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| LECTURES ON THE CURRY-HOWARD ISOMORPHISM, 149
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By
Morten Heine S rensen, M.Sc, Ph.D, University of Copenhagen, Denmark
Pawel Urzyczyn, prof. dr hab., Warsaw University, Poland
Included in series
Studies in Logic and the Foundations of Mathematics, 149
Description
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 transforms
proofs 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
Audience
Graduate students, lecturers and researchers in logic and theoretical computer science. Also for graduate students, lecturers and researchers in philosophy and mathematics.
Contents
Preface
Acknowledgements
1. Typefree lambda-calculus
2. Intuitionistic logic
3. Simply typed lambdacalculus
4. The
Curry-Howard isomorphism
5. Proofs as combinators
6. Classical logic and control operators
7. Sequent calculus
8. First-order
logic
9. First-order arithmetic
10. G del's system T
11. Second-order logic and polymorphism
12. Second-order arithmetic
13. Dependent types
14. Pure type systems and the lambda-cube
A Mathematical Background
B Solutions and hints to selected
exercises
Bibliography
Index
| Bibliographic details |
Hardbound, 456 pages, publication date: JUL-2006
ISBN-13: 978-0-444-52077-7
ISBN-10: 0-444-52077-5
Imprint: ELSEVIER
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Last update: 30 Nov 2009
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