arXiv subjects

Keng Meng Ng

Publications and source records attributed to Keng Meng Ng.

At least 19 recordsLinked to original sources

arXiv2026

Conjunctive reducibilities and completeness

In this article we study the notion of completeness for conjunctive reducibilities. We investigate the relationship between $c$-completeness and $r$-completeness of computably enumerable (c.e.) sets with respect to various strong reducibilities $\le_r$. By using simplicity properties of sets, we prove that there exist c.e. sets that are simultaneously $Q$-complete and $bd$-complete, yet fail to be $c$-complete. Similarly, there exist c.e. sets that are simultaneously $Q$-complete and $bwtt$-complete (respectively, $btt$-complete) but not $c$-complete. Furthermore, we study two restrictions of $c$-reducibility, namely $c_1$- and $c_{1,N}$-reducibility, and show that they are distinct on the c.e. sets. Nevertheless, we prove that the notions of completeness for $c$, $c_1$, and $c_{1,N}$ coincide.

math.LO↗

arXiv2026

sp-Homogeneous Linear Orderings

We study linear orderings expanded by functions for successor and predecessor. The successor and predecessor on linear orderings capture the relatively intrinsically computably enumerable information about orderings in much the same way that dependence captures that for vector spaces. In particular, the sp-homogeneous and weakly sp-homogeneous linear orderings are those which are (ultra-)homogeneous or weakly homogeneous with this additional structure. We demonstrate that these orderings are always relatively $Δ_4$ categorical and determine exactly which ones are (uniformly) relatively $Δ_3$ categorical. We also provide a classification for sp-homogeneity and weak sp-homogeneity. We establish that this is the best possible classification by showing that the set of sp-homogeneous linear orderings is $Π_5^0$ complete, and that the set of weakly sp-homogeneous linear orderings is $Σ_6^0$ complete. These results are obtained in two different ways, one using a hands-on computability theoretic approach and another using more abstract descriptive set theory.

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arXiv2026

Primitive recursive categoricity spectra

We study the primitive recursive analogue of computable categoricity spectra for various natural classes of structures. We show that these notions coincide for all relatively $Δ_{2}^{0}$-categorical equivalence structures and linear orders, relatively $Δ_{3}^{0}$-categorical Boolean algebras, and computably categorical tree as partial orders.

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arXiv2026

Primitive recursive categoricity spectra of functional structures

For the notion of degree of categoricity, we study an analogous notion for punctual structures. We show that such notions coincide for non-$Δ_{1}^{0}$-categorical injection structures, and construct an example of a $Δ_{1}^{0}$-categorical injection structure for which these notions differ. Additionally, we also show that in every non-zero c.e.~Turing degree, there exists a PR-degree that is low for punctual isomorphism (to be defined), and also a PR-degree that is a degree of punctual categoricity.

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arXiv2025

Computable classifications of continuous, transducer, and regular functions

We develop a systematic algorithmic framework that unites global and local classification problems using index sets. We prove that the classification problem for continuous (binary) regular functions among almost everywhere linear, pointwise linear-time Lipschitz functions is $Σ^0_2$-complete. (Every regular function is pointwise linear-time Lipschitz.) We show that a function $f\colon [0,1] \rightarrow \mathbb{R}$ is (binary) transducer if and only if it is continuous regular. As one of many consequences, our $Σ^0_2$-completeness result covers the class of transducer functions as well. Finally, we show that the Banach space $C[0,1]$ of real-valued continuous functions admits an arithmetical classification among separable Banach spaces. Our proofs combine methods of abstract computability theory, automata theory, and functional analysis.

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arXiv2024

The singleton degrees of the $Σ^0_2$ sets are not dense

Answering an open question raised by Cooper, we show that there exist $Δ^0_2$ sets $D$ and $E$ such that the singleton degree of $E$ is a minimal cover of the singleton degree of $D$. This shows that the $Σ^{0}_{2}$ singleton degrees, and the $Δ^{0}_{2}$ singleton degrees, are not dense (and consequently the $Π^0_2$ $Q$-degrees, and the $Δ^{0}_{2}$ $Q$-degrees, are not dense). Moreover $D$ and $E$ can be built to lie in the same enumeration degree.

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arXiv2024

The subTuring degrees

In this article, we introduce a notion of reducibility for partial functions on the natural numbers, which we call subTuring reducibility. One important aspect is that the subTuring degrees correspond to the structure of the realizability subtoposes of the effective topos. We show that the subTuring degrees (that is, the realizability subtoposes of the effective topos) form a dense non-modular (thus, non-distributive) lattice. We also show that there is a nonzero join-irreducible subTuring degree (which implies that there is a realizability subtopos of the effective topos that cannot be decomposed into two smaller realizability subtoposes).

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arXiv2024

The computational content of multidimensional discontinuity

The Weihrauch degrees are a tool to gauge the computational difficulty of mathematical problems. Often, what makes these problems hard is their discontinuity. We look at discontinuity in its purest form, that is, at otherwise constant functions that make a single discontinuous step along each dimension of their underlying space. This is an extension of previous work of Kihara, Pauly, Westrick from a single dimension to multiple dimensions. Among other results, we obtain strict hierarchies in the Weihrauch degrees, one of which orders mathematical problems by the richness of the truth-tables determining how discontinuous steps influence the output.

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arXiv2024

Finite final segments of the d.c.e. Turing degrees

We prove that every finite distributive lattice is isomorphic to a final segment of the d.c.e. Turing degrees (i.e., the degrees of differences of computably enumerable sets). As a corollary, we are able to infer the undecidability of the EAE-theory of the d.c.e. degrees in the language of partial ordering.

