Light affine logic

Abstract

Much effort has been recently devoted to the study of polytime formal (and especially logical) systems. The purpose of such systems is manyfold. On the theoretical side, they provide a better understanding of what is the logical essence of polytime reduction (and other complexity classes). On the practical side, via the well known Curry-Howard correspondence, they yield sophisticated typing systems, where types provide (statically) an accurate upper bound on the complexity of the computation. Even more, the type annotations give essential information on the "efficient way" to reduce the term. The most promising of these logical systems is Girard's light linear logic. In this paper, we introduce a slight variation of LLL, by adding full weakening (for this reason, we call it light affine logic). This modification does not alter the good complexity properties of LLL: cut-elimination is still polytime. On the other side, the logical system is much simpler: we reduce it from 21 to just 11 rules, and with simpler, traditional sequents. Rephrasing Girard, we could thus say that the abuse of contraction may have damaging complexity effects, but the abstinence from weakening leads to inessential syntactical complications.