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Call-by-name reduction and cut-elimination in classical logic. (English) Zbl 1136.03036
Summary: We present a version of Herbelin’s $$\overline{\lambda}\mu$$-calculus in the call-by-name setting to study the precise correspondence between normalization and cut-elimination in classical logic. Our translation of $$\lambda \mu$$-terms into a set of terms in the calculus does not involve any administrative redexes, in particular $$\eta$$-expansion on $$\mu$$-abstraction. The isomorphism preserves $$\beta ,\mu$$-reduction, which is simulated by a local-step cut-elimination procedure in the typed case, where the reduction system strictly follows the “cut = redex” paradigm. We show that the underlying untyped calculus is confluent and enjoys the PSN (preservation of strong normalization) property for the isomorphic image of $$\lambda \mu$$-calculus, which in turn yields a confluent and strongly normalizing local-step cut-elimination procedure for classical logic.

##### MSC:
 03F05 Cut-elimination and normal-form theorems 03B05 Classical propositional logic 03B40 Combinatory logic and lambda calculus 03B70 Logic in computer science
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##### References:
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