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 and Graham-Lengrand to provide exact bounds. Our typing judgements carry 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 λ-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 & Graham-Lengrand via a uniform technique and a complexity-based perspective. The two main novelties are exact split bounds for the leftmost strategy – the only known strategy that evaluates terms to full normal forms and provides 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.

中文翻译：

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严格的类型和拆分边界，完全开发

已经使用了多种类型——也就是非幂等交集类型。获得由 de Carvalho 开创的高阶程序的定量界限。值得注意的是，它们同时限制了评估步骤的数量和结果的大小。最近的结果表明，步数可以作为一种合理的时间复杂度度量。然而，与此同时，这些结果表明多类型提供了相当宽松的复杂性界限，因为结果的大小可能比步数大得多。从这个观察开始，我们改进和概括了 Bernadet 和 Graham-Lengrand 引入的一种技术，以提供精确的界限。我们的打字判断带有计数器，一个测量评估长度，另一个测量结果大小。为了强调该方法的模块化，我们为四种评估策略提供了准确的界限，两者都在λ-演算（头部，最左外层和最大评估）和线性替换演算（线性头部评估）。我们的工作旨在捕捉文献中的结果并以新的结果扩展它们。关于文献，它通过统一的技术和基于复杂性的视角统一了 de Carvalho 和 Bernadet & Graham-Lengrand。两个主要的新颖之处是最左边策略的精确分割界限——唯一已知的将术语评估为完全范式并提供合理的复杂性度量的策略——以及隐藏在多种类型后面的计算设备的观察是远距离替换的概念，由线性代换演算实现。