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Completing Herbelin’s programme
"... In 1994 Herbelin started and partially achieved the programme of showing that, for intuitionistic implicational logic, there is a CurryHoward interpretation of sequent calculus into a variant of the λcalculus, specifically a variant which manipulates formally “applicative contexts” and inverts t ..."
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In 1994 Herbelin started and partially achieved the programme of showing that, for intuitionistic implicational logic, there is a CurryHoward interpretation of sequent calculus into a variant of the λcalculus, specifically a variant which manipulates formally “applicative contexts” and inverts the associativity of “applicative terms”. Herbelin worked with a fragment of sequent calculus with constraints on left introduction. In this paper we complete Herbelin’s programme for full sequent calculus, that is, sequent calculus without the mentioned constraints, but where permutative conversions necessarily show up. This requires the introduction of a lambdalike calculus for full sequent calculus and an extension of natural deduction that gives meaning to “applicative contexts” and “applicative terms”. Such extension is a calculus with modus ponens and primitive substitution that refines von Plato’s natural deduction; it is also a “coercion calculus”, in the sense of Cervesato and Pfenning. The prooftheoretical outcome is noteworthy: the puzzling relationship between cut and substitution is settled; and cutelimination in sequent calculus is proven isomorphic to normalisation in the proposed natural deduction system. The isomorphism is the mapping that inverts the associativity of applicative terms.
Lectures on the curryhoward isomorphism
, 1998
"... The CurryHoward isomorphism states an amazing correspondence between systems of formal logic as encountered in proof theory and computational calculi as found in type theory. For instance, minimal propositional logic corresponds to simply typed λcalculus, firstorder logic corresponds to dependent ..."
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The CurryHoward isomorphism states an amazing correspondence between systems of formal logic as encountered in proof theory and computational calculi as found in type theory. For instance, minimal propositional logic corresponds to simply typed λcalculus, firstorder logic corresponds to dependent types, secondorder logic corresponds to polymorphic types, etc. The isomorphism has many aspects, even at the syntactic level: formulas correspond to types, proofs correspond to terms, provability corresponds to inhabitation, proof normalization corresponds to term reduction, etc. But there is much more to the isomorphism than this. For instance, it is an old idea—due to Brouwer, Kolmogorov, and Heyting, and later formalized by Kleene’s realizability interpretation—that a constructive proof of an implication is a procedure that transforms proofs of the antecedent into proofs of the succedent; the CurryHoward isomorphism gives syntactic representations of such procedures. These notes give an introduction to parts of proof theory and related
Cut Formulae and Logic Programming
"... . In this paper we present a mechanism to define names for proofwitnesses of formulae and thus to use Gentzen's cutrule in logic programming. We consider a program to be a set of logical formulae together with a list of such definitions. Occurrences of the defined names guide the proofsearch by i ..."
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. In this paper we present a mechanism to define names for proofwitnesses of formulae and thus to use Gentzen's cutrule in logic programming. We consider a program to be a set of logical formulae together with a list of such definitions. Occurrences of the defined names guide the proofsearch by indicating when an instance of the cutrule should be attempted. By using the cutrule there are proofs that can be made dramatically shorter. We explain how this idea of using the cutrule can be applied to the logic of hereditary Harrop formulae. 1 Introduction The computation mechanisms both for logic and for functional programming are searches for cutfree proofs. First, in pure logic programming the achievement of a goal G w.r.t. a program P can be seen 1 as the search for a proof in Gentzen's intuitionistic sequent calculus LJ [Gen69], of the sequent P ) G, that by Gentzen's cutelimination theorem can be cutfree [Bee89], [Mil90]; a term found as a witness to a proof contains among...