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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.
##### MSC:
 68N18 Functional programming and lambda calculus 03B40 Combinatory logic and lambda calculus 03F05 Cut-elimination and normal-form theorems
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