Luca San Mauro

TU Wien | TU Wien

Computability theorists have introduced multiple hierarchies to measure the complexity of sets of natural numbers. The Kleene Hierarchy classifies sets according to the first-order complexity of their defining formulas. The Ershov Hierarchy classifies limit computable sets with respect to the number of mistakes that are needed to approximate them....

Artificial intelligence plays an important role in contemporary medicine. In this short note, we emphasize that philosophy played a role in the development of artificial intelligence. We argue that research in computability theory, the theoretical foundation of modern computer science, was motivated by a philosophical question: can we characterize...

This paper studies algorithmic learning theory applied to algebraic structures. In previous papers, we have defined our framework, where a learner, given a family of structures, receives larger and larger pieces of an arbitrary copy of a structure in the family and, at each stage, is required to output a conjecture about the isomorphism type of suc...

We answer several questions about the computable Friedman-Stanley jump on equivalence relations. This jump, introduced by Clemens, Coskey, and Krakoff, deepens the natural connection between the study of computable reduction and its Borel analog studied deeply in descriptive set theory.

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 : ω → ω that induces an injective map from R-equivalence classes to S...

We examine the degree structure $\operatorname {\mathrm {\mathbf {ER}}}$ of equivalence relations on $\omega $ under computable reducibility. We examine when pairs of degrees have a least upper bound. In particular, we show that sufficiently incomparable pairs of degrees do not have a least upper bound but that some incomparable degrees do, and we...

We make some beginning observations about the category 𝔼q of equivalence relations on the set of natural numbers, where a morphism between two equivalence relations R and 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 𝔼q, s...

We study algorithmic learning of algebraic structures. In our framework, a learner receives larger and larger pieces of an arbitrary copy of a computable structure and, at each stage, is required to output a conjecture about the isomorphism type of such a structure. The learning is successful if the conjectures eventually stabilize to a correct gue...

In this short paper we analyze whether assuming that mathematical objects are "thin" in Linnebo's sense simplifies the epistemol-ogy of mathematics. Towards this end we introduce the notion of transparency and we show that not all thin objects are transparent. We end by arguing that, far from being a weakness of thin objects, the lack of transparen...

Computability theorists have introduced multiple hierarchies to measure the complexity of sets of natural numbers. The Kleene Hierarchy classifies sets according to the first-order complexity of their defining formulas. The Ershov Hierarchy classifies \(\varDelta ^0_2\) sets with respect to the number of mistakes that are needed to approximate them...

We offer a novel picture of mathematical language from the perspective of speech act theory. There are distinct speech acts within mathematics (not just assertions), and, as we intend to show, distinct illocutionary force indicators as well. Even mathematics in its most formalized version cannot do without some such indicators. This goes against a...

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 $f \colon \omega \to \omega$ that induces an injective map from...

In this paper we focus on the logicality of language, i.e. the idea that the language system contains a deductive device to exclude analytic constructions. Puzzling evidence for the logicality of language comes from acceptable contradictions and tautologies. The standard response in the literature involves assuming that the language system only acc...

Computability theorists have introduced multiple hierarchies to measure the complexity of sets of natural numbers. The Kleene Hierarchy classifies sets according to the first-order complexity of their defining formulas. The Ershov Hierarchy classifies $\Delta^0_2$ sets with respect to the number of mistakes that are needed to approximate them. Biac...

In previous work, we have combined computable structure theory and algorithmic learning theory to study which families of algebraic structures are learnable in the limit (up to isomorphism). In this paper, we measure the computational power that is needed to learn finite families of structures. In particular, we prove that, if a family of structure...

In previous work, we have combined computable structure theory and algorithmic learning theory to study which families of algebraic structures are learnable in the limit (up to isomorphism). In this paper, we measure the computational power that is needed to learn finite families of structures. In particular, we prove that, if a family of structure...

We examine the degree structure $\mathbf{ER}$ of equivalence relations on $\omega$ under computable reducibility. We examine when pairs of degrees have a join. In particular, we show that sufficiently incomparable pairs of degrees do not have a join but that some incomparable degrees do, and we characterize the degrees which have a join with every...

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...

In this brief article we present the following paradox: one cannot assume that mathematicians are trustworthy when they express their mathematical (dis)beliefs, while also maintaining four basic theses about natural and mathematical language. We carefully present the very natural hypotheses on which this paradox is based and then we show how to ded...

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 \(\leqslant _c\). This gives rise to a rich degree structure. In this paper, we lift the study of c-degrees to the \(\Delta ^0_2\) case. In doing so, we rely on the Ershov...

