Lectures on the Curry-Howard Isomorphism

by Morten Heine Sørensen M.Sc, Ph.D, Pawel Urzyczyn prof. dr hab.

Overview

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 lambda-calculus, first-order logic corresponds to dependent types, second-order logic corresponds to polymorphic types, sequent calculus is related to explicit substitution, 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 more to the isomorphism than this. For instance, it is an old idea—-due to Brouwer, Kolmogorov, and Heyting—-that a constructive proof of an implication is a procedure that transformsproofs of the antecedent into proofs of the succedent; the Curry-Howard isomorphism gives syntactic representations of such procedures. The Curry-Howard isomorphism also provides theoretical foundations for many modern proof-assistant systems (e.g. Coq).This book give an introduction to parts of proof theory and related aspects of type theory relevant for the Curry-Howard isomorphism. It can serve as an introduction to any or both of typed lambda-calculus and intuitionistic logic.

Key features- The Curry-Howard Isomorphism treated as common theme- Reader-friendly introduction to two complementary subjects: Lambda-calculus and constructive logics- Thorough study of the connection between calculi and logics- Elaborate study of classical logics and control operators- Account of dialogue games for classical and intuitionistic logic- Theoretical foundations of computer-assisted reasoning

Product Details

ISBN-13:	9780444520777
Publisher:	Elsevier Science
Publication date:	09/14/2006
Series:	Studies in Logic and the Foundations of Mathematics , #149
Pages:	456
Product dimensions:	6.00(w) x 9.00(h) x (d)

Table of Contents

PrefaceAcknowledgements1. Typefree lambda-calculus2. Intuitionistic logic3. Simply typed lambdacalculus4. The Curry-Howard isomorphism5. Proofs as combinators6. Classical logic and control operators7. Sequent calculus8. First-order logic9. First-order arithmetic10. Gödel's system T11. Second-order logic and polymorphism12. Second-order arithmetic13. Dependent types14. Pure type systems and the lambda-cubeA Mathematical BackgroundB Solutions and hints to selected exercisesBibliographyIndex