Searcharxiv⌕ Search

arXiv subjects

Arno Pauly

Publications and source records attributed to Arno Pauly.

At least 19 recordsLinked to original sources
arXiv2025

The weakness of finding descending sequences in ill-founded linear orders

We explore the Weihrauch degree of the problems ``find a bad sequence in a non-well quasi order'' ($\mathsf{BS}$) and ``find a descending sequence in an ill-founded linear order'' ($\mathsf{DS}$). We prove that $\mathsf{DS}$ is strictly Weihrauch reducible to $\mathsf{BS}$, correcting our mistaken claim in [arXiv:2010.03840]. This is done by separating their respective first-order parts. On the other hand, we show that $\mathsf{BS}$ and $\mathsf{DS}$ have the same finitary and deterministic parts, confirming that $\mathsf{BS}$ and $\mathsf{DS}$ have very similar uniform computational strength. We prove that König's lemma $\mathsf{KL}$ and the problem $\mathsf{wList}_{2^{\mathbb{N}},\leqω}$ of enumerating a given non-empty countable closed subset of $2^{\mathbb{N}}$ are not Weihrauch reducible to $\mathsf{DS}$ or $\mathsf{BS}$, resolving two main open questions raised in [arXiv:2010.03840]. We also answer the question, raised in [arXiv:1804.10968], on the existence of a ``parallel quotient'' operator, and study the behavior of $\mathsf{BS}$ and $\mathsf{DS}$ under the quotient with some known problems.

math.LO↗
arXiv2025

The Weihrauch degree of finding Nash equilibria in multiplayer games

Is there an algorithm that takes a game in normal form as input, and outputs a Nash equilibrium? If the payoffs are integers, the answer is yes, and lot of work has been done in its computational complexity. If the payoffs are permitted to be real numbers, the answer is no, for continuity reasons. It is worthwhile to investigate the precise degree of non-computability (the Weihrauch degree), since knowing the degree entails what other approaches are available (eg, is there a randomized algorithm with positive success change?). The two player case has already been fully classified, but the multiplayer case remains open and is addressed here. Our approach involves classifying the degree of finding roots of polynomials, and lifting this to systems of polynomial inequalities via cylindrical algebraic decomposition.

cs.LO↗
arXiv2025

Computably discrete represented spaces

In computable topology, a represented space is called computably discrete if its equality predicate is semidecidable. While any such space is classically isomorphic to an initial segment of the natural numbers, the computable-isomorphism types of computably discrete represented spaces exhibit a rich structure. We show that the widely studied class of computably enumerable equivalence relations (ceers) corresponds precisely to the computably Quasi-Polish computably discrete spaces. We employ computably discrete spaces to exhibit several separating examples in computable topology. We construct a computably discrete computably Quasi-Polish space admitting no decidable properties, a computably discrete and computably Hausdorff precomputably Quasi-Polish space admitting no computable injection into the natural numbers, a two-point space which is computably Hausdorff but not computably discrete, and a two-point space which is computably discrete but not computably Hausdorff. We further expand an example due to Weihrauch that separates computably regular spaces from computably normal spaces.

math.LO↗
arXiv2024

The equational theory of the Weihrauch lattice with multiplication

We study the equational theory of the Weihrauch lattice with multiplication, meaning the collection of equations between terms built from variables, the lattice operations $\sqcup$, $\sqcap$, the product $\times$, and the finite parallelization $(-)^*$ which are true however we substitute Weihrauch degrees for the variables. We provide a combinatorial description of these in terms of a reducibility between finite graphs, and moreover, show that deciding which equations are true in this sense is complete for the third level of the polynomial hierarchy.

cs.LO↗
arXiv2024

More on the indivisibility of $\mathbb{Q}$

We study the complexity of the computational task ``Given a colouring $c : \mathbb{Q} \to \mathbf{k}$, find a monochromatic $S \subseteq \mathbb{Q}$ such that $(S,<) \cong (\mathbb{Q},<)$''. The framework is Weihrauch reducibility. Our results answer some open questions recently raised by Gill, and by Dzhafarov, Solomon and Valenti.

math.LO↗
arXiv2024

Finding descending sequences through ill-founded linear orders

In this work we investigate the Weihrauch degree of the problem $\mathsf{DS}$ of finding an infinite descending sequence through a given ill-founded linear order, which is shared by the problem $\mathsf{BS}$ of finding a bad sequence through a given non-well quasi-order. We show that $\mathsf{DS}$, despite being hard to solve (it has computable inputs with no hyperarithmetic solution), is rather weak in terms of uniform computational strength. To make the latter precise, we introduce the notion of the deterministic part of a Weihrauch degree. We then generalize $\mathsf{DS}$ and $\mathsf{BS}$ by considering $\boldsymbolΓ$-presented orders, where $\boldsymbolΓ$ is a Borel pointclass or $\boldsymbolΔ^1_1$, $\boldsymbolΣ^1_1$, $\boldsymbolΠ^1_1$. We study the obtained $\mathsf{DS}$-hierarchy and $\mathsf{BS}$-hierarchy of problems in comparison with the (effective) Baire hierarchy and show that they do not collapse at any finite level.

math.LO↗
arXiv2024

Sequential discontinuity and first-order problems

We explore the low levels of the structure of the continuous Weihrauch degrees of first-order problems. In particular, we show that there exists a minimal discontinuous first-order degree, namely that of $\accn$, without any determinacy assumptions. The same degree is also revealed as the least sequentially discontinuous one, i.e. the least degree with a representative whose restriction to some sequence converging to a limit point is still discontinuous. The study of games related to continuous Weihrauch reducibility constitutes an important ingredient in the proof of the main theorem. We present some initial additional results about the degrees of first-order problems that can be obtained using this approach.

