A Compositional Cost Model for the λ-calculus

Abstract

We describe a (time) cost model for the (call-by-value) λ-calculus based on a natural presentation of its game semantics: the cost of computing a finite approximant to the denotation of a term (its evaluation tree) is the size of its smallest derivation in the semantics. This measure has an optimality property enabling compositional reasoning about cost bounds: for any term A, context C[_] and approximants a and c to the trees of A and C[A], the cost of computing c from C[A] is no more than the cost of computing a from A and c from C[a].Although the natural semantics on which it is based is nondeterministic, our cost model is reasonable: we describe a deterministic algorithm for recognizing evaluation tree approximants which satisfies it (up to a constant factor overhead) on a Random Access Machine. This requires an implementation of the λ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</inf> -calculus on the RAM which is completely lazy: compositionality of costs entails that work done to evaluate any part of a term cannot be duplicated. This is achieved by a novel implementation of graph reduction for nameless explicit substitutions, to which we compile the λ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</inf> -calculus via a series of linear cost reductions.