Multi types—aka non-idempotent intersection types—have been used to obtain quantitative bounds on higher-order programs, as pioneered by de Carvalho. Notably, they bound at the same time the number of evaluation steps and the size of the result. Recent results show that the number of steps can be taken as a reasonable time complexity measure. At the same time, however, these results suggest that multi types provide quite lax complexity bounds, because the size of the result can be exponentially bigger than the number of steps.

Starting from this observation, we refine and generalise a technique introduced by Bernadet & Lengrand to provide exact bounds for the maximal strategy. Our typing judgements carry two counters, one measuring evaluation lengths and the other measuring result sizes. In order to emphasise the modularity of the approach, we provide exact bounds for four evaluation strategies, both in the lambda-calculus (head, leftmost-outermost, and maximal evaluation) and in the linear substitution calculus (linear head evaluation).

Our work aims at both capturing the results in the literature and extending them with new outcomes. Concerning the literature, it unifies de Carvalho and Bernadet & Lengrand via a uniform technique and a complexity-based perspective. The two main novelties, instead, are exact split bounds for the leftmost strategy—the only strong strategy known to provide a %reasonable complexity measure—and the observation that the computing device hidden behind multi types is the notion of substitution at a distance, as implemented by the linear %substitution calculus.