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Tuple Interpretations for Higher-Order Complexity Kop, Cynthia Vale, Deivid Complexity higher-order term rewriting many-sorted term rewriting polynomial interpretations weakly monotonic algebras 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 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 nat may be associated to a pair ⟨ cost, 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 β and η reductions within our technique. Finally, we relate our method to runtime complexity and prove that specific interpretation shapes imply certain runtime complexity bounds. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Cynthia Kop and Deivid Vale 2021 Is Part Of LIPIcs, Volume 195, 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.FSCD.2021.31 urn:nbn:de:0030-drops-142692 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2021.31 eng https://creativecommons.org/licenses/by/4.0/legalcode