On the logical structure of choice and bar induction principles

Autor: Nuria Brede, Hugo Herbelin
Přispěvatelé: University of Potsdam, Design, study and implementation of languages for proofs and programs (PI.R2), Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Paris (UP)-Institut de Recherche en Informatique Fondamentale (IRIF (UMR_8243)), Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), University of Potsdam = Universität Potsdam, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Paris Cité (UPCité)-Institut de Recherche en Informatique Fondamentale (IRIF (UMR_8243)), Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)
Rok vydání: 2021
Předmět:
Zdroj: LICS
LICS 2021-36th Annual Symposium on Logic in Computer Science
LICS 2021-36th Annual Symposium on Logic in Computer Science, Jun 2021, Rome / Virtual, Italy
Logic in Computer Science
Logic in Computer Science, Jun 2021, Rome, Italy
Popis: We develop an approach to choice principles and their contrapositive bar-induction principles as extensionality schemes connecting an "intensional" or "effective" view of respectively ill-and well-foundedness properties to an "extensional" or "ideal" view of these properties. After classifying and analysing the relations between different intensional definitions of ill-foundedness and well-foundedness, we introduce, for a domain $A$, a codomain $B$ and a "filter" $T$ on finite approximations of functions from $A$ to $B$, a generalised form GDC$_{A,B,T}$ of the axiom of dependent choice and dually a generalised bar induction principle GBI$_{A,B,T}$ such that: GDC$_{A,B,T}$ intuitionistically captures the strength of$\bullet$ the general axiom of choice expressed as $\forall a\exists\beta R(a, b) \Rightarrow\exists\alpha\forall a R(\alpha,(a \alpha (a)))$ when $T$ is a filter that derives point-wise from a relation $R$ on $A x B$ without introducing further constraints,$\bullet$ the Boolean Prime Filter Theorem / Ultrafilter Theorem if $B$ is the two-element set $\mathbb{B}$ (for a constructive definition of prime filter),$\bullet$ the axiom of dependent choice if $A = \mathbb{N}$,$\bullet$ Weak K{\"o}nig's Lemma if $A = \mathbb{N}$ and $B = \mathbb{B}$ (up to weak classical reasoning): GBI$_{A,B,T}$ intuitionistically captures the strength of$\bullet$ G{\"o}del's completeness theorem in the form validity implies provability for entailment relations if $B = \mathbb{B}$,$\bullet$ bar induction when $A = \mathbb{N}$,$\bullet$ the Weak Fan Theorem when $A = \mathbb{N}$ and $B = \mathbb{B}$.Contrastingly, even though GDC$_{A,B,T}$ and GBI$_{A,B,T}$ smoothly capture several variants of choice and bar induction, some instances are inconsistent, e.g. when $A$ is $\mathbb{B}^\mathbb{N}$ and $B$ is $\mathbb{N}$.
Comment: LICS 2021 - 36th Annual Symposium on Logic in Computer Science, Jun 2021, Rome / Virtual, Italy
Databáze: OpenAIRE
Externí odkaz: