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Abstract. This paper explores the surprisingly rich design space for the simply typed lambda calculus with casts and a dynamic type. Such a calculus is the target intermediate language of the gradually typed lambda calculus but it is also interesting in its own right. In light of diverse requirements for casts, we develop a modular semantic framework, based on Henglein's Coercion Calculus, that instantiates a number of space-efficient, blame-tracking calculi, varying in what errors they detect and how they assign blame. Several of the resulting calculi extend work from the literature with either blame tracking or space efficiency, and in doing so reveal previously unknown connections. Furthermore, we introduce a new strategy for assigning blame under which casts that respect traditional subtyping are statically guaranteed to never fail. One particularly appealing outcome of this work is a novel cast calculus that is well-suited to gradual typing.

We introduce MetaML, a statically-typed multi-stage programming language extending Nielson and Nielson's two stage notation to an arbitrary number of stages. MetaML extends previous work by introducing four distinct staging annotations which generalize those published previously [25, 12, 7,6] We give a static semantics in which type checking is done once and for all before the first stage, and a dynamic semantics which introduces a new concept of cross-stage per= sistence, which requires that variables available in any stage are also available in all future stages.We illustrate that staging is a manual form of binding time analysis. We explain why, even in the presence of automatic binding time analysis, explicit annotations are useful, especially for programs with more than two stages.A thesis of this paper is that multi-stage languages are useful as programming languages in their own right, and should support features that make it possible for programmers to write staged computations without significantly changing their normal programming style. To illustrate this we provide a simple three stage example, and an extended twostage example elaborating a number of practical issues. IntroductionMulti-stage languages have recently been proposed aa intermediate representations for partial evaluation [12, 9, 10] and runtime code generation [7]. These languages generalize the well-known two-level notation of Nielson k Nielson [25] to an arbitrary number of levels.A major thesis of this paper is that multi-stage languages are useful not only as intermediate representations, but also as pragmmming languages in their own n"ght. Multi-stage programming is important whenever performance is important. But there is very little language support for writing multi-stage programs. This paper extends previous work on multi-stage programming with features that are of practical use to real programmers.