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arXiv2022

Computable topological groups

We investigate what it means for a (Hausdorff, second-countable) topological group to be computable. We compare several potential definitions in the literature. We relate these notions with the well-established definitions of effective presentability for discrete and profinite groups, and compare these results with similar results in computable topology. Most of these definitions can be separated by counter-examples. Remarkably, we prove that two such definitions are equivalent for locally compact Polish and abelian Polish groups. More specifically, we prove that in these broad classes of groups, every computable topological group admits a right-c.e.~(upper semi-computable) presentation with a left-invariant metric, and a computable dense sequence of points. In the locally compact case, we also show that if the group is additionally effectively locally compact, then we can produce an effectively proper left-invariant metric.

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arXiv2022

Limit Complexities, Minimal Descriptions, and $n$-Randomness

Let $K$ denote prefix-free Kolmogorov Complexity, and $K^A$ denote it relative to an oracle $A$. We show that for any $n$, $K^{\emptyset^{(n)}}$ is definable purely in terms of the unrelativized notion $K$. It was already known that 2-randomness is definable in terms of $K$ (and plain complexity $C$) as those reals which infinitely often have maximal complexity. We can use our characterization to show that $n$-randomness is definable purely in terms of $K$. To do this we extend a certain ``limsup'' formula from the literature, and apply Symmetry of Information. This extension entails a novel use of semilow sets, and a more precise analysis of the complexity of $Δ_2^0$ sets of mimimal descriptions.

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arXiv2021

Punctual equivalence relations and their (punctual) complexity

The complexity of equivalence relations has received much attention in the recent literature. The main tool for such endeavour is the following reducibility: given equivalence relations $R$ and $S$ on natural numbers, $R$ is computably reducible to $S$ if there is a computable function $f \colon ω\to ω$ that induces an injective map from $R$-equivalence classes to $S$-equivalence classes. In order to compare the complexity of equivalence relations which are computable, researchers considered also feasible variants of computable reducibility, such as the polynomial-time reducibility. In this work, we explore $\mathbf{Peq}$, the degree structure generated by primitive recursive reducibility on punctual equivalence relations (i.e., primitive recursive equivalence relations with domain $ω$). In contrast with all other known degree structures on equivalence relations, we show that $\mathbf{Peq}$ has much more structure: e.g., we show that it is a dense distributive lattice. On the other hand, we also offer evidence of the intricacy of $\mathbf{Peq}$, proving, e.g., that the structure is neither rigid nor homogeneous.

math.LO↗

arXiv2021

Foundations of Online Structure Theory II: The Operator Approach

We introduce a framework for online structure theory. Our approach generalises notions arising independently in several areas of computability theory and complexity theory. We suggest a unifying approach using operators where we allow the input to be a countable object of an arbitrary complexity. We give a new framework which (i) ties online algorithms with computable analysis, (ii) shows how to use modifications of notions from computable analysis, such as Weihrauch reducibility, to analyse finite but uniform combinatorics, (iii) show how to finitize reverse mathematics to suggest a fine structure of finite analogs of infinite combinatorial problems, and (iv) see how similar ideas can be amalgamated from areas such as EX-learning, computable analysis, distributed computing and the like. One of the key ideas is that online algorithms can be viewed as a sub-area of computable analysis. Conversely, we also get an enrichment of computable analysis from classical online algorithms.

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arXiv2020

A recursion theoretic foundation of computation over real numbers

We define a class of computable functions over real numbers using functional schemes similar to the class of primitive and partial recursive functions defined by Gödel and Kleene. We show that this class of functions can also be characterized by master-slave machines, which are Turing machine like devices. The proof of the characterization gives a normal form theorem in the style of Kleene. Furthermore, this characterization is a natural combination of two most influential theories of computation over real numbers, namely, the type-two theory of effectivity (TTE) (see, for example, Weihrauch) and the Blum-Shub-Smale model of computation (BSS). Under this notion of computability, the recursive (or computable) subsets of real numbers are exactly effective $Δ^0_2$ sets.

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arXiv2020

Enumeration degrees and non-metrizable topology

The enumeration degrees of sets of natural numbers can be identified with the degrees of difficulty of enumerating neighborhood bases of points in a universal second-countable $T_0$-space (e.g. the $ω$-power of the Sierpiński space). Hence, every represented second-countable $T_0$-space determines a collection of enumeration degrees. For instance, Cantor space captures the total degrees, and the Hilbert cube captures the continuous degrees by definition. Based on these observations, we utilize general topology (particularly non-metrizable topology) to establish a classification theory of enumeration degrees of sets of natural numbers.

math.GN↗

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.

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arXiv2015

Finitary reducibility on equivalence relations

We introduce the notion of finitary computable reducibility on equivalence relations on the natural numbers. This is a weakening of the usual notion of computable reducibility, and we show it to be distinct in several ways. In particular, whereas no equivalence relation can be $Π_{n+2}$-complete under computable reducibility, we show that, for every $n$, there does exist a natural equivalence relation which is $Π_{n+2}$-complete under finitary reducibility. We also show that our hierarchy of finitary reducibilities does not collapse, and illustrate how it sharpens certain known results. Along the way, we present several new results which use computable reducibility to establish the complexity of various naturally defined equivalence relations in the arithmetical hierarchy.

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arXiv2014

An analogy between cardinal characteristics and highness properties of oracles

We present an analogy between cardinal characteristics from set theory and highness properties from computability theory, which specify a sense in which a Turing oracle is computationally strong. While this analogy was first studied explicitly by Rupprecht in his PhD thesis, many prior results can be viewed from this perspective. After a comprehensive survey of the analogy for characteristics from Cichon's diagram, we extend it to Kurtz randomness and the analogue of the Specker-Eda number.

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