On the invariance of the unitary cost model for head reduction

The λ-calculus is a widely accepted computational model of higher-order functional programs, yet there is not any direct and universally accepted cost model for it. As a consequence, the computational difficulty of reducing λ-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 λ-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 the unitary cost model is invariant, if any.