11 research outputs found
Every metric space is separable in function realizability
We first show that in the function realizability topos every metric space is
separable, and every object with decidable equality is countable. More
generally, working with synthetic topology, every -space is separable and
every discrete space is countable. It follows that intuitionistic logic does
not show the existence of a non-separable metric space, or an uncountable set
with decidable equality, even if we assume principles that are validated by
function realizability, such as Dependent and Function choice, Markov's
principle, and Brouwer's continuity and fan principles
-locales in Formal Topology
A -frame is a poset with countable joins and finite meets in which
binary meets distribute over countable joins. The aim of this paper is to show
that -frames, actually -locales, can be seen as a branch of
Formal Topology, that is, intuitionistic and predicative point-free topology.
Every -frame is the lattice of Lindel\"of elements (those for which
each of their covers admits a countable subcover) of a formal topology of a
specific kind which, in its turn, is a presentation of the free frame over .
We then give a constructive characterization of the smallest (strongly) dense
-sublocale of a given -locale, thus providing a
``-version'' of a Boolean locale. Our development depends on the axiom
of countable choice.Comment: Paper presented at the conference Continuity, Computability,
Constructivity - From Logic to Algorithms (CCC 2017), Nancy, France, June
26-30 201
On the topological aspects of the theory of represented spaces
Represented spaces form the general setting for the study of computability
derived from Turing machines. As such, they are the basic entities for
endeavors such as computable analysis or computable measure theory. The theory
of represented spaces is well-known to exhibit a strong topological flavour. We
present an abstract and very succinct introduction to the field; drawing
heavily on prior work by Escard\'o, Schr\"oder, and others.
Central aspects of the theory are function spaces and various spaces of
subsets derived from other represented spaces, and -- closely linked to these
-- properties of represented spaces such as compactness, overtness and
separation principles. Both the derived spaces and the properties are
introduced by demanding the computability of certain mappings, and it is
demonstrated that typically various interesting mappings induce the same
property.Comment: Earlier versions were titled "Compactness and separation for
represented spaces" and "A new introduction to the theory of represented
spaces
Quotienting the delay monad by weak bisimilarity
The delay datatype was introduced by Capretta (Logical Methods in Computer Science, 1(2), article 1, 2005) as a means to deal with partial functions (as in computability theory) in Martin-Löf type theory. The delay datatype is a monad. It is often desirable to consider two delayed computations equal, if they terminate with equal values, whenever one of them terminates. The equivalence relation underlying this identification is called weak bisimilarity. In type theory, one commonly replaces quotients with setoids. In this approach, the delay datatype quotiented by weak bisimilarity is still a monad-a constructive alternative to the maybe monad. In this paper, we consider the alternative approach of Hofmann (Extensional Constructs in Intensional Type Theory, Springer, London, 1997) of extending type theory with inductive-like quotient types. In this setting, it is difficult to define the intended monad multiplication for the quotiented datatype. We give a solution where we postulate some principles, crucially proposition extensionality and the (semi-classical) axiom of countable choice. With the aid of these principles, we also prove that the quotiented delay datatype delivers free ω-complete pointed partial orders (ωcppos). Altenkirch et al. (Lecture Notes in Computer Science, vol. 10203, Springer, Heidelberg, 534-549, 2017) demonstrated that, in homotopy type theory, a certain higher inductive-inductive type is the free ωcppo on a type X essentially by definition; this allowed them to obtain a monad of free ωcppos without recourse to a choice principle. We notice that, by a similar construction, a simpler ordinary higher inductive type gives the free countably complete join semilattice on the unit type 1. This type suffices for constructing a monad, which is isomorphic to the one of Altenkirch et al. We have fully formalized our results in the Agda dependently typed programming language
Journées Francophones des Langages Applicatifs 2018
National audienceLes 29èmes journées francophones des langages applicatifs (JFLA) se déroulent en 2018 à l'observatoire océanographique de Banyuls-sur-Mer. Les JFLA réunissent chaque année, dans un cadre convivial, concepteurs, développeurs et utilisateurs des langages fonctionnels, des assistants de preuve et des outils de vérification de programmes en présentant des travaux variés, allant des aspects les plus théoriques aux applications industrielles.Cette année, nous avons sélectionné 9 articles de recherche et 8 articles courts. Les thématiques sont variées : preuve formelle, vérification de programmes, modèle mémoire, langages de programmation, mais aussi théorie de l'homotopieet blockchain