arXiv subjects

Pedro Resende

Publications and source records attributed to Pedro Resende.

At least 19 recordsLinked to original sources

arXiv2025

Scott locales

We prove some facts about locales $L$ equipped with the Scott topology $\Omega(L)$, in particular studying a canonical frame homomorphism $\phi:\Omega(L)\to L$ which is motivated by an application to cognitive science. Such a topological locale $L$ is called a Scott locale if the inclusion of primes $p:\Sigma(L)\to L$ is continuous. We prove that the spectrum $\Sigma(L)$ of a Scott locale $L$ is necessarily $T_1$, and that preregular locales (a generalization of regular locales) are Scott locales. If $L$ is the topology of a topological space $X$ we find a (necessarily unique) continuous map $f:X\to L$ such that $f^{-1}=\phi$ and compare it with the points-to-primes map $p:X\to L$, showing that $f=p$ if and only if $X$ is preregular, and that a sober space $X$ is Hausdorff if and only if $X$ is $T_1$ and $f(X)\subseteq\Sigma(L)$.

math.CT↗

arXiv2023

SAT-Based Algorithms for Regular Graph Pattern Matching

Graph matching is a fundamental problem in pattern recognition, with many applications such as software analysis and computational biology. One well-known type of graph matching problem is graph isomorphism, which consists of deciding if two graphs are identical. Despite its usefulness, the properties that one may check using graph isomorphism are rather limited, since it only allows strict equality checks between two graphs. For example, it does not allow one to check complex structural properties such as if the target graph is an arbitrary length sequence followed by an arbitrary size loop. We propose a generalization of graph isomorphism that allows one to check such properties through a declarative specification. This specification is given in the form of a Regular Graph Pattern (ReGaP), a special type of graph, inspired by regular expressions, that may contain wildcard nodes that represent arbitrary structures such as variable-sized sequences or subgraphs. We propose a SAT-based algorithm for checking if a target graph matches a given ReGaP. We also propose a preprocessing technique for improving the performance of the algorithm and evaluate it through an extensive experimental evaluation on benchmarks from the CodeSearchNet dataset.

cs.AI↗

arXiv2022

A physical approach to qualia and the emergence of conscious observers in qualia space

I propose that qualia are physical because they are directly observable, and revisit the contentious link between consciousness and quantum measurements from a new perspective -- one that does not rely on observers or wave function collapse but instead treats physical measurements as fundamental in a sense resonant with Wheeler's it-from-bit. Building on a mathematical definition of measurement space in physics, I reinterpret it as a model of qualia, effectively equating the measurement problem of quantum mechanics with the hard problem of consciousness. The resulting framework falls within panpsychism, and offers potential solutions to the combination problem. Moreover, some of the mathematical structure of measurement spaces, taken for granted in physics, needs justification for qualia, suggesting that the apparent solidity of physical reality is deeply rooted in how humans process information.

physics.hist-ph↗

arXiv2021

Duplicated Code Pattern Mining in Visual Programming Languages

Visual Programming Languages (VPLs), coupled with the high-level abstractions that are commonplace in visual programming environments, enable users with less technical knowledge to become proficient programmers. However, the lower skill floor required by VPLs also entails that programmers are more likely to not adhere to best practices of software development, producing systems with high technical debt, and thus poor maintainability. Duplicated code is one important example of such technical debt. In fact, we observed that the amount of duplication in the OutSystems VPL code bases can reach as high as $39\%$. Duplicated code detection in text-based programming languages is still an active area of research with important implications regarding software maintainability and evolution. However, to the best of our knowledge, the literature on duplicated code detection for VPLs is very limited. We propose a novel and scalable duplicated code pattern mining algorithm that leverages the visual structure of VPLs in order to not only detect duplicated code, but also highlight duplicated code patterns that explain the reported duplication. The performance of the proposed approach is evaluated on a wide range of real-world mobile and web applications developed using OutSystems.

cs.SE↗

arXiv2021

An abstract theory of physical measurements

The question of what should be meant by a measurement is tackled from a mathematical perspective whose physical interpretation is that a measurement is a fundamental process via which a finite amount of classical information is produced. This translates into an algebraic and topological definition of measurement space that caters for the distinction between quantum and classical measurements and allows a notion of observer to be derived.

quant-ph↗

arXiv2020

On the geometry of physical measurements: topological and algebraic aspects

We study the mathematical structure of the notion of measurement space, which extends aspects of noncommutative topology that are based on quantale theory. This yields a geometric model of physical measurements that provides a realist picture, yet also operational, such that measurements and classical information arise interdependently as primitive concepts. A derived notion of classical observer caters for a mathematical formulation of Bohr's classical/quantum divide. Two important classes of measurement spaces are obtained, respectively from C*-algebras and from second-countable locally compact open sober topological groupoids. The latter yield measurements of classical type and relate to Schwinger's notion of selective measurement. We show that the measurement space associated to the reduced C*-algebra of any second-countable locally compact Hausdorff \'etale groupoid is canonically equipped with a classical observer, and we establish a correspondence between properties of the observer and properties of the groupoid.

