This paper gives new confluence proofs for several lambda calculi with permutation-like reduction, including lambda calculi corresponding to intuitionistic and classical natural deduction with disjunction and permutative conversions, and a lambda calculus with explicit substitutions. For lambda calculi with permutative conversion, naïve parallel reduction technique does not work, and traditional notion of residuals is required as Ando pointed out. This paper shows that the difficulties can be avoided by extending the technique proposed by Dehornoy and van Oostrom, called the Z (...) theorem: existence of a mapping on terms with the Z property concludes the confluence. Since it is still hard to directly define a mapping with the Z property for the lambda calculi with permutative conversions, this paper extends the Z theorem to compositional functions, called compositional Z, and shows that we can adopt it to the calculi. (shrink)

This paper studies a new classical natural deduction system, presented as a typed calculus named View the MathML sourceλ̲μlet. It is designed to be isomorphic to Curien and Herbelinʼs View the MathML sourceλ¯μμ˜-calculus, both at the level of proofs and reduction, and the isomorphism is based on the correct correspondence between cut in sequent calculus, and substitution in natural deduction. It is a combination of Parigotʼs λμ-calculus with the idea of “coercion calculus” due to Cervesato and Pfenning, accommodating let-expressions in (...) a surprising way: they expand Parigotʼs syntactic class of named terms.This calculus and the mentioned isomorphism Θ offer three missing components of the proof theory of classical logic: a canonical natural deduction system; a robust process of “read-back” of calculi in the sequent calculus format into natural deduction syntax; a formalization of the usual semantics of the View the MathML sourceλ¯μμ˜-calculus, that explains co-terms and cuts as, respectively, contexts and hole-filling instructions. View the MathML sourceλ̲μlet is not yet another classical calculus, but rather a canonical reflection in natural deduction of the impeccable treatment of classical logic by sequent calculus; and Θ provides the “read-back” map and the formalized semantics, based on the precise notions of context and “hole-expression” provided by View the MathML sourceλ̲μlet.We use “read-back” to achieve a precise connection with Parigotʼs λμ, and to derive λ-calculi for call-by-value combining control and let-expressions in a logically founded way. Finally, the semantics Θ, when fully developed, can be inverted at each syntactic category. This development gives us license to see sequent calculus as the semantics of natural deduction; and uncovers a new syntactic concept in View the MathML sourceλ¯μμ˜ , with which one can give a new definition of η-reduction. (shrink)