On natural deduction in classical first-order logic: Curry–Howard correspondence, strong normalization and Herbrand's theorem

We present a new Curry–Howard correspondence for classical first-order natural deduction. We add to the lambda calculus an operator which represents, from the viewpoint of programming, a mechanism for raising and catching multiple exceptions, and from the viewpoint of logic, the excluded middle over...

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Vydáno v:Theoretical computer science Ročník 625; s. 125 - 146
Hlavní autoři: Aschieri, Federico, Zorzi, Margherita
Médium: Journal Article
Jazyk:angličtina
Vydáno: Elsevier B.V 25.04.2016
Elsevier
Témata:
Byproducts
Calculus
Classical first-order logic
Computer Science
Curry–Howard correspondence
Deduction
Delimited exceptions
Herbrand's theorem
Logic
Logic in Computer Science
Mathematical analysis
Mathematics
Natural deduction
Programming
Theorem proving
Theorems
Classical first-order logic
Natural deduction
Delimited exceptions
Curry–Howard correspondence
Herbrand's theorem
ISSN:0304-3975, 1879-2294
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Shrnutí:We present a new Curry–Howard correspondence for classical first-order natural deduction. We add to the lambda calculus an operator which represents, from the viewpoint of programming, a mechanism for raising and catching multiple exceptions, and from the viewpoint of logic, the excluded middle over arbitrary prenex formulas. The machinery will allow to extend the idea of learning – originally developed in Arithmetic – to pure logic. We prove that our typed calculus is strongly normalizing and show that proof terms for simply existential statements reduce to a list of individual terms forming an Herbrand disjunction. A by-product of our approach is a natural-deduction proof and a computational interpretation of Herbrand's Theorem.
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ISSN:0304-3975
1879-2294
DOI:10.1016/j.tcs.2016.02.028