math-ph↗

arXiv2020

Actions of \'etale-covered groupoids

By restricting to a class of localic open groupoids $G$ which, similarly to Lie groupoids, possess appropriate covers $\widehat G\to G$ by \'etale groupoids, we extend results about groupoid actions and quantales that were previously proved for \'etale groupoids but do not seem to work for arbitrary open groupoids. In particular we obtain a characterization of the category of $G$-actions as a category of quantale modules on $\mathcal O(\widehat G)$ that satisfy a condition related to the quantale $\mathcal O(G)$. This leads to a simple description of $G$-sheaves and the classifying topos of $G$ in terms of Hilbert $\mathcal O(\widehat G)$-modules. The bicategory whose 1-cells are the groupoid bi-actions is bi-equivalent to a corresponding bicategory of quantales and bimodules.

math.CT↗

arXiv2018

Effective equivalence relations and principal quantales

Stably supported quantales generalize pseudogroups and provide an algebraic context in which to study the correspondences between inverse semigroups and \'etale groupoids. Here we study a further generalization where a non-unital version of supported quantale carries the algebraic content of such correspondences to the setting of open groupoids. A notion of principal quantale is introduced which, in the case of groupoid quantales, corresponds precisely to effective equivalence relations.

math.CT↗

arXiv2018

Functoriality of groupoid quantales. II

Taking advantage of the quantale-theoretic description of \'etale groupoids we study principal bundles, Hilsum-Skandalis maps, and Morita equivalence in terms of modules on inverse quantal frames. The Hilbert module description of quantale sheaves leads naturally to a formulation of Morita equivalence in terms of bimodules that resemble imprimitivity bimodules of C*-algebras.

math.CT↗

arXiv2017

Quantales and Fell bundles

We study Fell bundles on groupoids from the viewpoint of quantale theory. Given any saturated upper semicontinuous Fell bundle $π:E\to G$ on an étale groupoid $G$ with $G_0$ locally compact Hausdorff, equipped with a suitable completion C*-algebra $A$ of its convolution algebra, we obtain a map of involutive quantales $p:\mathrm{Max}\ A\toΩ(G)$, where $\mathrm{Max}\ A$ consists of the closed linear subspaces of $A$ and $Ω(G)$ is the topology of $G$. We study various properties of $p$ which mimick, to various degrees, those of open maps of topological spaces. These are closely related to properties of $G$, $π$, and $A$, such as $G$ being Hausdorff, principal, or topological principal, or $π$ being a line bundle. Under suitable conditions, which include $G$ being Hausdorff, but without requiring saturation of the Fell bundle, $A$ is an algebra of sections of the bundle if and only if it is the reduced C*-algebra $C_r^*(G,E)$. We also prove that $\mathrm{Max}\ A$ is stably Gelfand. This implies the existence of a pseudogroup $\mathcal{I}_B$ and of an étale groupoid $\mathfrak B$ associated canonically to any sub-C*-algebra $B\subset A$. We study a correspondence between Fell bundles and sub-C*-algebras based on these constructions, and compare it to the construction of Weyl groupoids from Cartan subalgebras.

math.OA↗

arXiv2017

The many groupoids of a stably Gelfand quantale

We study the projections of an arbitrary stably Gelfand quantale $Q$ and show that each projection determines a pseudogroup $S\subset Q$ (and a corresponding localic étale groupoid $G$) together with a map of involutive quantales $p:Q\to\mathcal L^{\bigvee}(S)\ [=\mathcal O(G)]$. As an application we obtain a simplified axiomatization of inverse quantal frames (= quantales of étale groupoids) whereby such a quantale is shown to be the same as a unital stably Gelfand quantal frame whose partial units cover $Q$.

math.RA↗

arXiv2017

Open maps of involutive quantales

By a map $p:Q\to X$ of involutive quantales is meant a homomorphism $p^*:X\to Q$. Calling a map $p$ weakly open if $p^*$ has a left adjoint $p_!$ which satisfies the Frobenius reciprocity condition (i.e., $p_!$ is a homomorphism of $X$-modules), we say that $p$ is open if it is stably weakly open. We also study a two-sided version, FR2, of the Frobenius reciprocity condition, and show that the weakly open surjections that satisfy FR2 are open. Maps of the latter kind arise in the study of Fell bundles on groupoids.

