10 research outputs found
Continuation-passing Style Models Complete for Intuitionistic Logic
A class of models is presented, in the form of continuation monads
polymorphic for first-order individuals, that is sound and complete for minimal
intuitionistic predicate logic. The proofs of soundness and completeness are
constructive and the computational content of their composition is, in
particular, a -normalisation-by-evaluation program for simply typed
lambda calculus with sum types. Although the inspiration comes from Danvy's
type-directed partial evaluator for the same lambda calculus, the there
essential use of delimited control operators (i.e. computational effects) is
avoided. The role of polymorphism is crucial -- dropping it allows one to
obtain a notion of model complete for classical predicate logic. The connection
between ours and Kripke models is made through a strengthening of the
Double-negation Shift schema
An interpretation of the Sigma-2 fragment of classical Analysis in System T
We show that it is possible to define a realizability interpretation for the
-fragment of classical Analysis using G\"odel's System T only. This
supplements a previous result of Schwichtenberg regarding bar recursion at
types 0 and 1 by showing how to avoid using bar recursion altogether. Our
result is proved via a conservative extension of System T with an operator for
composable continuations from the theory of programming languages due to Danvy
and Filinski. The fragment of Analysis is therefore essentially constructive,
even in presence of the full Axiom of Choice schema: Weak Church's Rule holds
of it in spite of the fact that it is strong enough to refute the formal
arithmetical version of Church's Thesis
Perspectives for proof unwinding by programming languages techniques
In this chapter, we propose some future directions of work, potentially
beneficial to Mathematics and its foundations, based on the recent import of
methodology from the theory of programming languages into proof theory. This
scientific essay, written for the audience of proof theorists as well as the
working mathematician, is not a survey of the field, but rather a personal view
of the author who hopes that it may inspire future and fellow researchers
Delimited control operators prove Double-negation Shift
We propose an extension of minimal intuitionistic predicate logic, based on
delimited control operators, that can derive the predicate-logic version of the
Double-negation Shift schema, while preserving the disjunction and existence
properties
Normalization by realizability also evaluates
National audienceFor those of us that generally live in the world of syntax, semanticproof methods such as realizability or logical relations /parametricity sometimes feel like magic. Why do they work? At whichpoint in the proof is "the real work" done?Bernardy and Lasson express realizability and parametricity models assyntactic model -- but the abstraction/adequacy theorems are stillexplained as meta-level proofs. Hoping to better understand the prooftechnique, we look at those theorems as programs themselves. How doesa normalization proof using realizability actually computes thosenormal forms?This detective work is an early stage and we propose a first attemptin a simple setting. Instead of arbitrary Pure Type Systems, we usethe minimal negative propositional logic (arrows only). Instead ofstarting from the simply-typed lambda-calculus, we work onsequent-style terms in a simple subset of the Curien-HerbelinL calculus.Pour ceux d'entre nous qui vivent dans le monde de la syntaxe, les techniques de preuve sémantiques, comme la réalisabilité, les relations logiques ou la paramétricité ont parfois un arrière-goût de méthode magique. Pourquoi fonctionnent-elles ? Quel est le point clé de la preuve ? Bernardy et Lasson expriment la parametricité comme une construction de modèle syntaxique, par traduction bien typée, mais leurs théorèmes d'abstracion et adéquation restent des résultats au méta-niveau. Dans l'espoir de mieux comprendre ces résultats, nous étudiont leurs preuves comme des programmes. Les preuves de normalization par réalisabilité calculent-elles effectivement des formes normales, comment et à quel moment ? Ce travail de détective est encore à ses débuts, et nous proposons une première tentative dans un cadre très simple : au lieu de Pure Type Systems (PTS), nous utilisons le lambda-calcul simplement typé
An analysis of the constructive content of Henkin's proof of G\"odel's completeness theorem
G{\"o}del's completeness theorem for classical first-order logic is one of
the most basic theorems of logic. Central to any foundational course in logic,
it connects the notion of valid formula to the notion of provable formula.We
survey a few standard formulations and proofs of the completeness theorem
before focusing on the formal description of a slight modification of Henkin's
proof within intuitionistic second-order arithmetic.It is standard in the
context of the completeness of intuitionistic logic with respect to various
semantics such as Kripke or Beth semantics to follow the Curry-Howard
correspondence and to interpret the proofs of completeness as programs which
turn proofs of validity for these semantics into proofs of derivability.We
apply this approach to Henkin's proof to phrase it as a program which
transforms any proof of validity with respect to Tarski semantics into a proof
of derivability.By doing so, we hope to shed an effective light on the relation
between Tarski semantics and syntax: proofs of validity are syntactic objects
with which we can compute.Comment: R{\'e}dig{\'e} en 4 {\'e}tapes: 2013, 2016, 2022, 202
Russian Constructivism in a Prefascist Theory
International audienc
Une Dialectica matérialiste
In this thesis, we give a computational interpretation to Gödel's Dialectica translation, in a fashion inspired by classical realizability. In particular, it can be shown that the Dialectica translation manipulates stacks of the Krivine machine as first-class objects and that the main effect at work lies in the accumulation of those stacks at each variable use. The original translation suffers from a handful of defects due to hacks used by Gödel to work around historical limitations. Once these defects are solved, the translation naturally extends to much more expressive settings such as dependent type theory. A few variants are studied thanks to the linear decomposition, and relationships with other translations such as forcing and CPS are scrutinized.Cette thèse fournit une interprétation calculatoire de la traduction dite Dialectica de Gödel, dans une démarche inspirée par la réalisabilité classique. On peut en particulier montrer que Dialectica manipule des piles de la machine de Krivine comme objets de première classe et que le principal effet de cette traduction consiste à accumuler ces piles à chaque utilisation de variables. La traduction d'origine souffre d'une certaine quantité de défauts dus aux hacks utilisés par Gödel pour contourner des limitations historiques. Une fois ces problèmes résolus, la traduction s'étend naturellement à des paradigmes beaucoup plus expressifs tels que la théorie des types dépendants. On étudie d'autres variantes par la suite grâce à la décomposition linéaire, ainsi que lien de parenté avec d'autres traductions tels que le forcing et les CPS