arXiv subjects

Luca San Mauro

Publications and source records attributed to Luca San Mauro.

30 records · Page 2Linked to original sources

arXiv2021

A note on the category of equivalence relations

We make some beginning observations about the category $\mathbb{E}\mathrm{q}$ of equivalence relations on the set of natural numbers, where a morphism between two equivalence relations $R,S$ is a mapping from the set of $R$-equivalence classes to that of $S$-equivalence classes, which is induced by a computable function. We also consider some full subcategories of $\mathbb{E}\mathrm{q}$, such as the category $\mathbb{E}\mathrm{q}(Σ^0_1)$ of computably enumerable equivalence relations (called ceers), the category $\mathbb{E}\mathrm{q}(Π^0_1)$ of co-computably enumerable equivalence relations, and the category $\mathbb{E}\mathrm{q}(\mathrm{Dark}^*)$ whose objects are the so-called dark ceers plus the ceers with finitely many equivalence classes. Although in all these categories the monomorphisms coincide with the injective morphisms, we show that in $\mathbb{E}\mathrm{q}(Σ^0_1)$ the epimorphisms coincide with the onto morphisms, but in $\mathbb{E}\mathrm{q}(Π^0_1)$ there are epimorphisms that are not onto. Moreover, $\mathbb{E}\mathrm{q}$, $\mathbb{E}\mathrm{q}(Σ^0_1)$, and $\mathbb{E}\mathrm{q}(\mathrm{Dark}^*)$ are closed under finite products, binary coproducts, and coequalizers, but we give an example of two morphisms in $\mathbb{E}\mathrm{q}(Π^0_1)$ whose coequalizer in $\mathbb{E}\mathrm{q}$ is not an object of $\mathbb{E}\mathrm{q}(Π^0_1)$.

math.CT↗

arXiv2020

Word problems and ceers

This note addresses the issue as to which ceers can be realized by word problems of computably enumerable (or, simply, c.e.) structures (such as c.e. semigroups, groups, and rings), where being realized means to fall in the same reducibility degree (under the notion of reducibility for equivalence relations usually called "computable reducibility"), or in the same isomorphism type (with the isomorphism induced by a computable function), or in the same strong isomorphism type (with the isomorphism induced by a computable permutation of the natural numbers). We observe for instance that every ceer is isomorphic to the word problem of some c.e. semigroup, but (answering a question of Gao and Gerdes) not every ceer is in the same reducibility degree of the word problem of some finitely presented semigroup, nor is it in the same reducibility degree of some non-periodic semigroup. We also show that the ceer provided by provable equivalence of Peano Arithmetic is in the same strong isomorphism type as the word problem of some non-commutative and non-Boolean c.e. ring.

math.LO↗

arXiv2020

Learning families of algebraic structures from informant

We combine computable structure theory and algorithmic learning theory to study learning of families of algebraic structures. Our main result is a model-theoretic characterization of the class $\mathbf{InfEx}_{\cong}$, consisting of the structures whose isomorphism types can be learned in the limit. We show that a family of structures $\mathfrak{K}$ is $\mathbf{InfEx}_{\cong}$-learnable if and only if the structures from $\mathfrak{K}$ can be distinguished in terms of their $Σ^{\mathrm{inf}}_2$-theories. We apply this characterization to familiar cases and we show the following: there is an infinite learnable family of distributive lattices; no pair of Boolean algebras is learnable; no infinite family of linear orders is learnable.

math.LO↗

arXiv2020

Comparing the isomorphism types of equivalence structures and preorders

A general theme of computable structure theory is to investigate when structures have copies of a given complexity $Γ$. We discuss such problem for the case of equivalence structures and preorders. We show that there is a $Π^0_1$ equivalence structure with no $Σ^0_1$ copy, and in fact that the isomorphism types realized by the $Π^0_1$ equivalence structures coincide with those realized by the $Δ^0_2$ equivalence structures. We also construct a $Σ^0_1$ preorder with no $Π^0_1$ copy.

math.LO↗

arXiv2019

Minimal Equivalence Relations in Hyperarithmetical and Analytical Hierarchies

A standard tool for classifying the complexity of equivalence relations on $ω$ is provided by computable reducibility. This reducibility gives rise to a rich degree structure. The paper studies equivalence relations, which induce minimal degrees with respect to computable reducibility. Let $Γ$ be one of the following classes: $Σ^0_α$, $Π^0_α$, $Σ^1_n$, or $Π^1_n$, where $α\geq 2$ is a computable ordinal and $n$ is a non-zero natural number. We prove that there are infinitely many pairwise incomparable minimal equivalence relations that are properly in $Γ$.

math.LO↗

arXiv2019

Degrees of bi-embeddable categoricity

We investigate the complexity of embeddings between bi-embeddable structures. In analogy with categoricity spectra, we define the bi-embeddable categoricity spectrum of a structure $\mathcal A$ as the family of Turing degrees that compute embeddings between any computable bi-embeddable copies of $\mathcal A$; the degree of bi-embeddable categoricity of $\mathcal A$ is the least degree in this spectrum (if it exists). We extend many known results about categoricity spectra to the case of bi-embeddability. In particular, we exhibit structures without degree of bi-embeddable categoricity, and we show that every degree d.c.e. above $\mathbf{0}^{(α)}$ for $α$ a computable successor ordinal and $\mathbf{0}^{(λ)}$ for $λ$ a computable limit ordinal is a degree of bi-embeddable categoricity. We also give examples of families of degrees that are not bi-embeddable categoricity spectra.

