# Lectures on the Curry-Howard Isomorphism, Volume 149

## 1st Edition

**Authors:**Morten Heine Sørensen Pawel Urzyczyn

**Print ISBN:**9780444545961

**eBook ISBN:**9780080478920

**Imprint:**Elsevier Science

**Published Date:**4th July 2006

**Page Count:**456

## 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

## Key Features

· The Curry-Howard Isomorphism treated as the 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

## Readership

Graduate students, lecturers and researchers in logic and theoretical computer science. Also for graduate students, lecturers and researchers in philosophy and mathematics.

## Table of Contents

Preface

Acknowledgements

- Typefree lambda-calculus
- Intuitionistic logic
- Simply typed lambdacalculus
- The Curry-Howard isomorphism
- Proofs as combinators
- Classical logic and control operators
- Sequent calculus
- First-order logic
- First-order arithmetic
- Gödel's system T
- Second-order logic and polymorphism
- Second-order arithmetic
- Dependent types
- Pure type systems and the lambda-cube

A Mathematical Background

B Solutions and hints to selected exercises

Bibliography

Index

## Details

- No. of pages:
- 456

- Language:
- English

- Copyright:
- © Elsevier Science 2006

- Published:
- 4th July 2006

- Imprint:
- Elsevier Science

- eBook ISBN:
- 9780080478920

- Hardcover ISBN:
- 9780444520777

- Paperback ISBN:
- 9780444545961

## About the Author

### Morten Heine Sørensen

### Affiliations and Expertise

University of Copenhagen, Denmark

### Pawel Urzyczyn

### Affiliations and Expertise

Warsaw University, Poland

## Reviews

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