Breaking with traditional antipragmatic assumptions, recent discussion in this journal submits evidence for pragmatic inferences in mathematical language. The observation is that the interpretation of crucial expressions, e.g., positive quantifiers, results from a truth-conditional enrichment of their literal meaning. While providing general insigh...

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”)...

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")...

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 learning type InfEx≅, consisting of the structures whose isomorphism types can be learned in the limit. We show that a family of structures is InfEx≅-learnable if...

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 A as the family of Turing degrees that compute embeddings between any computable bi-embeddable copies of A; the degree of bi-embeddable categoricity of A is the least de...

In this paper we argue, against a somewhat standard view, that pragmatic phenomena occur in mathematical language. We provide concrete examples supporting this thesis.

A general theme of computable structure theory is to investigate when structures have copies of a given complexity $\Gamma$. We discuss such problem for the case of equivalence structures and preorders. We show that there is a $\Pi^0_1$ equivalence structure with no $\Sigma^0_1$ copy, and in fact that the isomorphism types realized by the $\Pi^0_1$...

A standard tool for classifying the complexity of equivalence relations on $\omega$ 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 $\Gamma$ be one of the following classes: $\Sigma^0_{\...

We study the algorithmic complexity of embeddings between bi-embeddable equivalence structures. We define the notions of computable bi-embeddable categoricity, (relative) \(\Delta ^0_\alpha \) bi-embeddable categoricity, and degrees of bi-embeddable categoricity. These notions mirror the classical notions used to study the complexity of isomorphism...

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...

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 categoricit...

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 InfEx_{\iso}, consisting of the structures whose isomorphism types can be learned in the limit. We show that a family of structures is InfEx-learnable if and...

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}...

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...

This paper is part of a project that is based on the notion of a dialectical system, introduced by Magari as a way of capturing trial and error mathematics. In Amidei et al. (2016, Rev. Symb. Logic, 9, 1-26) and Amidei et al. (2016, Rev. Symb. Logic, 9, 299-324), we investigated the expressive and computational power of dialectical systems, and we...

We study the algorithmic complexity of isomorphic embeddings between computable structures.

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 Maga...

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 $\Delta^0_2$ case. In doing so, we rely on 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 $\Delta^0_2$ case. In doing so, we rely on the Ershov hierarchy...

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 equivalen...

We study degree spectra of structures relative to the bi-embeddability relation. The bi-embeddability spectrum of a structure is the family of Turing degrees of structures bi-embeddable with it. We introduce the notions of bi-embeddable triviality and basis of a spectrum. Using bi-embeddable triviality we show that several well known classes of deg...

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 equivalen...

We study the algorithmic complexity of embeddings between bi-embeddable equivalence structures. We define the notions of computable bi-embeddable categoricity, (relative) $\Delta^0_\alpha$ bi-embeddable categoricity, and degrees of bi-embeddable categoricity. These notions mirror the classical notions used to study the complexity of isomorphisms be...

We aim at providing a philosophical analysis of the notion of "proof by Church's Thesis", which is-in a nutshell-the conceptual device that permits to rely on informal methods when working in Computability Theory. This notion allows, in most cases, to not specify the background model of computation in which a given algorithm-or a construction-is fr...

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 equivalen...

We aim at providing a philosophical analysis of the notion of “proof by Church’s Thesis”, which is – in a nutshell – the conceptual device that permits to rely on informal methods when working in Computability Theory. This notion allows, in most cases, to not specify the background model of computation in which a given algorithm – or a construction...

This paper is a continuation of Amidei, Pianigiani, San Mauro, Simi, & Sorbi (2016), where we have introduced the quasidialectical systems, which are abstract deductive systems designed to provide, in line with Lakatos’ views, a formalization of trial and error mathematics more adherent to the real mathematical practice of revision than Magari’s or...

We define and study quasidialectical systems, which are an extension of Magari’s dialectical systems, designed to make Magari’s formalization of trial and error mathematics more adherent to the real mathematical practice of revision: our proposed extension follows, and in several regards makes more precise, varieties of empiricist positions à la La...

In mathematical literature, it is quite common to make reference to an informal notion of naturalness: axioms or definitions may be defined as “natural,” and part of a proof may deserve the same label (i.e., “in a natural way…”). Our aim is to provide a philosophical account of these occurrences. The paper is divided in two parts. In the first part...

We study computably enumerable equivalence relations (ceers) under the reducibility R ≤ S if there exists a computable function f such that, for every x, y, x R y if and only if f (x) S f (y). We show that the degrees of ceers under the equivalence relation generated by ≤ form a bounded poset that is neither a lower semilattice, nor an upper semila...