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Documents authored by Vale, Deivid


Document
Certifying Higher-Order Polynomial Interpretations

Authors: Niels van der Weide, Deivid Vale, and Cynthia Kop

Published in: LIPIcs, Volume 268, 14th International Conference on Interactive Theorem Proving (ITP 2023)


Abstract
Higher-order rewriting is a framework in which one can write higher-order programs and study their properties. One such property is termination: the situation that for all inputs, the program eventually halts its execution and produces an output. Several tools have been developed to check whether higher-order rewriting systems are terminating. However, developing such tools is difficult and can be error-prone. In this paper, we present a way of certifying termination proofs of higher-order term rewriting systems. We formalize a specific method that is used to prove termination, namely the polynomial interpretation method. In addition, we give a program that processes proof traces containing a high-level description of a termination proof into a formal Coq proof script that can be checked by Coq. We demonstrate the usability of this approach by certifying higher-order polynomial interpretation proofs produced by Wanda, a termination analysis tool for higher-order rewriting.

Cite as

Niels van der Weide, Deivid Vale, and Cynthia Kop. Certifying Higher-Order Polynomial Interpretations. In 14th International Conference on Interactive Theorem Proving (ITP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 268, pp. 30:1-30:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{vanderweide_et_al:LIPIcs.ITP.2023.30,
  author =	{van der Weide, Niels and Vale, Deivid and Kop, Cynthia},
  title =	{{Certifying Higher-Order Polynomial Interpretations}},
  booktitle =	{14th International Conference on Interactive Theorem Proving (ITP 2023)},
  pages =	{30:1--30:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-284-6},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{268},
  editor =	{Naumowicz, Adam and Thiemann, Ren\'{e}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2023.30},
  URN =		{urn:nbn:de:0030-drops-184051},
  doi =		{10.4230/LIPIcs.ITP.2023.30},
  annote =	{Keywords: higher-order rewriting, Coq, termination, formalization}
}
Document
Cost-Size Semantics for Call-By-Value Higher-Order Rewriting

Authors: Cynthia Kop and Deivid Vale

Published in: LIPIcs, Volume 260, 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)


Abstract
Higher-order rewriting is a framework in which higher-order programs can be described by transformation rules on expressions. A computation occurs by transforming an expression into another using such rules. This step-by-step computation model induced by rewriting naturally gives rise to a notion of complexity as the number of steps needed to reduce expressions to a normal form, i.e., an expression that cannot be reduced further. The study of complexity analysis focuses on the development of automatable techniques to provide bounds to this number. In this paper, we consider a form of higher-order rewriting with a call-by-value evaluation strategy, so as to model call-by-value programs. We provide a cost-size semantics: a class of algebraic interpretations to map terms to tuples which bound both the reduction cost and the size of normal forms.

Cite as

Cynthia Kop and Deivid Vale. Cost-Size Semantics for Call-By-Value Higher-Order Rewriting. In 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 260, pp. 15:1-15:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{kop_et_al:LIPIcs.FSCD.2023.15,
  author =	{Kop, Cynthia and Vale, Deivid},
  title =	{{Cost-Size Semantics for Call-By-Value Higher-Order Rewriting}},
  booktitle =	{8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)},
  pages =	{15:1--15:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-277-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{260},
  editor =	{Gaboardi, Marco and van Raamsdonk, Femke},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2023.15},
  URN =		{urn:nbn:de:0030-drops-179993},
  doi =		{10.4230/LIPIcs.FSCD.2023.15},
  annote =	{Keywords: Call-by-Value Evaluation, Complexity Theory, Higher-Order Rewriting}
}
Document
Tuple Interpretations for Higher-Order Complexity

Authors: Cynthia Kop and Deivid Vale

Published in: LIPIcs, Volume 195, 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)


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.

Cite as

Cynthia Kop and Deivid Vale. Tuple Interpretations for Higher-Order Complexity. In 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 195, pp. 31:1-31:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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@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/entities/document/10.4230/LIPIcs.FSCD.2021.31},
  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}
}
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