Typing and Optimal reduction for λ-calculus in variants of Linear logic for Implicit computational complexity

Lambda-calculus has been introduced to study the mathematical functions from a computa-tional point of view. It has then been used as a basis for the design of functional programminglanguages. Knowing whether there exists a provably most efficient method to reduce lambda-terms, and evaluate the complexity of this operation in general are still open questions.In this thesis, we use the tools of typing, of Linear logic, of type inference and of Optimalreduction to explore those questions.We present a type inference algorithm for Dual light affine logic (dlal), a type system whichcharacterises the polynomial time complexity class. The algorithm takes in input a system Ftyped lambda-term, and outputs a typing in dlal if there exists one. An implementation isprovided.Then, we extend a type system based on Elementary affine logic with subtyping, in orderto automatise the cœrcions placement. We show that subtyping captures indeed the cœrcions,and we give a fully-fledged type inference algorithm for this extended system.Finally, we adapt Lamping’s Optimal reduction algorithm to the lambda-terms typable inSoft linear logic (sll), also characterising polynomial time. We prove a complexity bound onthe reduction of any Sharing graph, and that lambda-terms typable in sll can be correctlyreduced with our ad-hoc Optimal reduction algorithm.