Title:

The atomic lambdamu calculus

A cornerstone of theoretical computer science is the CurryHoward correspondence where formulas are types, proofs are programs, and proof normalization is computation. In this framework we introduce the atomic λμcalculus, an interpretation of a classical deep inference proof system. It is based on two extensions of the λcalculus, the λμcalculus and the atomic λcalculus. The former interprets classical logic, featuring continuationlike constructs, while the latter interprets intuitionistic deep inference, featuring explicit sharing operators. The main property of the atomic λcalculus is reduction on individual constructors, derived from atomicity in deep inference. We thus work on open deduction, a deep inference formalism, allowing composition with connectives and with derivations, and using the medial rule to obtain atomicity. One challenge is to find a suitable formulation for deriving a computational interpretation of classical natural deduction. A second design challenge leads us to work on a variant of the λμcalculus, the ΛμScalculus, adding streams and dropping names. We show that our calculus has preservation of strong normalization (PSN), confluence, fullylazy sharing, and subject reduction in the typed case. There are two challenges with PSN. First, we need to show that sharing reductions strongly normalize, underlining that only β, μreductions create divergence. Our proof is new and follows a graphical approach to terms close to the idea of sharing. Second, infinite reductions of the atomic calculus can appear in weakenings, creating infinite atomic paths corresponding to finite ΛμSpaths. Our solution is to separate the proof into two parts, isolating the problem of sharing from that of weakening. We first translate into anintermediate weakening calculus, which unfolds shared terms while keeping weakened ones, and preserves infinite reductions. We then design a reduction strategy preventing infinite paths from falling into weakenings.
