  We investigate a simply typed modal $\lambda$-calculus,
$\lambda^{\to\square}$, due to Pfenning, Wong and Davies, where we define a
well-typed term with respect to a context stack that captures the possible
world semantics in a syntactic way. It provides logical foundation for
multi-staged meta-programming. Our main contribution in this paper is a
normalization by evaluation (NbE) algorithm for $\lambda^{\to\square}$ which we
prove sound and complete. The NbE algorithm is a moderate extension to the
standard presheaf model of simply typed $\lambda$-calculus. However, central to
the model construction and the NbE algorithm is the observation of Kripke-style
substitutions on context stacks which brings together two previously separate
concepts, structural modal transformations on context stacks and substitutions
for individual assumptions. Moreover, Kripke-style substitutions allow us to
give a formulation for contextual types, which can represent open code in a
meta-programming setting. Our work lays the foundation for extending the
logical foundation by Pfenning, Wong, and Davies towards building a practical,
dependently typed foundation for meta-programming.
