Inria / Raweb 2004
Project-Team: CALLIGRAMME

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## Section: Scientific Foundations

Keywords: sequent calculus, proof nets, lambda calculus, Curry-Howard isomorphism, type theory, denotational semantics.

## Proof Nets, Sequent Calculus and Typed Lambda Calculi

Participants: Guillaume Bonfante, Philippe de Groote, Bruno Guillaume, François Lamarche, Guy Perrier, Sylvain Pogodalla, Lutz Straßburger.

The aim of this action is the development of the theoretical tools that we use in our other research actions. We are interested, in particular, in the notion of formal proof itself, as much from a syntactical point of view (sequential derivations, proof nets, -terms), as from a semantical point of view.

Proof nets are graphical representations (in the sense of graph theory) of proofs in linear logic. Their role is very similar to lambda terms for more traditional logics; as a matter of fact there are several back-and-forth translations that relate several classes of lambda terms with classes of proof nets. In addition to their strong geometric character, another difference between proof nets and lambda terms is that the proof net structure of a proof of formula T can be considered as structure which is added to T, as a coupling between the atomic formula nodes of the usual syntactic tree graph of T. Since not all couplings correspond to proofs of T there is a need to distinguish the ones that do actually correspond to proofs; this is called a correctness criterion.

The discovery of new correctness criteria remains an important research problem, as much for Girard's original linear logic as for the field of non-commutative logics. Some criteria are better adapted to some applications than others. In particular, in the case of automatic proof search, correctness criteria can be used as invariants during the inductive process of proof construction.

The theory of proof nets also presents a dynamic character: cut elimination. This embodies a notion of normalization (or evaluation) akin to -reduction in the -calculus.

As we said above, until the invention of proof nets, the principal tool for representing proofs in constructive logics was the -calculus. This is due to the Curry-Howard isomorphism, which establishes a correspondence between natural deduction systems for intuitionistic logics and typed -calculi.

Although the Curry-Howard isomorphism owes its existence to the functional character of intuitionistic logic, it can be extended to fragments of classical logic. It turns out that some constructions that one meets in functional progamming languages, such as control operators, can presently only be explained by the use of deduction rules that are related to proof by contradiction 

This extension of the Curry-Howard isomorphism to classical logic and its applications has a perennial place as research field in the project.