedvedev equivalentif there exist partial computable functionals fromXintoYand vice versa. TheMedvedev degreeofXis the equivalence class ofXunder Medvedev equivalence. There is an extensive recursion-theoretic literature on the lattices ℰsand ℰwof Medvedev degrees and Muchnik degrees of non-empty effectively closed subsets of {0,1}ℕ. We now prove that ℰsand ℰwconsist precisely of the Medvedev degrees and Muchnik degrees of two-dimensional subshifts of finite type. We apply this result to obtain an infinite collection of two-dimensional subshifts of finite type which are, in a certain sense, mutually incompatible.

Effectively closed mass problems and intuitionism

Let Ps denote the lattice of all strong degrees (or Medvedev degrees) of nonempty Π10 (or effectively closed) subsets of the Cantor space 2ω, and let Pw denote the lattice of all weak degrees (or Muchnik degrees) of nonempty Π10 subsets of 2ω. We show that Ps is not a Brouwer algebra and Pw is not a Heyting algebra. These give solutions to problems presented by Simpson in 2008 and by Terwijn in 2006.

Coding true arithmetic in the Medvedev degrees of [formula omitted] classes

Let E s denote the lattice of Medvedev degrees of non-empty Π 1 0 subsets of 2 ω , and let E w denote the lattice of Muchnik degrees of non-empty Π 1 0 subsets of 2 ω . We prove that the first-order theory of E s as a partial order is recursively isomorphic to the first-order theory of true arithmetic. Our coding of arithmetic in E s also shows that the Σ 3 0 -theory of E s as a lattice and the Σ 4 0 -theory of E s as a partial order are undecidable. Moreover, we show that the degree of E s as a lattice is 0 ‴ in the sense that 0 ‴ computes a presentation of E s and that every presentation of E s computes 0 ‴ . Finally, we show that the Σ 3 0 -theory of E w as a lattice and the Σ 4 0 -theory of E w as a partial order are undecidable.

Coding true arithmetic in the Medvedev and Muchnik degrees

AbstractWe prove that the first-order theory of the Medvedev degrees, the first-order theory of the Muchnik degrees, and the third-order theory of true arithmetic are pairwise recursively isomorphic (obtained independently by Lewis, Nies, and Sorbi [7]). We then restrict our attention to the degrees of closed sets and prove that the following theories are pairwise recursively isomorphic: the first-order theory of the closed Medvedev degrees, the first-order theory of the compact Medvedev degrees, the first-order theory of the closed Muchnik degrees, the first-order theory of the compact Muchnik degrees, and the second-order theory of true arithmetic. Our coding methods also prove that neither the closed Medvedev degrees nor the compact Medvedev degrees are elementarily equivalent to either the closed Muchnik degrees or the compact Muchnik degrees.

Mass problems associated with effectively closed sets

The study of mass problems and Muchnik degrees was originally motivated by Kolmogorov's non-rigorous 1932 interpretation of intuitionism as a calculus of problems. The purpose of this paper is to summarize recent investigations into the lattice of Muchnik degrees of nonempty effectively closed sets in Euclidean space. Let $\mathcal{E}_\mathrm{w}$ be this lattice. We show that $\mathcal{E}_\mathrm{w}$ provides an elegant and useful framework for the classification of certain foundationally interesting problems which are algorithmically unsolvable. We exhibit some specific degrees in $\mathcal{E}_\mathrm{w}$ which are associated with such problems. In addition, we present some structural results concerning the lattice $\mathcal{E}_\mathrm{w}$. One of these results answers a question which arises naturally from the Kolmogorov interpretation. Finally, we show how $\mathcal{E}_\mathrm{w}$ can be applied in symbolic dynamics, toward the classification of tiling problems and $\boldsymbol{Z}^d$-subshifts of finite type.

Mass Problems and Measure-Theoretic Regularity

AbstractA well known fact is that every Lebesgue measurable set is regular, i.e., it includes an Fσ set of the same measure. We analyze this fact from a metamathematical or foundational standpoint. We study a family of Muchnik degrees corresponding to measuretheoretic regularity at all levels of the effective Borel hierarchy. We prove some new results concerning Nies's notion of LR-reducibility. We build some ω-models of RCA0 which are relevant for the reverse mathematics of measure-theoretic regularity.

Mass Problems and Intuitionism

Let Pw be the lattice of Muchnik degrees of nonempty � 0 subsets of 2 ! . The lattice Pw has been studied extensively in previous publications. In this note we prove that the lattice Pw is not Brouwerian.

Hyperimmunity in 2\sp ℕ

We investigate the notion of hyperimmunity with respect to how it can be applied to Π{\sp 0}{\sb 1} classes and their Muchnik degrees. We show that hyperimmunity is a strong enough concept to prove the existence of Π{\sp 0}{\sb 1} classes with intermediate Muchnik degree—in contrast to Post's attempts to construct intermediate c.e. degrees.

An extension of the recursively enumerable Turing degrees

Consider the countable semilattice ℛT consisting of the recursively enumerable Turing degrees. Although ℛT is known to be structurally rich, a major source of frustration is that no specific, natural degrees in ℛT have been discovered, except the bottom and top degrees, 0 and 0′. In order to overcome this difficulty, we embed ℛT into a larger degree structure which is better behaved. Namely, consider the countable distributive lattice 𝒫w consisting of the weak degrees (also known as Muchnik degrees) of mass problems associated with non-empty Π01 subsets of 2ω. It is known that 𝒫w contains a bottom degree 0 and a top degree 1 and is structurally rich. Moreover, 𝒫w contains many specific, natural degrees other than 0 and 1. In particular, we show that in 𝒫w one has 0 < d < r1 < ∈ f(r2, 1) < 1. Here, d is the weak degree of the diagonally non-recursive functions, and rn is the weak degree of the n-random reals. It is known that r1 can be characterized as the maximum weak degree of a Π01 subset of 2ω of positive measure. We now show that∈f(r2, 1) can be characterized as the maximum weak degree of a Π01 subset of 2ω, the Turing upward closure of which is of positive measure. We exhibit a natural embedding of ℛT into 𝒫w which is one-to-one, preserves the semilattice structure of ℛT, carries 0 to 0, and carries 0′ to 1. Identifying ℛT with its image in 𝒫w, we show that all of the degrees in ℛT except 0 and 1 are incomparable with the specific degrees d, r1, and∈f(r2, 1) in 𝒫w.