## Abstract

Many languages are beginning to integrate dynamic and static typing. Siek and Taha offered gradual typing as an approach to this integration that provides the benefits of a coherent and full-span migration between the two disciplines. However, the literature lacks a general methodology for designing gradually typed languages. Our first contribution is to provide such a methodology insofar as the static aspects of gradual typing are concerned: deriving the gradual type system and the compilation to the cast calculus.

Based on this methodology, we present the Gradualizer, an algorithm that generates a gradual type system from a well-formed type system (expressed as a logic program) and also generates a compiler to the cast calculus. Our algorithm handles a large class of type systems and generates systems that are correct with respect to the formal criteria of gradual typing. Finally, we report on an implementation of the Gradualizer that takes type systems expressed in $\lambda$-prolog and outputs their gradually typed version (and compiler to the cast calculus) in $\lambda$-prolog. We also report on ongoing work towards a methodology for deriving the operational semantics for executing gradual programs.