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Monadic translation of classical sequent calculus. (English) Zbl 1311.68042
Summary: We study monadic translations of the call-by-name (cbn) and call-by-value (cbv) fragments of the classical sequent calculus \(\overline{\lambda}\mu\tilde\mu\) due to Curien and Herbelin, and give modular and syntactic proofs of strong normalisation. The target of the translations is a new meta-language for classical logic, named monadic \({\lambda}{\mu}\). This language is a monadic reworking of Parigot’s \({\lambda}{\mu}\)-calculus, where the monadic binding is confined to commands, thus integrating the monad with the classical features. Also, its \({\mu}\)-reduction rule is replaced by a rule expressing the interaction between monadic binding and \({\mu}\)-abstraction.
Our monadic translations produce very tight simulations of the respective fragments of \(\overline{\lambda}\mu\tilde\mu\) within monadic \({\lambda}{\mu}\), with reduction steps of \(\overline{\lambda}\mu\tilde\mu\) being translated in a \(1-1\) fashion, except for \({\beta}\) steps, which require two steps. The monad of monadic \({\lambda}{\mu}\) can be instantiated to the continuations monad so as to ensure strict simulation of monadic \({\lambda}{\mu}\) within simply typed \({\lambda}\)-calculus with \({\beta}\)- and \({\eta}\)-reduction. Through strict simulation, the strong normalisation of simply typed \({\lambda}\)-calculus is inherited by monadic \({\lambda}{\mu}\), and then by cbn and cbv \(\overline{\lambda}\mu\tilde\mu\), thus reproving strong normalisation in an elementary syntactical way for these fragments of \(\overline{\lambda}\mu\tilde\mu\), and establishing it for our new calculus. These results extend to second-order logic, with polymorphic \({\lambda}\)-calculus as the target, giving new strong normalisation results for classical second-order logic in sequent calculus style.
CPS translations of cbn and cbv \(\overline{\lambda}\mu\tilde\mu\) with the strict simulation property are obtained by composing our monadic translations with the continuations-monad instantiation. In an appendix to the paper, we investigate several refinements of the continuations-monad instantiation in order to obtain in a modular way improvements of the CPS translations enjoying extra properties like simulation by cbv \({\beta}\)-reduction or reduction of administrative redexes at compile time.
68N18 Functional programming and lambda calculus
03B40 Combinatory logic and lambda calculus
03F05 Cut-elimination and normal-form theorems
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