Light Linear Logic with Controlled Weakening

Abstract

Starting from Girard’s seminal paper on light linear logic (LLL), a number of works investigated on systems derived from linear logic to capture polynomial time computation within the computation-as-cut-elimination paradigm. The original syntax of LLL is too complicated, mainly because one has to deal with sequents which not just consist of formulas but also of ‘blocks’ of formulas. We circumvent the complications of ‘blocks’ by introducing a new modality $\nabla$ which is exclusively in charge of ‘additive blocks’. The most interesting feature of this purely multiplicative $\nabla$ is the possibility of the second-order encodings of additive connectives. The resulting system (with the traditional syntax), called Easy-LLL, is still powerful to represent any deterministic polynomial time computations in purely logical terms. Unlike the original LLL, Easy-LLL admits polynomial time strong normalization, namely, cut elimination terminates in a unique way in polytime by any choice of cut reduction strategies.