Effective decomposition of σ-continuous Borel functions

Debs, Gabriel

doi:10.4064/fm224-2-4

Cited by 3 publications

(6 citation statements)

References 7 publications

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“…In this case we write y ≤ M x. 6 The continuous degree of x is defined as usual to be its equivalence class under…”

Section: Generalized Turing Degree Theorymentioning

confidence: 99%

“…In this case we write y ≤ M x. 6 The continuous degree of x is defined as usual to be its equivalence class under ≡ M := ≤ M ∩ ≥ M . We say that a point x ∈ X is total if there is z ∈ 2 ω such that x ≡ M z.…”

Section: Generalized Turing Degree Theorymentioning

confidence: 99%

“…That is, if x, y ∈ N then x ≤ M y if and only if x ≤ T y. 6 The relation ≤ M can be characterized in terms of the Medvedev reducibility between sets of names of x and y, see [16]. This is the justification for our notation.…”

Section: Generalized Turing Degree Theorymentioning

confidence: 99%

“…In recent years, researchers have made remarkable progress on extending the Jayne-Rogers theorem (cf. [3,4,6,11,15,21,22,23,25,29]). One prominent result among them is by Semmes [25], who showed that a function f : N → N on the Baire space N := ω ω is a second-level Borel function (i.e., f −1 Σ 0 3 ⊆ Σ 0 3 ) if and only if it is decomposable into countably many continuous functions with G δ domains, and that a function f : N → N is a Borel function at level (1,2)…”

Section: Introductionmentioning

confidence: 99%

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Turing degrees in Polish spaces and decomposability of Borel functions

Gregoriades

Kihara

2020

J. Math. Log.

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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 (e.g. 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.

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“…In this case we write y ≤ M x. 6 The continuous degree of x is defined as usual to be its equivalence class under…”

Section: Generalized Turing Degree Theorymentioning

confidence: 99%

Section: Generalized Turing Degree Theorymentioning

confidence: 99%

Section: Generalized Turing Degree Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Turing degrees in Polish spaces and decomposability of Borel functions

Gregoriades

Kihara

2020

J. Math. Log.

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“…Although Luzin's problem has been solved negatively, in recent years, the notion of σ-continuity itself has received increasing attention in descriptive set theory and related areas. In these areas, researchers have accomplished an enormous amount of work connecting finite-level Borel functions and Borel-piecewise continuous functions (see [9,10,12,15,24,26,29,36,46,48,51]). These works have also led us to the discovery that the notion of piecewise continuity plays a crucial role in the study of the hierarchy of Borel isomorphisms (see [23,30]).…”

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confidence: 99%

Borel-Piecewise Continuous Reducibility for Uniformization Problems

Kihara

2017

Log.Meth.Comput.Sci.

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Abstract. 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 setvalued functions with compact graphs. This method is also applicable for separating various non-constructive principles in the Weihrauch lattice.

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Comparing Computability in Two Topologies

Amir¹,

Hoyrup²

2023

J. symb. log.

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Computable analysis provides ways of representing points in a topological space, and therefore of defining a notion of computable points of the space. In this article, we investigate when two topologies on the same space induce different sets of computable points.We first study a purely topological version of the problem, which is to understand when two topologies are not σ-homeomorphic. We obtain a characterization leading to an effective version, and we prove that two topologies satisfying this condition induce different sets of computable points. Along the way, we propose an effective version of the Baire category theorem which captures the construction technique, and enables one to build points satisfying properties that are co-meager w.r.t. a topology, and are computable w.r.t. another topology. Finally, we generalize the result to three topologies and give an application to prove that certain sets do not have computable type, i.e. have a homeomorphic copy that is semicomputable but not computable.

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Effective decomposition of σ-continuous Borel functions

Abstract: We prove that if a ∆ 1 1 function f with Σ 1 1 domain X is σ-continuous then one can find a ∆ 1 1 covering (An)n∈ω of X such that f |An is continuous for all n. This is an effective version of a recent result by Pawlikowski and Sabok, generalizing an earlier result of Solecki.

Cited by 3 publications

References 7 publications

Turing degrees in Polish spaces and decomposability of Borel functions

Turing degrees in Polish spaces and decomposability of Borel functions

Borel-Piecewise Continuous Reducibility for Uniformization Problems

Comparing Computability in Two Topologies

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