Modeling sharing and recursion for weak reduction strategies using explicit substitution

Abstract

We present the λσ w a -calculus, a formal synthesis of the concepts of sharing and explicit substitution for weak reduction. We show how λσ w a , can be used as a foundation of implementations of functional programming languages by modeling the essential ingredients of such implementations, namely weak reduction strategies, recursion, space leaks, recursive data structures, and parallel evaluation. We use λσ w a , to give a unified model of several computational aspects. First, we give a precise account of the major reduction strategies used in functional programming and the consequences of choosing λ-graph-reduction vs. environment-based evaluation. Second, we show how to add constructors and explicit recursion to give a precise account of recursive functions and data structures even with respect to space complexity. Third, we formalize the notion of space leaks in λσ w a and use this to define a space leak free calculus; this suggests optimisations for call-by-need reduction that prevent space leaking and enables us to prove that the “trimming” performed by the STG machine does not leak space. In summary we give a formal account of several implementation techniques used by state of the art implementations of functional programming languages.