Lectures on the Curry-Howard Isomorphism

Lectures on the Curry-Howard Isomorphism (1998)

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by Morten Heine B. Sørensen , Pawel Urzyczyn

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@MISC{Sørensen98lectureson,
    author = {Morten Heine B. Sørensen and Pawel Urzyczyn},
    title = {Lectures on the Curry-Howard Isomorphism},
    year = {1998}
}

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Abstract

The Curry-Howard 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-calculus, first-order logic corresponds to dependent types, second-order 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 Curry-Howard isomorphism gives syntactic representations of such procedures. These notes give an introduction to parts of proof theory and related aspects of type theory relevant for the Curry-Howard isomorphism.

Keyphrases

curry-howard isomorphism proof theory realizability interpretation computational calculus minimal propositional logic corresponds many aspect amazing correspondence type theory relevant second-order logic corresponds proof normalization corresponds constructive proof first-order logic corresponds syntactic level old idea formula correspond term reduction proof correspond related aspect type theory formal logic syntactic representation provability corresponds