Light logics and optimal reduction: Completeness and complexity

Patrick Baillot (1) , Paolo Coppola (2) , Ugo Dal Lago (3, 4)

1 LIP - Laboratoire de l'Informatique du Parallélisme
2 DIMI - Dipartimento di Matematica e Informatica - Universita Udine
3 DISI - Dipartimento di Scienze dell'Informazione [Bologna]
4 FOCUS - Foundations of Component-based Ubiquitous Systems

Typing of lambda-terms in elementary and light affine logic (EAL and LAL, respectively) has been studied for two different reasons: on the one hand the evaluation of typed terms using LAL (EAL, respectively) proof-nets admits a guaranteed polynomial (elementary, respectively) bound; on the other hand these terms can also be evaluated by optimal reduction using the abstract version of Lamping's algorithm. The first reduction is global while the second one is local and asynchronous. We prove that for LAL (EAL, respectively) typed terms, Lamping's abstract algorithm also admits a polynomial (elementary, respectively) bound. We also give a proof of its soundness and completeness (for EAL and LAL with type fixpoints), by using a simple geometry of interaction model (context semantics).

Mots clés: Lambda calculus; Linear logic; Optimal reduction; Implicit computational complexity; Proof-nets; Light linear logic

Domaines: Logique en informatique [cs.LO]

Patrick Baillot, Paolo Coppola, Ugo Dal Lago. Light logics and optimal reduction: Completeness and complexity. Information and Computation, 2011, 209 (2), pp.118-142. ⟨10.1016/j.ic.2010.10.002⟩. ⟨hal-00798315⟩