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Takayuki Kihara

Publications and source records attributed to Takayuki Kihara.

At least 37 records · Page 2Linked to original sources
arXiv2020

The Brouwer invariance theorems in reverse mathematics

In his book, John Stillwell wrote "finding the exact strength of the Brouwer invariance theorems seems to me one of the most interesting open problems in reverse mathematics." In this article, we solve Stillwell's problem by showing that (some forms of) the Brouwer invariance theorems are equivalent to weak König's lemma over the base system ${\sf RCA}_0$. In particular, there exists an explicit algorithm which, whenever weak König's lemma is false, constructs a topological embedding of $\mathbb{R}^4$ into $\mathbb{R}^3$.

math.LO↗
arXiv2020

Searching for an analogue of ATR in the Weihrauch lattice

There are close similarities between the Weihrauch lattice and the zoo of axiom systems in reverse mathematics. Following these similarities has often allowed researchers to translate results from one setting to the other. However, amongst the big five axiom systems from reverse mathematics, so far ATR_0 has no identified counterpart in the Weihrauch degrees. We explore and evaluate several candidates, and conclude that the situation is complicated.

cs.LO↗
arXiv2019

Wadge-like degrees of Borel bqo-valued functions

We unite two well known generalisations of the Wadge theory. The first one considers more general reducing functions than the continuous functions in the classical case, and the second one extends Wadge reducibility from sets (i.e., $\{0,1\}$-valued functions) to $Q$-valued functions, for a better quasiorder $Q$. In this article, we consider more general reducibilities on the $Q$-valued functions and generalise some results of L. Motto Ros in the first direction and of T. Kihara and A. Montalbán in the second direction: Our main result states that the structure of the $\mathbfΔ^0_α$-degrees of $\mathbfΔ^0_{α+γ}$-measurable $Q$-valued functions is isomorphic to the $\mathbfΔ^0_β$-degrees of $\mathbfΔ^0_{β+γ}$-measurable $Q$-valued functions, and these are isomorphic to the generalized homomorphism order on the $γ$-th iterated $Q$-labeled forests.

math.LO↗
arXiv2019

A comparison of various analytic choice principles

We investigate computability theoretic and descriptive set theoretic contents of various kinds of analytic choice principles by performing detailed analysis of the Medvedev lattice of $Σ^1_1$-closed sets. Among others, we solve an open problem on the Weihrauch degree of the parallelization of the $Σ^1_1$-choice principle on the integers. Harrington's unpublished result on a jump hierarchy along a pseudo-well-ordering plays a key role in solving the problem.

math.LO↗
arXiv2019

Topological reducibilities for discontinuous functions and their structures

In this article, we give a full description of a topological many-one degree structure of real-valued functions, recently introduced by Day-Downey-Westrick. We also point out that their characterization of the Bourgain rank of a Baire-one function of compact Polish domain can be extended to noncompact Polish domain. Finally, we clarify the relationship between the Martin conjecture and Day-Downey-Westrick's topological Turing-like reducibility, also known as parallelized continuous strong Weihrauch reducibility, for single-valued functions: Under the axiom of determinacy, we show that the continuous Weihrauch degrees of parallelizable single-valued functions are well-ordered; and moreover, if $f$ is has continuous Weihrauch rank $α$, then $f'$ has continuous Weihrauch rank $α+1$, where $f'(x)$ is defined as the Turing jump of $f(x)$.

math.LO↗
arXiv2019

Convex choice, finite choice and sorting

We study the Weihrauch degrees of closed choice for finite sets, closed choice for convex sets and sorting infinite sequences over finite alphabets. Our main results are: One, that choice for finite sets of cardinality $i + 1$ is reducible to choice for convex sets in dimension $j$, which in turn is reducible to sorting infinite sequences over an alphabet of size $k + 1$, iff $i \leq j \leq k$. Two, that convex choice in dimension two is not reducible to the product of convex choice in dimension one with itself. Three, that sequential composition of one-dimensional convex choice is not reducible to convex choice in any dimension. The latter solves an open question raised at a Dagstuhl seminar on Weihrauch reducibility in 2015. Our proofs invoke Kleene's recursion theorem, and we describe in some detail how Kleene's recursion theorem gives rise to a technique for proving separations of Weihrauch degrees.

