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

. 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"

The 7th International Conference on Typed Lambda Calculi and Applications (TLCA 2005) was held in Nara (Japan) from 21 to 23 April 2005, as part of the Joint Conference on Rewriting, Deduction and Programming (RDP 2005). This book contains the contributed papers, and extended abstracts of two invited talks, given by Thierry Coquand and Susumu Hayashi. A short abstract of the joint RDP invited lecture by Amy Felty is also included. The 27 contributed papers were selected from 61 submissions of generally very high quality, and the Program Committee had a hard time making the selection. The editor would like to thank everyone who submitted a paper and to express his regret that many interesting works could not be included. The editor also wishes to thank the invited speakers, the members of the Program and Organizing Committees, the Publicity Chair, and the referees for their joint e?ort towards the success of the conference. The support from the Nara Convention Bureau is gratefully acknowledged. The typed lambda calculus continues to be an important tool in logic and theoretical computer science. Since 1993, the research progress in this area has been documented by the TLCA proceedings. The present volume contributes to this tradition.

This book constitutes the refereed proceedings of the 31st International Symposium on Mathematical Foundations of Computer Science, MFCS 2006. The book presents 62 revised full papers together with the full papers or abstracts of 7 invited talks. All current aspects in theoretical computer science and its mathematical foundations are addressed, from algorithms and data structures, to complexity, automata, semantics, logic, formal specifications, models of computation, concurrency theory, computational geometry and more.