A Curry-Howard Correspondence for Intuitionistic Normal Modal Logic

Abstract

This paper provides a call-by-name and a call-by-value term calculus, both of which have a Curry-Howard correspondence to the box fragment of the intuitionistic modal logic IK. The call-by-name calculus is an extension of the simply typed call-by-name λ-calculus, and sound and complete for the categorical semantics. We show the strong normalizability and the confluency of the call-by-name calculus. On the other hand, the call-by-value calculus is an extension of the λ_c-calculus. We also show the strong normalizability and the confluency of the call-by-value one. This paper shows that a subcalculus of the call-by-value calculus is characterized by a CPS transformation into the call-by-name calculus. Moreover, we extend the call-by-name calculus to the modal logic IS4.