math.LO↗
arXiv2018

On a metric generalization of the $tt$-degrees and effective dimension theory

In this article, we study an analogue of $tt$-reducibility for points in computable metric spaces. We characterize the notion of the metric $tt$-degree in the context of first-level Borel isomorphism. Then, we study this concept from the perspectives of effective topological dimension theory and of effective fractal dimension theory.

math.LO↗
arXiv2017

Point degree spectra of represented spaces

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 non-isomorphic. This provides new examples of Banach algebras of real valued Baire class two functions on metrizable compacta, and strengthen various theorems in infinite dimensional topology such as Pol's solution to Alexandrov's old problem.

math.GN↗
arXiv2017

On the structure of the Wadge degrees of BQO-valued Borel functions

In this article, we give a full description of the Wadge degrees of Borel functions from $ω^ω$ to a better quasi ordering $\mathcal{Q}$. More precisely, for any countable ordinal $ξ$, we show that the Wadge degrees of $\mathbfΔ^0_{1+ξ}$-measurable functions $ω^ω\to\mathcal{Q}$ can be represented by countable joins of the $ξ$-th transfinite nests of $\mathcal{Q}$-labeled well-founded trees.

math.LO↗
arXiv2017

Effective forcing with Cantor manifolds

A set $A$ of integers is called total if there is an algorithm which, given an enumeration of $A$, enumerates the complement of $A$, and called cototal if there is an algorithm which, given an enumeration of the complement of $A$, enumerates $A$. Many variants of totality and cototality have been studied in computability theory. In this note, by an effective forcing construction with strongly infinite dimensional Cantor manifolds, which can be viewed as an effectivization of Zapletal's "half-Cohen" forcing (i.e., the forcing with Henderson compacta), we construct a set of integers whose enumeration degree is cototal, almost total, but neither cylinder-cototal nor telograph-cototal.

math.LO↗
arXiv2016

Borel-piecewise continuous reducibility for uniformization problems

We study a fine hierarchy of Borel-piecewise continuous functions, especially, between closed-piecewise continuity and $G_δ$-piecewise continuity. Our aim is to understand how a priority argument in computability theory is connected to the notion of $G_δ$-piecewise continuity, and then we utilize this connection to obtain separation results on subclasses of $G_δ$-piecewise continuous reductions for uniformization problems on set-valued functions with compact graphs. This method is also applicable for separating various non-constructive principles in the Weihrauch lattice.

math.LO↗
arXiv2016

Decomposing Borel functions using the Shore-Slaman join theorem

Jayne and Rogers proved that every function from an analytic space into a separable metric space is decomposable into countably many continuous functions with closed domains if and only if the preimage of each $F_σ$ set under it is again $F_σ$. Many researchers conjectured that the Jayne-Rogers theorem can be generalized to all finite levels of Borel functions. In this paper, by using the Shore-Slaman join theorem on the Turing degrees, we show the following variant of the Jayne-Rogers theorem at finite and transfinite levels of the hierarchy of Borel functions: For all countable ordinals $α$ and $β$ with $α\leqβ<α\cdot 2$, every function between Polish spaces having small transfinite inductive dimension is decomposable into countably many Baire class $γ$ functions with $\mathbfΔ^0_{β+1}$ domains such that $γ+α\leqβ$ if and only if the preimage of each $\mathbfΣ^0_{α+1}$ set under that function is $\mathbfΣ^0_{β+1}$, and the transformation of a $\mathbfΣ^0_{α+1}$ set into the $\mathbfΣ^0_{β+1}$ preimage is continuous.

math.LO↗
arXiv2016

The uniform Martin's conjecture for many-one degrees

We study functions from reals to reals which are uniformly degree-invariant from Turing-equivalence to many-one equivalence, and compare them "on a cone." We prove that they are in one-to-one correspondence with the Wadge degrees, which can be viewed as a refinement of the uniform Martin's conjecture for uniformly invariant functions from Turing- to Turing-equivalence. Our proof works in the general case of many-one degrees on $\mathcal{Q}^ω$ and Wadge degrees of functions $ω^ω\to\mathcal{Q}$ for any better quasi ordering $\mathcal{Q}$.

math.LO↗
arXiv2016

Dividing by zero - how bad is it, really?