math.LO↗
arXiv2024

Embeddability of graphs and Weihrauch degrees

We study the complexity of the following related computational tasks concerning a fixed countable graph G: 1. Does a countable graph H provided as input have a(n induced) subgraph isomorphic to G? 2. Given a countable graph H that has a(n induced) subgraph isomorphic to G, find such a subgraph. The framework for our investigations is given by effective Wadge reducibility and by Weihrauch reducibility. Our work follows on "Reverse mathematics and Weihrauch analysis motivated by finite complexity theory" (Computability, 2021) by BeMent, Hirst and Wallace, and we answer several of their open questions.

math.LO↗
arXiv2023

On the Weihrauch degree of the additive Ramsey theorem

We characterize the strength, in terms of Weihrauch degrees, of certain problems related to Ramsey-like theorems concerning colourings of the rationals and of the natural numbers. The theorems we are chiefly interested in assert the existence of almost-homogeneous sets for colourings of pairs of rationals respectively natural numbers satisfying properties determined by some additional algebraic structure on the set of colours. In the context of reverse mathematics, most of the principles we study are equivalent to $Σ^0_2$-induction over $\mathrm{RCA}_0$. The associated problems in the Weihrauch lattice are related to $\mathrm{TC}_\mathbb{N}^*$, $(\mathrm{LPO}')^*$ or their product, depending on their precise formalizations.

cs.LO↗
arXiv2023

Minimal covers in the Weihrauch degrees

In this paper, we study the existence of minimal covers and strong minimal covers in the Weihrauch degrees. We characterize when a problem $f$ is a minimal cover or strong minimal cover of a problem $h$. We show that strong minimal covers only exist in the cone below $\mathsf{id}$ and that the Weihrauch lattice above $\mathsf{id}$ is dense. From this, we conclude that the degree of $\mathsf{id}$ is first-order definable in the Weihrauch degrees and that the first-order theory of the Weihrauch degrees is computably isomorphic to third-order arithmetic.

math.LO↗
arXiv2021

A Computability Perspective on (Verified) Machine Learning

There is a strong consensus that combining the versatility of machine learning with the assurances given by formal verification is highly desirable. It is much less clear what verified machine learning should mean exactly. We consider this question from the (unexpected?) perspective of computable analysis. This allows us to define the computational tasks underlying verified ML in a model-agnostic way, and show that they are in principle computable.

cs.LG↗
arXiv2021

Adaptive Neighbourhoods for the Discovery of Adversarial Examples

Deep Neural Networks (DNNs) have often supplied state-of-the-art results in pattern recognition tasks. Despite their advances, however, the existence of adversarial examples have caught the attention of the community. Many existing works have proposed methods for searching for adversarial examples within fixed-sized regions around training points. Our work complements and improves these existing approaches by adapting the size of these regions based on the problem complexity and data sampling density. This makes such approaches more appropriate for other types of data and may further improve adversarial training methods by increasing the region sizes without creating incorrect labels.

cs.LG↗
arXiv2020

Luzin's (N) and randomness reflection

We show that a computable function $f:\mathbb R\rightarrow\mathbb R$ has Luzin's property (N) if and only if it reflects $Π^1_1$-randomnes, if and only if it reflects $Δ^1_1(\mathcal O)$-randomness, and if and only if it reflects $\mathcal O$-Kurtz randomness, but reflecting Martin-Löf randomness or weak-2-randomness does not suffice. Here a function $f$ is said to reflect a randomness notion $R$ if whenever $f(x)$ is $R$-random, then $x$ is $R$-random as well. If additionally $f$ is known to have bounded variation, then we show $f$ has Luzin's (N) if and only if it reflects weak-2-randomness, and if and only if it reflects $\emptyset'$-Kurtz randomness. This links classical real analysis with algorithmic randomness.

math.LO↗
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↗
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

Computing Haar Measures

According to Haar's Theorem, every compact group $G$ admits a unique (regular, right and) left-invariant Borel probability measure $μ_G$. Let the Haar integral (of $G$) denote the functional $\int_G:\mathcal{C}(G)\ni f\mapsto \int f\,dμ_G$ integrating any continuous function $f:G\to\mathbb{R}$ with respect to $μ_G$. This generalizes, and recovers for the additive group $G=[0;1)\mod 1$, the usual Riemann integral: computable (cmp. Weihrauch 2000, Theorem 6.4.1), and of computational cost characterizing complexity class #P$_1$ (cmp. Ko 1991, Theorem 5.32). We establish that in fact every computably compact computable metric group renders the Haar integral computable: once asserting computability using an elegant synthetic argument, exploiting uniqueness in a computably compact space of probability measures; and once presenting and analyzing an explicit, imperative algorithm based on 'maximum packings' with rigorous error bounds and guaranteed convergence. Regarding computational complexity, for the groups $\mathcal{SO}(3)$ and $\mathcal{SU}(2)$ we reduce the Haar integral to and from Euclidean/Riemann integration. In particular both also characterize #P$_1$. Implementation and empirical evaluation using the iRRAM C++ library for exact real computation confirms the (thus necessary) exponential runtime.

cs.LO↗
arXiv2019

Ramsey's theorem and products in the Weihrauch degrees

We study the positions in the Weihrauch lattice of parallel products of various combinatorial principles related to Ramsey's theorem. Among other results, we obtain an answer to a question of Brattka, by showing that Ramsey's theorem for pairs ($\mathsf{RT}^2_2$) is strictly Weihrauch below the parallel product of the stable Ramsey's theorem for pairs and the cohesive principle ($\mathsf{SRT}^2_2 \times \mathsf{COH}$).

math.LO↗