Research objectives and content:
Girard's Light and Elementary Linear Logic (LLL and ELL) are logics in which a fixed complexity bound is intrinsic to the logic. Namely, a function is representable in LLL iff it is polytime w.r.t. its input, while a function is representable in ELL iff it is elementary. Such bounds are intrinsic to the logic as cut-elimination for LLL and ELL proof-net is have the corresponding bounds: in LLL the reduction of a proof-net is polynomial in its size, while in LLL it is elementary. Hence, the proof associated to a function representable in LLL is not only a proof of its polynomiality, but also a program computing it in a polynomial time.
2-sequents give a natural framework in which to express the structural constraints of ELL and LLL. The hope is that they might help in removing some of the technical problems that lead to the actual involved syntax of LLL. Furthemore, the approach based on levels that 2-sequents imply might help in the comprehension of the semantics of LLL and ELL.
Sharing-graphs are a new and promising technique for the efficient implementation of functional languages originated from Levy's family reduction. The constraints that lead to the complexity bound of LLL and ELL seem to have a natural counterpart in terms of Levy's families. Training content (objective, benefit and expected impact):
1. To find a general framework in which the structural constraints that led Girard to the formulation of ELL and LLL still ensure the same complexity bounds.
2. To reformulate ELL and LLL in terms of Masini's 2-sequents. The objective is to find a good syntax leading to a practical) usable functional programming language.
3. To study sharing-graph reduction of the nets originated from the previos calculi.
Links with industry / industrial relevance (22)