By Takeshi Tsukada and Atsushi Igarashi. Logical Methods in Computer Science, 6(4:8):1-43, December 2010.
An earlier version appeared in Proceedings of 9th International Conference on Typed Lambda-Calculi and Applications (TLCA2009), volume 5608 of Lecture Notes in Computer Science, pages 341-355, Brasília, Brazil, July, 2009. Springer-Verlag.
Taha and Nielsen have developed a multi-stage calculus λα with a sound type system using the notion of environment classifiers. They are special identifiers, with which code fragments and variable declarations are annotated, and their scoping mechanism is used to ensure statically that certain code fragments are closed and safely runnable.
In this paper, we investigate the Curry-Howard isomorphism for environment classifiers by developing a typed λ-calculus λ|>. It corresponds to multi-modal logic that allows quantification by transition variables---a counterpart of classifiers---which range over (possibly empty) sequences of labeled transitions between possible worlds. This interpretation will reduce the ``run'' construct---which has a special typing rule in λα---and embedding of closed code into other code fragments of different stages---which would be only realized by the cross-stage persistence operator in λα---to merely a special case of classifier application. We prove that λ|> enjoys basic properties including subject reduction, confluence, and strong normalization and that the execution of a well-typed λ|> program is properly staged. Finally, we show that the proof system augmented with a classical axiom is sound and complete with respect to a Kripke semantics of the logic.