math.LO↗

arXiv2019

Bi-embeddability spectra and bases of spectra

We study degree spectra of structures with respect to the bi-embeddability relation. The bi-embeddability spectrum of a structure is the family of Turing degrees of its bi-embeddable copies. To facilitate our study we introduce the notions of bi-embeddable triviality and basis of a spectrum. Using bi-embeddable triviality we show that several known families of degrees are bi-embeddability spectra of structures. We then characterize the bi-embeddability spectra of linear orderings and study bases of bi-embeddability spectra of strongly locally finite graphs.

math.LO↗

arXiv2019

Limit Learning Equivalence Structures

While most research in Gold-style learning focuses on learning formal languages, we consider the identification of computable structures, specifically equivalence structures. In our core model the learner gets more and more information about which pairs of elements of a structure are related and which are not. The aim of the learner is to find (an effective description of) the isomorphism type of the structure presented in the limit. In accordance with language learning we call this learning criterion InfEx-learning (explanatory learning from informant). Our main contribution is a complete characterization of which families of equivalence structures are InfEx-learnable. This characterization allows us to derive a bound of $\mathbf{0''}$ on the computational complexity required to learn uniformly enumerable families of equivalence structures. We also investigate variants of InfEx-learning, including learning from text (where the only information provided is which elements are related, and not which elements are not related) and finite learning (where the first actual conjecture of the learner has to be correct). Finally, we show how learning families of structures relates to learning classes of languages by mapping learning tasks for structures to equivalent learning tasks for languages.

math.LO↗

arXiv2018

Trial and error mathematics: Dialectical systems and completions of theories

This paper is part of a project that is based on the notion of dialectical system, introduced by Magari as a way of capturing trial and error mathematics. In previous work, we investigated the expressive and computational power of dialectical systems, and we compared them to a new class of systems, that of quasidialectical systems, that enrich Magari's systems with a natural mechanism of revision. In the present paper we consider a third class of systems, that of $p$-dialectical systems, that naturally combine features coming from the two other cases. We prove several results about $p$-dialectical systems and the sets that they represent. Then we focus on the completions of first-order theories. In doing so, we consider systems with connectives, i.e. systems that encode the rules of classical logic. We show that any consistent system with connectives represents the completion of a given theory. We prove that dialectical and $q$-dialectical systems coincide with respect to the completions that they can represent. Yet, $p$-dialectical systems are more powerful: we exhibit a $p$-dialectical system representing a completion of Peano Arithmetic which is neither dialectical nor $q$-dialectical.

math.LO↗

arXiv2018

Classifying equivalence relations in the Ershov hierarchy

Computably enumerable equivalence relations (ceers) received a lot of attention in the literature. The standard tool to classify ceers is provided by the computable reducibility $\leq_c$. This gives rise to a rich degree-structure. In this paper, we lift the study of $c$-degrees to the $Δ^0_2$ case. In doing so, we rely on the Ershov hierarchy. For any notation $a$ for a non-zero computable ordinal, we prove several algebraic properties of the degree-structure induced by $\leq_c$ on the $Σ^{-1}_{a}\smallsetminus Π^{-1}_a$ equivalence relations. A special focus of our work is on the (non)existence of infima and suprema of $c$-degrees.

math.LO↗

arXiv2018

Measuring the complexity of reductions between equivalence relations

Computable reducibility is a well-established notion that allows to compare the complexity of various equivalence relations over the natural numbers. We generalize computable reducibility by introducing degree spectra of reducibility and bi-reducibility. These spectra provide a natural way of measuring the complexity of reductions between equivalence relations. We prove that any upward closed collection of Turing degrees with a countable basis can be realised as a reducibility spectrum or as a bi-reducibility spectrum. We show also that there is a reducibility spectrum of computably enumerable equivalence relations with no countable basis and a reducibility spectrum of computably enumerable equivalence relations which is downward dense, thus has no basis.

math.LO↗

arXiv2017

Degrees of bi-embeddable categoricity of equivalence structures

We study the algorithmic complexity of embeddings between bi-embeddable equivalence structures. We define the notions of computable bi-embeddable categoricity, (relative) $Δ^0_α$ bi-embeddable categoricity, and degrees of bi-embeddable categoricity. These notions mirror the classical notions used to study the complexity of isomorphisms between structures. We show that the notions of $Δ^0_α$ bi-embeddable categoricity and relative $Δ^0_α$ bi-embeddable categoricity coincide for equivalence structures for $α=1,2,3$. We also prove that computable equivalence structures have degree of bi-embeddable categoricity $\mathbf{0},\mathbf{0}'$, or $\mathbf{0}''$. We obtain results on index sets of computable equivalence structure with respect to bi-embeddability.

math.LO↗