Medvedev Degrees Research Articles

Strong reducibilities and set theory

We study Medvedev reducibility in the context of set theory — specifically, forcing and large cardinal hypotheses. Answering a question of Hamkins and Li [7], we show that the Medvedev degrees of countable ordinals are far from linearly ordered in multiple ways, our main result here being that there is a club of ordinals which is an antichain with respect to Medvedev reducibility. We then generalize these results to arbitrary “reasonably-definable” reducibilities, under appropriate set-theoretic hypotheses.We then turn from ordinals to general structures. We show that some of the results above yield characterizations of counterexamples to Vaught's conjecture; another applies to all situations, assigning an ordinal to any reasonable class of structures and “measure” on that class. We end by discussing some directions for future research.

Translation-like actions by $\mathbb{Z}$, the subgroup membership problem, and Medvedev degrees of effective subshifts

We show that every infinite, locally finite, and connected graph admits a translation-like action by \mathbb{Z} , and that this action can be taken to be transitive exactly when the graph has either one or two ends. The actions constructed satisfy d(v,v\ast1)\leq3 for every vertex v . This strengthens a theorem by Brandon Seward. We also study the effective computability of translation-like actions on groups and graphs. We prove that every finitely generated infinite group with decidable word problem admits a translation-like action by \mathbb{Z} which is computable and satisfies an extra condition which we call decidable orbit membership problem. As a nontrivial application of our results, we prove that for every finitely generated infinite group with decidable word problem, effective subshifts attain all \Pi_{1}^{0} Medvedev degrees. This extends a classification proved by Joseph Miller for \mathbb{Z}^{d},\,d\geq1 .

On New Notions of Algorithmic Dimension, Immunity, and Medvedev Degree

We prove various results connected together by the common thread of computability theory.First, we investigate a new notion of algorithmic dimension, the inescapable dimension, which lies between the effective Hausdorff and packing dimensions. We also study its generalizations, obtaining an embedding of the Turing degrees into notions of dimension.We then investigate a new notion of computability theoretic immunity that arose in the course of the previous study, that of a set of natural numbers with no co-enumerable subsets. We demonstrate how this notion of $\Pi ^0_1$ -immunity is connected to other immunity notions, and construct $\Pi ^0_1$ -immune reals throughout the high/low and Ershov hierarchies. We also study those degrees that cannot compute or cannot co-enumerate a $\Pi ^0_1$ -immune set.Finally, we discuss a recently discovered truth-table reduction for transforming a Kolmogorov–Loveland random input into a Martin-Löf random output by exploiting the fact that at least one half of such a KL-random is itself ML-random. We show that there is no better algorithm relying on this fact, in the sense that there is no positive, linear, or bounded truth-table reduction which does this. We also generalize these results to the problem of outputting randomness from infinitely many inputs, only some of which are random.Abstract prepared by David J. Webb.E-mail: dwebb@math.hawaii.eduURL: https://arxiv.org/pdf/2209.05659.pdf

Point Degree Spectra of Represented Spaces

Abstract We introduce the point degree spectrum of a represented space as a substructure of the Medvedev degrees, which integrates the notion of Turing degrees, enumeration degrees, continuous degrees and so on. The notion of point degree spectrum creates a connection among various areas of mathematics, including computability theory, descriptive set theory, infinite-dimensional topology and Banach space theory. Through this new connection, for instance, we construct a family of continuum many infinite-dimensional Cantor manifolds with property C whose Borel structures at an arbitrary finite rank are mutually nonisomorphic. This resolves a long-standing question by Jayne and strengthens various theorems in infinite-dimensional topology such as Pol’s solution to Alexandrov’s old problem.

Comparing the degrees of enumerability and the closed Medvedev degrees

We compare the degrees of enumerability and the closed Medvedev degrees and find that many situations occur. There are nonzero closed degrees that do not bound nonzero degrees of enumerability, there are nonzero degrees of enumerability that do not bound nonzero closed degrees, and there are degrees that are nontrivially both degrees of enumerability and closed degrees. We also show that the compact degrees of enumerability exactly correspond to the cototal enumeration 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.

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.

Randomness extraction and asymptotic Hamming distance

We obtain a non-implication result in the Medvedev degrees by studying sequences that are close to Martin-L\"of random in asymptotic Hamming distance. Our result is that the class of stochastically bi-immune sets is not Medvedev reducible to the class of sets having complex packing dimension 1.

