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Andrea Sorbi

Publications and source records attributed to Andrea Sorbi.

18 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

Intuitionism and computing with partial information

There exist initial segments of both the Dyment lattice and the Dyment-Muchnik lattice that yield Brouwer algebras modeling exactly the intuitionistic propositional calculus. For the Dyment-Muchnik lattice, this result is obtained by constructing a splitting class of enumeration degrees. In contrast, the full Dyment lattice and the full Dyment-Muchnik lattice model the intuitionistic propositional calculus plus the weak law of excluded middle. We also observe that certain naturally definable classes of enumeration degrees, which are downwards closed under enumeration reducibility, fail to form splitting classes.

math.LO↗
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.

math.LO↗
arXiv2023

Classifying word problems of finitely generated algebras via computable reducibility

We contribute to a recent research program which aims at revisiting the study of the complexity of word problems, a major area of research in combinatorial algebra, through the lens of the theory of computably enumerable equivalence relations (ceers), which has considerably grown in recent times. To pursue our analysis, we rely on the most popular way of assessing the complexity of ceers, that is via computable reducibility on equivalence relations, and its corresponding degree structure (the c-degrees). On the negative side, building on previous work of Kasymov and Khoussainov, we individuate a collection of c-degrees of ceers which cannot be realized by the word problem of any finitely generated algebra of finite type. On the positive side, we show that word problems of finitely generated semigroups realize a collection of c-degrees which embeds rich structures and is large in several reasonable ways.

math.LO↗
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

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

The Theory of Ceers Computes True Arithmetic

We show that the theory of the partial order of computably enumerable equivalence relations (ceers) under computable reduction is 1-equivalent to true arithmetic. We show the same result for the structure comprised of the dark ceers and the structure comprised of the light ceers. We also show the same for the structure of $\mathcal{I}$-degrees in the dark, light, or complete structure. In each case, we show that there is an interpretable copy of $(\mathbb{N},+,\cdot)$.

math.LO↗
arXiv2020

Self-full ceers and the uniform join operator

A computably enumerable equivalence relation (ceer) $X$ is called self-full if whenever $f$ is a reduction of $X$ to $X$ then the range of $f$ intersects all $X$-equivalence classes. It is known that the infinite self-full ceers properly contain the dark ceers, i.e. the infinite ceers which do not admit an infinite computably enumerable transversal. Unlike the collection of dark ceers, which are closed under the operation of uniform join, we answer a question from \cite{joinmeet} by showing that there are self-full ceers $X$ and $Y$ so that their uniform join $X\oplus Y$ is non-self-full. We then define and examine the hereditarily self-full ceers, which are the self-full ceers $X$ so that for any self-full $Y$, $X\oplus Y$ is also self-full: we show that they are closed under uniform join, and that every non-universal degree in $\textrm{Ceers}_{/{\mathcal{I}}}$ have infinitely many incomparable hereditarily self-full strong minimal covers. In particular, every non-universal ceer is bounded by a hereditarily self-full ceer. Thus the hereditarily self-full ceers form a properly intermediate class in between the dark ceers and the infinite self-full ceers which is closed under $\oplus$.

