342 research outputs found
A Classical Realizability Model arising from a Stable Model of Untyped Lambda Calculus
We study a classical realizability model (in the sense of J.-L. Krivine)
arising from a model of untyped lambda calculus in coherence spaces. We show
that this model validates countable choice using bar recursion and bar
induction
Models of Intuitionistic Set Theory in Subtoposes of Nested Realizability Toposes
With every pca and subpca we associate the
nested realizability topos within
which we identify a class of small maps giving rise to a model of
intuitionistic set theory within . For
every subtopos of such a nested realizability topos we construct
an induced class of small maps in giving rise to
a model of intuitionistic set theory within . This covers relative
realizability toposes, modified relative realizability toposes, the modified
realizability topos and van den Berg's recent Herbrand topos
Classical logic, continuation semantics and abstract machines
One of the goals of this paper is to demonstrate that denotational semantics is useful for operational issues like implementation of functional languages by abstract machines. This is exemplified in a tutorial way by studying the case of extensional untyped call-by-name λ-calculus with Felleisen's control operator 𝒞. We derive the transition rules for an abstract machine from a continuation semantics which appears as a generalization of the ¬¬-translation known from logic. The resulting abstract machine appears as an extension of Krivine's machine implementing head reduction. Though the result, namely Krivine's machine, is well known our method of deriving it from continuation semantics is new and applicable to other languages (as e.g. call-by-value variants). Further new results are that Scott's D∞-models are all instances of continuation models. Moreover, we extend our continuation semantics to Parigot's λμ-calculus from which we derive an extension of Krivine's machine for λμ-calculus. The relation between continuation semantics and the abstract machines is made precise by proving computational adequacy results employing an elegant method introduced by Pitts
Fibred Categories a la Jean Benabou
These are notes about the theory of Fibred Categories as I have learned it
from Jean Benabou. I also have used results from the Thesis of Jean-Luc Moens
from 1982 in those sections where I discuss the fibered view of geometric
morphisms. Thus, almost all of the contents is not due to me but most of it
cannot be found in the literature since Benabou has given many talks on it but
most of his work on fibered categories is unpublished. But I am solely
responsible for the mistakes and for misrepresentations of his views. And
certainly these notes do not cover all the work he has done on fibered
categories. I just try to explain the most important notions he has come up
with in a way trying to be as close as possible to his intentions and
intuitions. I started these notes in 1999 when I gave a course on some of the
material at a workshop in Munich. They have developed quite a lot over the
years and I have tried to include most of the things I want to remember.Comment: Have added an appendix describing a fibrational account of Lawvere's
notion of (stably) precohesive geometric morphism
Computability in basic quantum mechanics
The basic notions of quantum mechanics are formulated in terms of separable infinite dimensional Hilbert space H. In terms of the Hilbert lattice L of closed linear subspaces of H the notions of state and observable can be formulated as kinds of measures as in [21]. The aim of this paper is to show that there is a good notion of computability for these data structures in the sense of Weihrauch’s Type Two Effectivity (TTE) [26]. Instead of explicitly exhibiting admissible representations for the data types under consideration we show that they do live within the category QCB0 which is equivalent to the category AdmRep of admissible representations and continuously realizable maps between them. For this purpose in case of observables we have to replace measures by valuations which allows us to prove an effective version of von Neumann’s Spectral Theorem
A comonad for Grothendieck fibrations
We study the 2-category theory of Grothendieck fibrations in the 2-category
of functors \ct{Cat}^{\ct{2}}. After redrawing a few general results in that
context, we show that fibrations over a given base are pseudo-coalgebras for a
2-comonad on \ct{Cat} / \ct{B}. We use that result to explain how an
arbitrary fibration is equivalent to one with a splitting
Constructive toposes with countable sums as models of constructive set theory
AbstractWe define a constructive topos to be a locally cartesian closed pretopos. The terminology is supported by the fact that constructive toposes enjoy a relationship with constructive set theory similar to the relationship between elementary toposes and (impredicative) intuitionistic set theory. This paper elaborates upon one aspect of the relationship between constructive toposes and constructive set theory. We show that any constructive topos with countable coproducts provides a model of a standard constructive set theory, CZFExp (that is, the variant of Aczel’s Constructive Zermelo–Fraenkel set theory CZF obtained by weakening Subset Collection to the Exponentiation axiom). The model is constructed as a category of classes, using ideas derived from Joyal and Moerdijk’s programme of algebraic set theory. A curiosity is that our model always validates the axiom V=Vω1 (in an appropriate formulation). It follows that the full Separation schema is always refuted
- …