Une étude des sommes fortes : isomorphismes et formes normales. (A study of strong sums: isomorphisms and normal forms)

A study of strong sums: isomorphisms and normal forms The goal of this thesis is to study the sum and the zero within two principal frameworks: type isomorphisms and the normalization of λ-terms. Type isomorphisms have already been studied within the framework of the simply typed λ-calculus with surjective pairing but without sums. To handle the case with sums and zero, I first restricted the study to the case of linear isomorphisms, within the framework of linear logic, which led to a remarkably simple characterization of these isomorphisms, obtained thanks to a syntactic method on proof-nets. The more general framework of intuitionistic logic corresponds to the open problem of characterizing isomorphisms in bi-cartesian closed categories. I contributed to this study by showing that there is no finite axiomatization of these isomorphisms. To achieve this, I used some results in number theory regarding Alfred Tarski’s so-called “high school algebra” problem. The whole of this work brought about the problem of finding a canonical form to represent λterms, with the aim of either denying the existence of an isomorphism by a case study on the form of the term, or checking their existence in the case of the very complex functions I was brought to handle. This analysis led us to give an “extensional” definition of normal form for the λ-calculus with sums and zero, obtained by categorical methods using Grothendieck logical relations. Finally, I could obtain an “intentional” version of this result by using normalization by evaluation. By adapting the technique of type-directed partial evaluation, it is possible to produce a result in the new normal form, reducing considerably its size in the case of the type isomorphisms considered before.