We develop a class of algebraic interpretations for many-sorted and higher-order term rewriting systems that takes type information into account. Specifically, base-type terms are mapped to \emph{tuples} of natural numbers and higher-order terms to functions between those tuples. Tuples may carry information relevant to the type; for instance, a term of type $\mathsf{nat}$ may be associated to a pair $(\mathsf{cost}, \mathsf{size})$ representing its evaluation cost and size. This class of interpretations results in a more fine-grained notion of complexity than runtime or derivational complexity, which makes it particularly useful to obtain complexity bounds for higher-order rewriting systems. We show that rewriting systems compatible with tuple interpretations admit finite bounds on derivation height. Furthermore, we demonstrate how to mechanically construct tuple interpretations and how to orient $\beta$ and $\eta$ reductions within our technique. Finally, we relate our method to runtime complexity and prove that specific interpretation shapes imply certain runtime complexity bounds.

中文翻译：

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高阶重写的元组解释

我们为考虑类型信息的多种分类和高阶术语重写系统开发了一类代数解释。具体而言，将基本类型项映射到自然数的\ emph {tuples}，将高阶项映射到这些元组之间的函数。元组可以携带与类型有关的信息；例如，类型$ \ mathsf {nat} $的项可能与代表其评估成本和大小的对$（\ mathsf {cost}，\ mathsf {size}）$相关联。与运行时或派生复杂度相比，此类解释会导致更复杂的概念，这对于获取高阶重写系统的复杂度范围特别有用。我们表明，与元组解释兼容的重写系统允许在派生高度上有有限范围。此外，我们将演示如何机械地构造元组解释，以及如何在我们的技术中确定$ \ beta $和$ \ eta $约简的方向。最后，我们将方法与运行时复杂度相关联，并证明特定的解释形状暗示了某些运行时复杂度界限。