A Survey of Mučnik and Medvedev Degrees

AbstractWe survey the theory of Mučnik (weak) and Medvedev (strong) degrees of subsets of ωω with particular attention to the degrees of subsets of ω2. Sections 1–6 present the major definitions and results in a uniform notation. Sections 7–16 present proofs, some more complete than others, of the major results of the subject together with much of the required background material.

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.

Two notes on subshifts

We prove two unrelated results about subshifts. First, we give a condition on the lengths of forbidden words that is sufficient to guarantee that the corresponding subshift is nonempty. The condition implies that, for example, any sequence of binary words of lengths 5, 6, 7, . . . is avoidable. As another application, we derive a result of Durand, Levin and Shen [2, 3] that there are infinite sequences such that every substring has high Kolmogorov complexity. In particular, for any d < 1, there is a b ∈ N and an infinite binary sequence X such that if τ is a substring of X, then τ has Kolmogorov complexity greater than d |τ | − b. The second result says that from the standpoint of computability theory, any behavior possible from an arbitrary effectively closed subset of nN (i.e., a Π1 class) is exhibited by an effectively closed subshift. In technical terms, every Π1 Medvedev degree contains a Π 0 1 subshift. This answers a question of Simpson [10].

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.

Characterizing the Join-Irreducible Medvedev Degrees

We characterize the join-irreducible Medvedev degrees as the degrees of complements of Turing ideals, thereby solving a problem posed by Sorbi. We use this characterization to prove that there are Medvedev degrees above the second-least degree that do not bound any join-irreducible degrees above this second-least degree. This solves a problem posed by Sorbi and Terwijn. Finally, we prove that the filter generated by the degrees of closed sets is not prime. This solves a problem posed by Bianchini and Sorbi.

Topological aspects of the Medvedev lattice

We study the Medvedev degrees of mass problems with distinguished topological properties, such as denseness, closedness, or discreteness. We investigate the sublattices generated by these degrees; the prime ideal generated by the dense degrees and its complement, a prime filter; the filter generated by the nonzero closed degrees and the filter generated by the nonzero discrete degrees. We give a complete picture of the relationships of inclusion holding between these sublattices, these filters, and this ideal. We show that the sublattice of the closed Medvedev degrees is not a Brouwer algebra. We investigate the dense degrees of mass problems that are closed under Turing equivalence, and we prove that the dense degrees form an automorphism base for the Medvedev lattice. The results hold for both the Medvedev lattice on the Baire space and the Medvedev lattice on the Cantor space.

The ∀∃-theory of the effectively closed Medvedev degrees is decidable

We show that there is a computable procedure which, given an ∀∃-sentence \({\varphi}\) in the language of the partially ordered sets with a top element 1 and a bottom element 0, computes whether \({\varphi}\) is true in the Medvedev degrees of \({\Pi^0_1}\) classes in Cantor space, sometimes denoted by \({\mathcal{P}_s}\) .

K-Triviality of Closed Sets and Continuous Functions

We investigate the notion of K-triviality for closed sets and continuous functions in 2ℕ. For every K-trivial degree d, there exists a closed set of degree d and a continuous function of degree d. Every K-trivial closed set contains a K-trivial real. There exists a K-trivial Π10 class with no computable elements. A closed set is K-trivial if and only if it is the set of zeroes of some K-trivial continuous function. We give a density result for the Medvedev degrees of K-trivial Π10 sets. If W ≤TA′, then W can compute a path through every A′-decidable random closed set if and only if W ≡TA′.

A note on universality in multidimensional symbolic dynamics

We show that in the category of effective $Z$ dynamical systems there is a universal system, i.e. one that factors onto every other effective system. In particular, for d $\geq 3$ there exist d-dimensional shifts of finite type which are universal for 1-dimensional subactions of SFTs. On the other hand, we show that there is no universal effective $Z^d$-system for $d>1$, and in particular SFTs cannot be universal for subactions of rank $d>1$. As a consequence, a decrease in entropy and Medvedev degree and periodic data are not sufficient for a factor map to exists between SFTs. We also discuss dynamics of cellular automata on their limit sets and show that (except for the unavoidable presence of a periodic point) they can model a large class of physical systems.