Light linear logics with controlled weakening: Expressibility, confluent strong normalization

Starting from Girard’s seminal paper on light linear logic (LLL), a number of works investigated 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 do not just consist of formulas but also of ‘blocks’ of formulas.We circumvent the complications of ‘blocks’ by introducing a new modality ∇ which is exclusively in charge of ‘additive blocks’. One of the most interesting features of this purely multiplicative ∇ 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 together with the Church–Rosser property, namely, cut elimination terminates in a unique way in polytime by any choice of cut reduction strategies.