Polarized proof-nets and λμ-calculus

Abstract

We first define polarized proof-nets, an extension of MELL proof-nets for the polarized fragment of linear logic; the main difference with usual proof-nets is that we allow structural rules on any negative formula. The essential properties (confluence, strong normalization in the typed case) of polarized proof-nets are proved using a reduction preserving translation into usual proof-nets. We then give a reduction preserving encoding of Parigot's λμ-terms for classical logic as polarized proof-nets. It is based on the intuitionistic translation: A→B⇝!A⊸B, so that it is a straightforward extension of the usual translation of λ-calculus into proof-nets. We give a reverse encoding which sequentializes any polarized proof-net as a λμ-term. In the last part of the paper, we extend the σ-equivalence for λ-calculus to λμ-calculus. Interestingly, this new σ-equivalence relation identifies normal λμ-terms. We eventually show that two terms are equivalent iff they are translated as the same polarized proof-net; thus the set of polarized proof-nets represents the quotient of λμ-calculus by σ-equivalence.