arXiv subjects

Manlio Valenti

Publications and source records attributed to Manlio Valenti.

13 recordsLinked to original sources

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

Chains and antichains in the Weihrauch lattice

We study the existence and the distribution of "long" chains in the Weihrauch degrees, mostly focusing on chains with uncountable cofinality. We characterize when such chains have an upper bound and prove that there are no cofinal chains (of any order type) in the Weihrauch degrees. Furthermore, we show that the existence of coinitial sequences of non-zero degrees is equivalent to $\mathrm{CH}$. Finally, we explore the extendibility of antichains, providing some necessary conditions for maximality.

math.LO↗

arXiv2024

A jump operator on the Weihrauch degrees

A partial order $(P,\le)$ admits a jump operator if there is a map $j\colon P \to P$ that is strictly increasing and weakly monotone. Despite its name, the jump in the Weihrauch lattice fails to satisfy both of these properties: it is not degree-theoretic and there are functions $f$ such that $f\equiv_{\mathrm{W}} f'$. This raises the question: is there a jump operator in the Weihrauch lattice? We answer this question positively and provide an explicit definition for an operator on partial multi-valued functions that, when lifted to the Weihrauch degrees, induces a jump operator. This new operator, called the totalizing jump, can be characterized in terms of the total continuation, a well-known operator on computational problems. The totalizing jump induces an injective endomorphism of the Weihrauch degrees. We study some algebraic properties of the totalizing jump and characterize its behavior on some pivotal problems in the Weihrauch lattice.

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

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 K\"onig's lemma $\mathsf{KL}$ and the problem $\mathsf{wList}_{2^{\mathbb{N}},\leq\omega}$ 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↗

arXiv2023

The tree pigeonhole principle in the Weihrauch degrees

We study versions of the tree pigeonhole principle, $\mathsf{TT}^1$, in the context of Weihrauch-style computable analysis. The principle has previously been the subject of extensive research in reverse mathematics. Two outstanding questions from the latter investigation are whether $\mathsf{TT}^1$ is $\Pi^1_1$-conservative over the ordinary pigeonhole principle, $\mathsf{RT}^1$, and whether it is equivalent to any first-order statement of second-order arithmetic. Using the recently introduced notion of the first-order part of an instance-solution problem, we formulate, and answer in the affirmative, the analogue of the first question for Weihrauch reducibility. We then use this, in combination with other results, to answer in the negative the analogue of the second question. Our proofs develop a new combinatorial machinery for constructing and understanding solutions to instances of $\mathsf{TT}^1$.

math.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↗

arXiv2023

Algebraic properties of the first-order part of a problem

In this paper we study the notion of first-order part of a computational problem, first introduced by Dzhafarov, Solomon, and Yokoyama, which captures the "strongest computational problem with codomain $\mathbb{N}$ that is Weihrauch reducible to $f$". This operator is very useful to prove separation results, especially at the higher levels of the Weihrauch lattice. We explore the first-order part in relation with several other operators already known in the literature. We also introduce a new operator, called unbounded finite parallelization, which plays an important role in characterizing the first-order part of parallelizable problems. We show how the obtained results can be used to explicitly characterize the first-order part of several known problems.

math.LO↗

arXiv2022

The Weihrauch lattice at the level of $\boldsymbol{\Pi}_1^1\mathsf{-CA}_0$: the Cantor-Bendixson theorem

This paper continues the program connecting reverse mathematics and computable analysis via the framework of Weihrauch reducibility. In particular, we consider problems related to perfect subsets of Polish spaces, studying the perfect set theorem, the Cantor-Bendixson theorem and various problems arising from them. In the framework of reverse mathematics these theorems are equivalent respectively to $\mathsf{ATR}_0$ and $\boldsymbol{\Pi}_1^1\mathsf{-CA}_0$, the two strongest subsystems of second order arithmetic among the so-called big five. As far as we know, this is the first systematic study of problems at the level of $\boldsymbol{\Pi}_1^1\mathsf{-CA}_0$ in the Weihrauch lattice. We show that the strength of some of the problems we study depends on the topological properties of the Polish space under consideration, while others have the same strength once the space is rich enough.

math.LO↗

arXiv2022

Categorifying computable reducibilities

This paper presents categorical formulations of Turing, Medvedev, Muchnik, and Weihrauch reducibilities in Computability Theory, utilizing Lawvere doctrines. While the first notions lend themselves to a smooth categorical presentation, essentially dualizing the traditional idea of realizability doctrines, Weihrauch reducibility and its extensions to represented and multi-represented spaces require a separate investigation. Our abstract analysis of these concepts highlights a shared characteristic among all these reducibilities. Specifically, we demonstrate that all these doctrines stemming from computability concepts can be proven to be instances of completions of quantifiers for doctrines, analogous to what occurs for doctrines for realizability. As a corollary of these results, we will be able to formally compare Weihrauch reducibility with the dialectica doctrine constructed from a doctrine representing Turing degrees.

math.LO↗

arXiv2022

Effective aspects of Hausdorff and Fourier dimension

In this paper, we study Hausdorff and Fourier dimension from the point of view of effective descriptive set theory and Type-2 Theory of Effectivity. Working in the hyperspace $\mathbf{K}(X)$ of compact subsets of $X$, with $X=[0,1]^d$ or $X=\mathbb{R}^d$, we characterize the complexity of the family of sets having sufficiently large Hausdorff or Fourier dimension. This, in turn, allows us to show that family of all the closed Salem sets is $Π^0_3$-complete. One of our main tools is a careful analysis of the effectiveness of a classical theorem of Kaufman. We furthermore compute the Weihrauch degree of the functions computing Hausdorff and Fourier dimension of closed sets.

math.LO↗

arXiv2021

On the descriptive complexity of Salem sets

In this paper we study the notion of Salem set from the point of view of descriptive set theory. We first work in the hyperspace $\mathbf{K}([0,1])$ of compact subsets of $[0,1]$ and show that the closed Salem sets form a $\boldsymbolΠ^0_3$-complete family. This is done by characterizing the complexity of the family of sets having sufficiently large Hausdorff or Fourier dimension. We also show that the complexity does not change if we increase the dimension of the ambient space and work in $\mathbf{K}([0,1]^d)$. We then generalize the results by relaxing the compactness of the ambient space, and show that the closed Salem sets are still $\boldsymbolΠ^0_3$-complete when we endow the hyperspace of all closed subsets of $\mathbb{R}^d$ with the Fell topology. A similar result holds also for the Vietoris topology.

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.

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