151 research outputs found
Bibliography on Realizability
AbstractThis document is a bibliography on realizability and related matters. It has been collected by Lars Birkedal based on submissions from the participants in “A Workshop on Realizability Semantics and Its Applications”, Trento, Italy, June 30–July 1, 1999. It is available in BibTEX format at the following URL: http://www.cs.cmu.edu./~birkedal/realizability-bib.html
Impredicative Encodings of (Higher) Inductive Types
Postulating an impredicative universe in dependent type theory allows System
F style encodings of finitary inductive types, but these fail to satisfy the
relevant {\eta}-equalities and consequently do not admit dependent eliminators.
To recover {\eta} and dependent elimination, we present a method to construct
refinements of these impredicative encodings, using ideas from homotopy type
theory. We then extend our method to construct impredicative encodings of some
higher inductive types, such as 1-truncation and the unit circle S1
Parametricity and Local Variables
We propose that the phenomenon of local state may be understood in terms of Strachey\u27s concept of parametric (i.e., uniform) polymorphism. The intuitive basis for our proposal is the following analogy: a non-local procedure is independent of locally-declared variables in the same way that a parametrically polymorphic function is independent of types to which it is instantiated. A connection between parametricity and representational abstraction was first suggested by J. C. Reynolds. Reynolds used logical relations to formalize this connection in languages with type variables and user-defined types. We use relational parametricity to construct a model for an Algol-like language in which interactions between local and non-local entities satisfy certain relational criteria. Reasoning about local variables essentially involves proving properties of polymorphic functions. The new model supports straightforward validations of all the test equivalences that have been proposed in the literature for local-variable semantics, and encompasses standard methods of reasoning about data representations. It is not known whether our techniques yield fully abstract semantics. A model based on partial equivalence relations on the natural numbers is also briefly examined
Regular Functors and Relative Realizability Categories
Relative realizability toposes satisfy a universal property that involves
regular functors to other categories. We use this universal property to define
what relative realizability categories are, when based on other categories than
of the topos of sets. This paper explains the property and gives a construction
for relative realizability categories that works for arbitrary base Heyting
categories. The universal property shows us some new geometric morphisms to
relative realizability toposes too
Relational Parametricity for Computational Effects
According to Strachey, a polymorphic program is parametric if it applies a
uniform algorithm independently of the type instantiations at which it is
applied. The notion of relational parametricity, introduced by Reynolds, is one
possible mathematical formulation of this idea. Relational parametricity
provides a powerful tool for establishing data abstraction properties, proving
equivalences of datatypes, and establishing equalities of programs. Such
properties have been well studied in a pure functional setting. Many programs,
however, exhibit computational effects, and are not accounted for by the
standard theory of relational parametricity. In this paper, we develop a
foundational framework for extending the notion of relational parametricity to
programming languages with effects.Comment: 31 pages, appears in Logical Methods in Computer Scienc
What should a generic object be?
Jacobs has proposed definitions for (weak, strong, split) generic objects for
a fibered category; building on his definition of generic object and split
generic object, Jacobs develops a menagerie of important fibrational structures
with applications to categorical logic and computer science, including higher
order fibrations, polymorphic fibrations, -fibrations, triposes, and
others. We observe that a split generic object need not in particular be a
generic object under the given definitions, and that the definitions of
polymorphic fibrations, triposes, etc. are strict enough to rule out many
fundamental examples: for instance, the fibered preorder induced by a partial
combinatory algebra in realizability is not a tripos in the sense of Jacobs. We
argue for a new alignment of terminology that emphasizes the forms of generic
object that appear most commonly in nature, i.e. in the study of internal
categories, triposes, and the denotational semantics of polymorphic types. In
addition, we propose a new class of acyclic generic objects inspired by recent
developments in the semantics of homotopy type theory, generalizing the
realignment property of universes to the setting of an arbitrary fibration
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