math.LO↗
arXiv2019

Effective inseparability, lattices, and pre-ordering relations

We study effectively inseparable (e.i.) pre-lattices (i.e. structures of the form $L=\langle ω, \wedge, \lor, 0, 1, \leq_L\rangle$ where $ω$ denotes the set of natural numbers and the following hold: $\wedge, \lor$ are binary computable operations; $\leq_L$ is a c.e. pre-ordering relation, with $0 \leq_{L} x \leq_{L} 1$ for every $x$; the equivalence relation $\equiv_L$ originated by $\leq_L$ is a congruence on $L$ such that the corresponding quotient structure is a non-trivial bounded lattice; the $\equiv_L$-equivalence classes of $0$ and $1$ form an effectively inseparable pair), and show that if $L$ is an e.i. pre-lattice then $\le_{L}$ is universal with respect to all c.e. pre-ordering relations, i.e. for every c.e. pre-ordering relation $R$ there exists a computable function $f$ such that, for all $x,y$, $x \mathrel{R} y$ if and only if $f(x) \le_{L} f(y)$; in fact $\leq_L$ is locally universal, i.e. for every pair $a<_{L} b$ and every c.e. pre ordering relation $R$ one can find a reducing function $f$ from $R$ to $\le_{L}$ such that the range of $f$ is contained in the interval $\{x: a \leq_{L} x \leq_{L} b\}$. Also $\leq_L$ is uniformly dense, i.e. there exists a computable function $f$ such that for every $a,b$ if $a<_{L} b$ then $a<_{L} f(a,b) <_{L} b$, and if $a\equiv_{L} a'$ and $b \equiv_{L} b'$ then $f(a,b)\equiv_{L} f(a',b')$. Some consequences and applications of these results are discussed: in particular for $n \ge 1$ the c.e. pre-ordering relation on $Σ_{n}$ sentences yielded by the relation of provable implication of any c.e. consistent extension of Robinson's $Q$ or $R$ is locally universal and uniformly dense; and the c.e. pre-ordering relation of provable implication of Heyting Arithmetic is locally universal and uniformly dense.

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

Comparing the degrees of enumerability and the closed Medvedev degrees

We compare the degrees of enumerability and the closed Medvedev degrees and find that many situations occur. There are nonzero closed degrees that do not bound nonzero degrees of enumerability, there are nonzero degrees of enumerability that do not bound nonzero closed degrees, and there are degrees that are nontrivially both degrees of enumerability and closed degrees. We also show that the compact degrees of enumerability exactly correspond to the cototal enumeration degrees.

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

Joins and meets in the structure of Ceers

We study computably enumerable equivalence relations (abbreviated as ceers) under computable reducibility, and we investigate the resulting degree structure Ceers, which is a poset with a smallest and a greatest element. We point out a partition of the ceers into three classes: the finite ceers, the light ceers, and the dark ceers. These classes yield a partition of the degree structure as well, and in the language of posets the corresponding classes of degrees are first order definable within Ceers. There is no least, no maximal, no greatest dark degree, but there are infinitely many minimal dark degrees. We study joins and meets in Ceers, addressing the cases when two incomparable degrees of ceers X,Y have or do not have join or meet according to where X,Y are located in the classes of the aforementioned partition: in particular no pair of dark ceers has join, and no pair in which at least one ceer is dark has meet. We also exhibit examples of ceers X,Y having join which coincides with their uniform join, but also examples when their join is strictly less than the uniform join. We study join-irreducibility and meet-irreducibility. In particular we characterize the property of being meet-irreducible for a ceer E, by showing that it coincides with the property of E being self-full, i.e. every reducibility from E to itself is in fact surjective on its equivalence classes (this property properly extends darkness). We then study the quotient structure obtained by dividing the poset Ceers by the degrees of the finite ceers, and study joins and meets in this quotient structure. We look at automorphisms of Ceers, and show that there are continuum many automorphisms fixing the dark ceers, and continuum many automorphisms fixing the light ceers. Finally, we compute the complexity of the index sets of the classes of ceers studied in the paper.

math.LO↗
arXiv2016

Calibrating word problems of groups via the complexity of equivalence relations

(1) There is a finitely presented group with a word problem which is a uniformly effectively inseparable equivalence relation. (2) There is a finitely generated group of computable permutations with a word problem which is a universal co-computably enumerable equivalence relation. (3) Each c.e. truth-table degree contains the word problem of a finitely generated group of computable permutations.

math.LO↗
arXiv2011

Generalizations of the Weak Law of the Excluded Middle

We study a class of formulas generalizing the weak law of the excluded middle, and provide a characterization of these formulas in terms of Kripke frames and Brouwer algebras. We use these formulas to separate logics corresponding to factors of the Medvedev lattice.

math.LO↗
arXiv2010

Intuitionistic Logic and Muchnik Degrees

We prove that there is a factor of the Muchnik lattice that captures intuitionistic propositional logic. This complements a now classic result of Skvortsova for the Medvedev lattice.

math.LO↗