In computable analysis testing a real number for being zero is a fundamental example of a non-computable task. This causes problems for division: We cannot ensure that the number we want to divide by is not zero. In many cases, any real number would be an acceptable outcome if the divisor is zero - but even this cannot be done in a computable way. In this note we investigate the strength of the computational problem "Robust division": Given a pair of real numbers, the first not greater than the other, output their quotient if well-defined and any real number else. The formal framework is provided by Weihrauch reducibility. One particular result is that having later calls to the problem depending on the outcomes of earlier ones is strictly more powerful than performing all calls concurrently. However, having a nesting depths of two already provides the full power. This solves an open problem raised at a recent Dagstuhl meeting on Weihrauch reducibility. As application for "Robust division", we show that it suffices to execute Gaussian elimination.

cs.LO↗
arXiv2016

Turing degrees in Polish spaces and decomposability of Borel functions

We give a partial answer to an important open problem in descriptive set theory, the Decomposability Conjecture for Borel functions on an analytic subset of a Polish space to a separable metrizable space. Our techniques employ deep results from effective descriptive set theory and recursion theory. In fact it is essential to extend several prominent results in recursion theory (\eg the Shore-Slaman Join Theorem) to the setting of Polish spaces. As a by-product we give both positive and negative results on the Martin Conjecture on the degree preserving Borel functions between Polish spaces. Additionally we prove results about the transfinite version as well as the computable version of the Decomposability Conjecture, and we explore the idea of applying the technique of turning Borel-measurable functions into continuous ones.

math.LO↗
arXiv2013

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 $(Π^0_1)_2$-piecewise computability (where $(Π^0_1)_2$ denotes the difference of two $Π^0_1$ sets), error-bounded-learnability is equivalent to finite $Δ^0_2$-piecewise computability, and learnability is equivalent to countable $Π^0_1$-piecewise computability (equivalently, countable $Σ^0_2$-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.

math.LO↗
arXiv2013

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 $Π^0_1$ subset of Cantor space. We shed light on the fine structures inside these Muchnik degrees related to learnability and piecewise computability. As for nonempty $Π^0_1$ subsets of Cantor space, we show the existence of a finite-$Δ^0_2$-piecewise degree containing infinitely many finite-$(Π^0_1)_2$-piecewise degrees, and a finite-$(Π^0_2)_2$-piecewise degree containing infinitely many finite-$Δ^0_2$-piecewise degrees (where $(Π^0_n)_2$ denotes the difference of two $Π^0_n$ sets), whereas the greatest degrees in these three "finite-$Γ$-piecewise" degree structures coincide. Moreover, as for nonempty $Π^0_1$ subsets of Cantor space, we also show that every nonzero finite-$(Π^0_1)_2$-piecewise degree includes infinitely many Medvedev (i.e., one-piecewise) degrees, every nonzero countable-$Δ^0_2$-piecewise degree includes infinitely many finite-piecewise degrees, every nonzero finite-$(Π^0_2)_2$-countable-$Δ^0_2$-piecewise degree includes infinitely many countable-$Δ^0_2$-piecewise degrees, and every nonzero Muchnik (i.e., countable-$Π^0_2$-piecewise) degree includes infinitely many finite-$(Π^0_2)_2$-countable-$Δ^0_2$-piecewise degrees. Indeed, we show that any nonzero Medvedev degree and nonzero countable-$Δ^0_2$-piecewise degree of a nonempty $Π^0_1$ 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 are Brouwerian, where $Γ$ is any of the Wadge classes mentioned above.

math.LO↗