Exhaustible sets in higher-type computation

We say that a set is exhaustible if it admits algorithmic universal quantification for continuous predicates in finite time, and searchable if there is an algorithm that, given any continuous predicate, either selects an element for which the predicate holds or else tells there is no example. The Cantor space of infinite sequences of binary digits is known to be searchable. Searchable sets are exhaustible, and we show that the converse also holds for sets of hereditarily total elements in the hierarchy of continuous functionals; moreover, a selection functional can be constructed uniformly from a quantification functional. We prove that searchable sets are closed under intersections with decidable sets, and under the formation of computable images and of finite and countably infinite products. This is related to the fact, established here, that exhaustible sets are topologically compact. We obtain a complete description of exhaustible total sets by developing a computational version of a topological Arzela--Ascoli type characterization of compact subsets of function spaces. We also show that, in the non-empty case, they are precisely the computable images of the Cantor space. The emphasis of this paper is on the theory of exhaustible and searchable sets, but we also briefly sketch applications

Projecte d’un hivernacle de 80 m2 dissenyat amb criteris sostenibles

With this end-of-grade work, I intend to design and build a greenhouse of 80 square meters with sustainable criteria, located in the Masia sant Joan de la Muntanya in Pontons (Barcelona). The objective is designing and developing a greenhouse provided with solar energy (2153,3 Wh) and which the water used to irrigate the crops contained inside is provided just with the rainwater from its own roof and from the farmhouse. Previously, a building will be designed using wood as a construction material and following the patterns and needs similar to those of the area where it's located. For that purpose, I'll start with the results of the climate study, which being projected on the specific tool, will identify the strategies to be followed to develop a self-sufficient greenhouse. Once the study and design of the building is done, it will proceed to its construction.Amb aquest treball de fi de grau es pretén realitzar el disseny i construcció d'un hivernacle de 80 m2 amb criteris sostenibles, situat a les immediacions de la Masia sant Joan de la Muntanya, a Pontons (Barcelona). L'objectiu és dissenyar i desenvolupar un hivernacle el qual es proveeixi d'energia solar (2153,3 Wh) i que tingui un cultiu que es regui únicament amb les aigües pluvials recollides en la seva pròpia coberta i de la de la masia. Prèviament es dissenyarà un edifici usant la fusta com a material de construcció i seguint els patrons i necessitats similars als de la zona on s'ubica. Per això es partirà dels resultats del estudi climàtic, que projectats sobre les eines específiques, identificaran les estratègies a seguir per desenvolupar un hivernacle autosuficient. Un cop realitzat tot l'estudi i disseny i de l'edifici, es procedirà a la seva construcció.Con este trabajo de fin de grado se pretende realizar el diseño y la construcción de un invernadero de 80m2 con criterios sostenibles, situado en las inmediaciones de la Masia sant Joan de la Muntanya, en Pontons (Barcelona). El objetico es diseñar y desarrollar un invernadero el cual se provea energía solar (2153,3 Wh) y que tenga un cultivo el riego el cual provenga únicamente de la recogida de aguas pluviales de su propia cubierta y de la de la masia. Previamente se diseñará un edificio usando madera como material de construcción y siguiendo los patrones y necesidades similares a los de la zona donde se ubica. Para eso se partirá de los resultados del estudio climático, que proyectados sobres las herramientas específicas, identificaran las estrategias a seguir para desarrollar un invernadero autosuficiente. Una vez realizado el estudio y diseño del edificio, se procederá a su construcción.Objectius de Desenvolupament Sostenible::6 - Aigua Neta i Sanejamen

On generalized algebraic theories and categories with families

We give a syntax independent formulation of finitely presented generalized algebraic theories as initial objects in categories of categories with families (cwfs) with extra structure. To this end, we simultaneously define the notion of a presentation Σ of a generalized algebraic theory and the associated category CwFΣ of small cwfs with a Σ-structure and cwf-morphisms that preserve Σ-structure on the nose. Our definition refers to the purely semantic notion of uniform family of contexts, types, and terms in CwFΣ. Furthermore, we show how to syntactically construct an initial cwf with a Σ-structure. This result can be viewed as a generalization of Birkhoff’s completeness theorem for equational logic. It is obtained by extending Castellan, Clairambault, and Dybjer’s construction of an initial cwf. We provide examples of generalized algebraic theories for monoids, categories, categories with families, and categories with families with extra structure for some type formers of Martin-Löf type theory. The models of these are internal monoids, internal categories, and internal categories with families (with extra structure) in a small category with families. Finally, we show how to extend our definition to some generalized algebraic theories that are not finitely presented, such as the theory of contextual cwfs.publishedVersio

Notions of anonymous existence in Martin-Löf type theory

As the groupoid model of Hofmann and Streicher proves, identity proofs in intensional Martin-L\"of type theory cannot generally be shown to be unique. Inspired by a theorem by Hedberg, we give some simple characterizations of types that do have unique identity proofs. A key ingredient in these constructions are weakly constant endofunctions on identity types. We study such endofunctions on arbitrary types and show that they always factor through a propositional type, the truncated or squashed domain. Such a factorization is impossible for weakly constant functions in general (a result by Shulman), but we present several non-trivial cases in which it can be done. Based on these results, we define a new notion of anonymous existence in type theory and compare different forms of existence carefully. In addition, we show possibly surprising consequences of the judgmental computation rule of the truncation, in particular in the context of homotopy type theory. All the results have been formalized and verified in the dependently typed programming language Agda

Meteorología del vuelo en Alta montaña

La presente publicación recoge los temas desarrollados, los problemasplanteados, y las soluciones a ello apuntadas en las XIIIJornadas Científicas de la Asociación Meteorológica Española yI Simposio Nacional de Meteorología de Alta Montaña celebradosen Jaca del 4 al 9 de mayo de 1982, con el patrocinio del ExcelentísimoSr. Ministro de Transportes, Turismo y Comunicaciones encolaboración con el Instituto Nacional de Meteorología y bajo laPresidencia de Honor de S. M. el Rey Don Juan Carlos I

A universal characterization of the closed Euclidean interval

We propose a notion of interval object in a category with finite products, providing a universal property for closed and bounded real line segments. The universal property gives rise to an analogue of primitive recursion for defining computable functions on the interval. We use this to define basic arithmetic operations and to verify equations between them. We test the notion in categories of interest. In the category of sets, any closed and bounded interval of real numbers is an interval object. In the category of topological spaces, the interval objects are closed and bounded intervals with the Euclidean topology. We also prove that an interval object exists in any elementary topos with natural numbers object.</p

Notions of anonymous existence in Martin-Löf type theory

As the groupoid model of Hofmann and Streicher proves, identity proofs in intensional Martin-L\"of type theory cannot generally be shown to be unique. Inspired by a theorem by Hedberg, we give some simple characterizations of types that do have unique identity proofs. A key ingredient in these constructions are weakly constant endofunctions on identity types. We study such endofunctions on arbitrary types and show that they always factor through a propositional type, the truncated or squashed domain. Such a factorization is impossible for weakly constant functions in general (a result by Shulman), but we present several non-trivial cases in which it can be done. Based on these results, we define a new notion of anonymous existence in type theory and compare different forms of existence carefully. In addition, we show possibly surprising consequences of the judgmental computation rule of the truncation, in particular in the context of homotopy type theory. All the results have been formalized and verified in the dependently typed programming language Agda