On the Invariance of the Unitary Cost Model for Head Reduction (Long Version)

TL;DR

This paper proves that the unitary cost model remains invariant for head reduction in lambda calculus, extending previous results and using explicit substitutions to analyze reduction steps.

Contribution

It demonstrates the invariance of the unitary cost model specifically for head reduction, a case not previously established, using a novel linear calculus approach.

Findings

Unitary cost model is invariant for head reduction in lambda calculus.

Explicit substitutions facilitate the analysis of head reduction steps.

The approach offers a new tool for studying cost model invariance in normalization strategies.

Abstract

The lambda calculus is a widely accepted computational model of higher-order functional pro- grams, yet there is not any direct and universally accepted cost model for it. As a consequence, the computational difficulty of reducing lambda terms to their normal form is typically studied by reasoning on concrete implementation algorithms. In this paper, we show that when head reduction is the underlying dynamics, the unitary cost model is indeed invariant. This improves on known results, which only deal with weak (call-by-value or call-by-name) reduction. Invariance is proved by way of a linear calculus of explicit substitutions, which allows to nicely decompose any head reduction step in the lambda calculus into more elementary substitution steps, thus making the combinatorics of head-reduction easier to reason about. The technique is also a promising tool to attack what we see as the main open problem, namely understanding for which normalizing strategies derivation complexity is an invariant cost model, if any.