734 research outputs found
Arithmetical proofs of strong normalization results for symmetric lambda calculi
International audienceWe give arithmetical proofs of the strong normalization of two symmetric -calculi corresponding to classical logic. The first one is the -calculus introduced by Curien & Herbelin. It is derived via the Curry-Howard correspondence from Gentzen's classical sequent calculus LK in order to have a symmetry on one side between ``program'' and ``context'' and on other side between ``call-by-name'' and ``call-by-value''. The second one is the symmetric -calculus. It is the -calculus introduced by Parigot in which the reduction rule , which is the symmetric of , is added. These results were already known but the previous proofs use candidates of reducibility where the interpretation of a type is defined as the fix point of some increasing operator and thus, are highly non arithmetical
Strong Normalization for HA + EM1 by Non-Deterministic Choice
We study the strong normalization of a new Curry-Howard correspondence for HA
+ EM1, constructive Heyting Arithmetic with the excluded middle on
Sigma01-formulas. The proof-term language of HA + EM1 consists in the lambda
calculus plus an operator ||_a which represents, from the viewpoint of
programming, an exception operator with a delimited scope, and from the
viewpoint of logic, a restricted version of the excluded middle. We give a
strong normalization proof for the system based on a technique of
"non-deterministic immersion".Comment: In Proceedings COS 2013, arXiv:1309.092
Why the Usual Candidates of Reducibility Do Not Work for the Symmetric λμ-calculus
AbstractThe symmetric λμ-calculus is the λμ-calculus introduced by Parigot in which the reduction rule μ′, which is the symmetric of μ, is added. We give examples explaining why the technique using the usual candidates of reducibility does not work. We also prove a standardization theorem for this calculus
Proofs and Refutations for Intuitionistic and Second-Order Logic
The ?^{PRK}-calculus is a typed ?-calculus that exploits the duality between the notions of proof and refutation to provide a computational interpretation for classical propositional logic. In this work, we extend ?^{PRK} to encompass classical second-order logic, by incorporating parametric polymorphism and existential types. The system is shown to enjoy good computational properties, such as type preservation, confluence, and strong normalization, which is established by means of a reducibility argument. We identify a syntactic restriction on proofs that characterizes exactly the intuitionistic fragment of second-order ?^{PRK}, and we study canonicity results
Termination of rewrite relations on -terms based on Girard's notion of reducibility
In this paper, we show how to extend the notion of reducibility introduced by
Girard for proving the termination of -reduction in the polymorphic
-calculus, to prove the termination of various kinds of rewrite
relations on -terms, including rewriting modulo some equational theory
and rewriting with matching modulo , by using the notion of
computability closure. This provides a powerful termination criterion for
various higher-order rewriting frameworks, including Klop's Combinatory
Reductions Systems with simple types and Nipkow's Higher-order Rewrite Systems
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