A typed context calculus

Abstract

This paper develops a typed calculus for contexts i.e., lambda terms with “holes”. In addition to ordinary lambda terms, the calculus contains labeled holes, hole abstraction and context application for manipulating first-class contexts. The primary operation for contexts is hole-filling, which captures free variables. This operation conflicts with substitution of the lambda calculus, and a straightforward mixture of the two results in an inconsistent system. We solve this problem by defining a type system that precisely specifies the variable-capturing nature of contexts and that keeps track of bound variable renaming. These mechanisms enable us to define a reduction system that properly integrates β-reduction and hole-filling. The resulting calculus is Church–Rosser and the type system has the subject reduction property. We believe that the context calculus will serve as a basis for developing a programming language with advanced features that call for manipulation of open terms.