On the Curry-Howard Interpretation of a Fragment of Classical Linear Logic with Subexponentials

We construct a partially-ordered hierarchy of delimited control operators similar to those of the CPS hierarchy of Danvy and Filinski [6]. However, instead of relying on nested CPS translations, these operators give directly a Curry-Howard interpretation of a fragment of linear logic extended with subexponentials, i.e., multiple pairs of ! and ?. We show how the fundamental problem of delimited control, from the perspective of logic, is the combination of classical and non-classical inference within one system and how subexponentials give a new approach to combining classical and intuitionistic logics. A natural deduction system called MC (multi-colored classical logic) is formulated with proof terms that include indexed control operators. We then define a call-by-value evaluation strategy for these terms.