Book on Curry-Howard Isomorphisms

Question

I would like to learn about the Curry-Howard Isomorphism because I want to know more about the connections between computability and logic. I have already read a book on first order logic and I know about natural deduction, so I do not want to learn about logic, but about the connections between logic and computability. Can you recommend few books for a self-learner? Best if they have exercises with answers or full solutions.

Answer 1

I recommend Lectures on the Curry-Howard Isomorphism, Volume 149 (Studies in Logic and the Foundations of Mathematics) by Morten Heine Sørensen and Pawel Urzyczyn.

From the Amazon review:

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