License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.FSCD.2021.31
URN: urn:nbn:de:0030-drops-142692
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14269/
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Kop, Cynthia ; Vale, Deivid

Tuple Interpretations for Higher-Order Complexity

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LIPIcs-FSCD-2021-31.pdf (0.9 MB)

Abstract

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.

BibTeX - Entry

@InProceedings{kop_et_al:LIPIcs.FSCD.2021.31,
  author =	{Kop, Cynthia and Vale, Deivid},
  title =	{{Tuple Interpretations for Higher-Order Complexity}},
  booktitle =	{6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)},
  pages =	{31:1--31:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-191-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{195},
  editor =	{Kobayashi, Naoki},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14269},
  URN =		{urn:nbn:de:0030-drops-142692},
  doi =		{10.4230/LIPIcs.FSCD.2021.31},
  annote =	{Keywords: Complexity, higher-order term rewriting, many-sorted term rewriting, polynomial interpretations, weakly monotonic algebras}
}

Keywords: Complexity, higher-order term rewriting, many-sorted term rewriting, polynomial interpretations, weakly monotonic algebras
Collection: 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)
Issue Date: 2021
Date of publication: 06.07.2021

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