Sharing and Linear Logic with Restricted Access (Extended Version)

Abstract

The two Girard translations provide two different means of obtaining embeddings of Intuitionistic Logic into Linear Logic, corresponding to different lambda-calculus calling mechanisms. The translations, mapping A -> B respectively to !A -o B and !(A -o B), have been shown to correspond respectively to call-by-name and call-by-value. In this work, we split the of-course modality of linear logic into two modalities, written "!" and "$\bullet$". Intuitively, the modality "!" specifies a subproof that can be duplicated and erased, but may not necessarily be "accessed", i.e. interacted with, while the combined modality "$!\bullet$" specifies a subproof that can moreover be accessed. The resulting system, called MSCLL, enjoys cut-elimination and is conservative over MELL. We study how restricting access to subproofs provides ways to control sharing in evaluation strategies. For this, we introduce a term-assignment for an intuitionistic fragment of MSCLL, called the $\lambda!\bullet$-calculus, which we show to enjoy subject reduction, confluence, and strong normalization of the simply typed fragment. We propose three sound and complete translations that respectively simulate call-by-name, call-by-value, and a variant of call-by-name that shares the evaluation of its arguments (similarly as in call-by-need). The translations are extended to simulate the Bang-calculus, as well as weak reduction strategies.