Dynamic Typing: Syntax and Proof Theory

We present the dynamically typed λ-calculus, an extension of the statically typed λcalculus with a special type Dyn and explicit dynamic type coercions corresponding to runtime type tagging and type check-and-untag operations. Programs in run-time typed languages can be interpreted in the dynamically typed λ-calculus via a nondeterministic completion process that inserts explicit coercions and type declarations such that a well-typed term results. We characterize when two different completions of the same run-time typed program are coherent with an equational theory that is independent of an underlying λ-theory. This theory is refined by orienting some equations to define safety and minimality of completions. Intuitively, a safe completion is one that does not produce an error at run-time which another completion would have avoided, and a minimal completion is a safe completion that executes fewest tagging and check-and-untag operations amongst all safe completions. We show that every every untyped λ-term has a safe completion at any type and that it is unique modulo a suitable congruence relation. Furthermore, we present a rewriting system for generating minimal completions. Assuming strong normalization of this rewriting system we show that every λI-term has a minimal completion at any type, which is furthermore unique modulo equality in the dynamically typed λ-calculus.