The Complexity of Abstract Machines

TL;DR

This paper explores the complexity and efficiency of abstract machines implementing lambda-calculus evaluation strategies, particularly focusing on the weak head (call-by-name) strategy, providing new insights and a tutorial perspective.

Contribution

It offers an original analysis of the overhead complexity of abstract machines for lambda-calculus, including new observations not previously published.

Findings

Analyzes the overhead complexity of abstract machines

Provides a tutorial on implementing the weak head strategy

Introduces new unpublished observations about machine efficiency

Abstract

The lambda-calculus is a peculiar computational model whose definition does not come with a notion of machine. Unsurprisingly, implementations of the lambda-calculus have been studied for decades. Abstract machines are implementations schema for fixed evaluation strategies that are a compromise between theory and practice: they are concrete enough to provide a notion of machine and abstract enough to avoid the many intricacies of actual implementations. There is an extensive literature about abstract machines for the lambda-calculus, and yet-quite mysteriously-the efficiency of these machines with respect to the strategy that they implement has almost never been studied. This paper provides an unusual introduction to abstract machines, based on the complexity of their overhead with respect to the length of the implemented strategies. It is conceived to be a tutorial, focusing on the case study of implementing the weak head (call-by-name) strategy, and yet it is an original re-elaboration of known results. Moreover, some of the observation contained here never appeared in print before.