88 research outputs found
Strong normalization of lambda-Sym-Prop- and lambda-bar-mu-mu-tilde-star- calculi
In this paper we give an arithmetical proof of the strong normalization of
lambda-Sym-Prop of Berardi and Barbanera [1], which can be considered as a
formulae-as-types translation of classical propositional logic in natural
deduction style. Then we give a translation between the
lambda-Sym-Prop-calculus and the lambda-bar-mu-mu-tilde-star-calculus, which is
the implicational part of the lambda-bar-mu-mu-tilde-calculus invented by
Curien and Herbelin [3] extended with negation. In this paper we adapt the
method of David and Nour [4] for proving strong normalization. The novelty in
our proof is the notion of zoom-in sequences of redexes, which leads us
directly to the proof of the main theorem
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
An estimation for the lengths of reduction sequences of the -calculus
Since it was realized that the Curry-Howard isomorphism can be extended to
the case of classical logic as well, several calculi have appeared as
candidates for the encodings of proofs in classical logic. One of the most
extensively studied among them is the -calculus of Parigot. In this
paper, based on the result of Xi presented for the -calculus Xi, we
give an upper bound for the lengths of the reduction sequences in the
-calculus extended with the - and -rules.
Surprisingly, our results show that the new terms and the new rules do not add
to the computational complexity of the calculus despite the fact that
-abstraction is able to consume an unbounded number of arguments by virtue
of the -rule
Strong normalization results by translation
We prove the strong normalization of full classical natural deduction (i.e.
with conjunction, disjunction and permutative conversions) by using a
translation into the simply typed lambda-mu-calculus. We also extend Mendler's
result on recursive equations to this system.Comment: Submitted to APA
The First-Order Hypothetical Logic of Proofs
The Propositional Logic of Proofs (LP) is a modal logic in which the modality □A is revisited as [[t]]A , t being an expression that bears witness to the validity of A . It enjoys arithmetical soundness and completeness, can realize all S4 theorems and is capable of reflecting its own proofs ( ⊢A implies ⊢[[t]]A , for some t ). A presentation of first-order LP has recently been proposed, FOLP, which enjoys arithmetical soundness and has an exact provability semantics. A key notion in this presentation is how free variables are dealt with in a formula of the form [[t]]A(i) . We revisit this notion in the setting of a Natural Deduction presentation and propose a Curry–Howard correspondence for FOLP. A term assignment is provided and a proof of strong normalization is given.Fil: Steren, Gabriela. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; ArgentinaFil: Bonelli, Eduardo Augusto. Universidad Nacional de Quilmes. Departamento de Ciencia y Tecnología; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin
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