An Operational Semantics of Lexically-Scoped Dynamic Variables

Lexical and dynamic scoping are the two primary approaches to variable binding in functional programming languages. While medieval Lisp dialects commonly featured dynamic scoping, most languages today emphasize lexical scoping. This is a sensible choice: lexical scoping enables local reasoning about programs at the source code level. Nonetheless, dynamic variables are more appropriate for certain use cases, including implicit parameter passing. But do we really have to choose between lexical and dynamic scoping? A few systems, including the fluid-let macro and parameter objects in Scheme provide lexically-scoped dynamic variables. The semantics of such systems has thus far been presented only as implementations in terms of unrestricted mutable variables (e.g., using the set! operation). This article introduces λLD, a simple extension to the call-byvalue λ-calculus that supports lexically-scoped dynamic variables. The semantics of λLD does not rely upon any form of mutable state; and it is equivalent to the λ-calculus on terms not involving dynamic variables. Terms, including those containing dynamic variables, can be independently simplified and later composed; this allows for concurrent evaluation. The semantics of λLD has been implemented as a rewrite theory in Maude, allowing examples to be executed. The article shows how λLD can be used to model other forms of dynamic variables, as well as unique values, records and (functional) objects.