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

Higher-dimensional realizability for intensional type theory

We develop realizability models of intensional type theory, based on groupoids, wherein realizers themselves carry higher-dimensional structure. In the spirit of realizability this is intended to formalise a higher-dimensional (topological, homotopical) BHK interpretation whereby evidence for an identification is a path. The parameter over which we build realizability models is a "realizer category". These are equipped with an interval qua internal co-groupoid, which facilitates a notion of homotopy in the ambient category, as well as a fundamental groupoid construction on it. In groupoidal realizability, objects of a base groupoid are realized by points in the fundamental groupoid of some object from the realizer category, and the isomorphisms from the base groupoid are realized by paths in that fundamental groupoid. We first explain why a naive formulation of groupoidal assemblies is not fit for modelling type theory; this motivates studying partitioned groupoidal assemblies. The main result of the thesis is that, when the realizer category is finitely complete in a suitable sense, the ensuing category of partitioned groupoidal assemblies is a path category with weak dependent products, hence a model of a version of intensional (1-truncated) type theory with dependent products and without function extensionality. When the underlying realizer category is "untyped", there exists an impredicative universe of 1-types, given by the modest fibrations

Toward a General Rewriting-Based Framework for Reducibility

Reducibility is a powerful proof method which applies to various properties of typed terms in different type systems. For strong normalization, different vari- ants are known, such as Girard's reducibility candidates, Tait's saturated sets and biorthogonals. They differ by the closure conditions imposed to types interpreta- tions, called here reducibility families. This paper is about the computational and observational properties underlying untyped reducibility. Our starting point is the comparison of reducibility families w.r.t. their ability to handle rewriting, for which their possible stability by union plays an important role. Indeed, usual saturated sets are generally stable by union, but with rewriting it can be difficult to define a uniform notion of saturated sets. On the other hand, rewriting is more naturally taken into account by reducibility candidates, but they are not always stable by union. It seems that for a given rewrite relation, the stability by union of reducibility candidates should imply the ability to naturally define corresponding saturated sets. In this paper, we seek to devise a general framework in which the above claim can be substantiated. In particular, this framework should be as simple as possible, while allowing the formulation of general notions of reducibility candidates and saturated sets. We present a notion of non-interaction which allows to define neutral terms and reducibility candidates in a generic way. This notion can be formulated in a very simple and general framework, based only on a rewrite relation and a set of contexts, called elimination contexts, required to satisfy some simple properties. This provides a convenient level of abstraction to prove fundamental properties of reducibility candidates, to compare them with biorthogonals, and to study their stability by union. Moreover, we propose a general form of saturated sets, issued from the stability by union of reducibility candidates

Proving Properties of Typed Lambda-Terms Using Realizability, Covers, and Sheaves (Preliminary Version)

We present a general method for proving properties of typed λ-terms. This method is obtained by introducing a semantic notion of realizability which uses the notion of a cover algebra (as in abstract sheaf theory). For this, we introduce a new class of semantic structures equipped with preorders, called pre-applicative structures. These structures need not be extensional. In this framework, a general realizability theorem can be shown. Kleene\u27s recursive realizability and a variant of Kreisel\u27s modified realizability both fit into this framework. Applying this theorem to the special case of the term model, yields a general theorem for proving properties of typed λ-terms, in particular, strong normalization and confluence. This approach clarifies the reducibility method by showing that the closure conditions on candidates of reducibility can be viewed as sheaf conditions. The above approach is applied to the simply-typed λ-calculus (with types →, x, +, and ⊥), and to the second-order (polymorphic λ-calculus (with types → and ∀2), for which it yields a new theorem

A program for the full axiom of choice

The theory of classical realizability is a framework for the Curry-Howard correspondence which enables to associate a program with each proof in Zermelo-Fraenkel set theory. But, almost all the applications of mathematics in physics, probability, statistics, etc. use Analysis i.e. the axiom of dependent choice (DC) or even the (full) axiom of choice (AC). It is therefore important to find explicit programs for these axioms. Various solutions have been found for DC, for instance the lambda-term called "bar recursion" or the instruction "quote" of LISP. We present here the first program for AC