An S4 extension of the λμμ-calculus

Abstract

In the context of the Curry-Howard correspondence, it is well-known that a typed calculus with control operator(s) corresponds to a formal system of classical logic, and various concrete examples have been discovered and studied since the early 90s. Among such work, Curien and Herbelin proposed the lambda-bar-mu-mu-tilde-calculus to exhibit the so-called duality of computation such as the (program/continuation)-duality and the (call-by-value/call-by-name)-duality. In their calculus, programs and continuations are first-class citizens formalized in a symmetric manner; and the call-by-value and call-by-name evaluation disciplines are characterized in some appropriate subcalculi, which are dual to each other. This work aims to introduce an extension of the lambda-bar-mu-mu-tilde-calculus with necessity and possibility modalities, which logically corresponds to classical S4 modal logic. In this paper, we show that the calculus is defined as a combination of existing calculi, and show that it satisfies desirable properties such as the subject reduction and the strong normalization. Two well-behaved subcalculi that realize the call-by-value and call-by-name disciplines are also discussed, following the work of Curien and Herbelin. As an application, we demonstrate that our calculus can be used to prove the strong normalization of the S4 modal lambda-mu-calculus of Kimura and Kakutani.