math.CT↗

arXiv2017

Open quotients of trivial vector bundles

Given an arbitrary topological complex vector space $A$, a quotient vector bundle for $A$ is a quotient of a trivial vector bundle $π_2:A\times X\to X$ by a fiberwise linear continuous open surjection. We show that this notion subsumes that of a Banach bundle over a locally compact Hausdorff space $X$. Hyperspaces consisting of linear subspaces of $A$, topologized with natural topologies that include the lower Vietoris topology and the Fell topology, provide classifying spaces for various classes of quotient vector bundles, in a way that generalizes the classification of locally trivial vector bundles by Grassmannians. If $A$ is normed, a finer hyperspace topology is introduced that classifies bundles with continuous norm, including Banach bundles, and such that bundles of constant finite rank must be locally trivial.

math.FA↗

arXiv2016

Linear structures on locales

We define a notion of morphism for quotient vector bundles that yields both a category $\textit{QVBun}$ and a contravariant global sections functor $C:\textit{QVBun}^{\textrm{op}}\to\textit{Vect}$ whose restriction to trivial vector bundles with fiber $F$ coincides with the contravariant functor $\textit{Top}^{\textrm{op}}\to\textit{Vect}$ of $F$-valued continuous functions. Based on this we obtain a linear extension of the adjunction between the categories of topological spaces and locales: (i) a linearized topological space is a spectral vector bundle, by which is meant a mildly restricted type of quotient vector bundle; (ii) a linearized locale is a locale $\triangle$ equipped with both a topological vector space $A$ and a $\triangle$-valued support map for the elements of $A$ satisfying a continuity condition relative to the spectrum of $\triangle$ and the lower Vietoris topology on $\operatorname{Sub} A$; (iii) we obtain an adjunction between the full subcategory of spectral vector bundles $\textit{QVBun}_Σ$ and the category of linearized locales $\textit{LinLoc}$, which restricts to an equivalence of categories between sober spectral vector bundles and spatial linearized locales. The spectral vector bundles are classified by a finer topology on $\operatorname{Sub} A$, called the open support topology, but there is no notion of universal spectral vector bundle for an arbitrary topological vector space $A$.

math.CT↗

arXiv2015

Invariant means on Boolean inverse monoids

The classical theory of invariant means, which plays an important role in the theory of paradoxical decompositions, is based upon what are usually termed `pseudogroups'. Such pseudogroups are in fact concrete examples of the Boolean inverse monoids which give rise to etale topological groupoids under non-commutative Stone duality. We accordingly initiate the theory of invariant means on arbitrary Boolean inverse monoids. Our main theorem is a characterization of when a Boolean inverse monoid admits an invariant mean. This generalizes the classical Tarski alternative proved, for example, by de la Harpe and Skandalis, but using different methods.

math.CT↗

arXiv2014

Functoriality of groupoid quantales. I

We provide three functorial extensions of the equivalence between localic etale groupoids and their quantales. The main result is a biequivalence between the bicategory of localic etale groupoids, with bi-actions as 1-cells, and a bicategory of inverse quantal frames whose 1-cells are bimodules. As a consequence, the category InvQuF of inverse quantale frames, whose morphisms are the (necessarily involutive) homomorphisms of unital quantales, is equivalent to a category of localic etale groupoids whose arrows are the algebraic morphisms in the sense of Buneci and Stachura. We also show that the subcategory of InvQuF with the same objects and whose morphisms preserve finite meets is dually equivalent to a subcategory of the category of localic etale groupoids and continuous functors whose morphisms, in the context of topological groupoids, have been studied by Lawson and Lenz.

math.CT↗

arXiv2011

Groupoid actions as quantale modules

For an arbitrary localic etale groupoid G we provide simple descriptions, in terms of modules over the quantale O(G) of the groupoid, of the continuous actions of G, including actions on open maps and sheaves. The category of G-actions is isomorphic to a corresponding category of O(G)-modules, and as a corollary we obtain a new quantale based representation of etendues.

math.CT↗

arXiv2011

Groupoid sheaves as quantale sheaves

Several notions of sheaf on various types of quantale have been proposed and studied in the last twenty five years. It is fairly standard that for an involutive quantale Q satisfying mild algebraic properties the sheaves on Q can be defined to be the idempotent self-adjoint Q-valued matrices. These can be thought of as Q-valued equivalence relations, and, accordingly, the morphisms of sheaves are the Q-valued functional relations. Few concrete examples of such sheaves are known, however, and in this paper we provide a new one by showing that the category of equivariant sheaves on a localic etale groupoid G (the classifying topos of G) is equivalent to the category of sheaves on its involutive quantale O(G). As a means towards this end we begin by replacing the category of matrix sheaves on Q by an equivalent category of complete Hilbert Q-modules, and we approach the envisaged example where Q is an inverse quantal frame O(G) by placing it in the wider context of stably supported quantales, on one hand, and in the wider context of a module theoretic description of arbitrary actions of étale groupoids, both of which may be interesting in their own right.

math.RA↗