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Documents authored by Park, Sewon

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Software
DOI: 10.4230/artifacts.22444
cAERN library

Authors: Michal Konečný, Sewon Park, and Holger Thies

Abstract
Cite as

Michal Konečný, Sewon Park, Holger Thies. cAERN library (Software, Source Code). Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@misc{dagstuhl-artifact-22444,
   title = {{cAERN library}}, 
   author = {Kone\v{c}n\'{y}, Michal and Park, Sewon and Thies, Holger},
   note = {Software, swhId: \href{https://archive.softwareheritage.org/swh:1:dir:91c89245541a8dbcad3ab085bb8682112e684311;origin=https://github.com/holgerthies/coq-aern;visit=swh:1:snp:4a136325144f9e2b906fa56a7cd796b6cfbcb691;anchor=swh:1:rev:fac208d7aa858884395cb77474788d4c8605c8ce}{\texttt{swh:1:dir:91c89245541a8dbcad3ab085bb8682112e684311}} (visited on 2024-11-28)},
   url = {https://github.com/holgerthies/coq-aern},
   doi = {10.4230/artifacts.22444},
}
Document
DOI: 10.4230/LIPIcs.ITP.2024.30
A Coq Formalization of Taylor Models and Power Series for Solving Ordinary Differential Equations

Authors: Sewon Park and Holger Thies

Published in: LIPIcs, Volume 309, 15th International Conference on Interactive Theorem Proving (ITP 2024)

Abstract
In exact real computation real numbers are manipulated exactly without round-off errors, making it well-suited for high precision verified computation. In recent work we propose an axiomatic formalization of exact real computation in the Coq theorem prover. The formalization admits an extended extraction mechanism that lets us extract computational content from constructive parts of proofs to efficient programs built on top of AERN, a Haskell library for exact real computation. Many processes in science and engineering are modeled by ordinary differential equations (ODEs), and often safety-critical applications depend on computing their solutions correctly. The primary goal of the current work is to extend our framework to spaces of functions and to support computation of solutions to ODEs and other essential operators. In numerical mathematics, the most common way to represent continuous functions is to use polynomial approximations. This can be modeled by so-called Taylor models, that encode a function as a polynomial and a rigorous error-bound over some domain. We define types of classical functions that do not hold any computational content and formalize Taylor models to computationally approximate those classical functions. Classical functions are defined in a way to admit classical principles in their constructions and verification. We define various basic operations on Taylor models and verify their correctness based on the classical functions that they approximate. We then shift our interest to analytic functions as a generalization of Taylor models where polynomials are replaced by infinite power series. We use the formalization to develop a theory of non-linear polynomial ODEs. From the proofs we can extract certified exact real computation programs that compute solutions of ODEs on some time interval up to any precision.
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Sewon Park and Holger Thies. A Coq Formalization of Taylor Models and Power Series for Solving Ordinary Differential Equations. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 30:1-30:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{park_et_al:LIPIcs.ITP.2024.30,
  author =	{Park, Sewon and Thies, Holger},
  title =	{{A Coq Formalization of Taylor Models and Power Series for Solving Ordinary Differential Equations}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{30:1--30:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-337-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{309},
  editor =	{Bertot, Yves and Kutsia, Temur and Norrish, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2024.30},
  URN =		{urn:nbn:de:0030-drops-207581},
  doi =		{10.4230/LIPIcs.ITP.2024.30},
  annote =	{Keywords: Exact real computation, Taylor models, Analytic functions, Computable analysis, Program extraction}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2023.59
Formalizing Hyperspaces for Extracting Efficient Exact Real Computation

Authors: Michal Konečný, Sewon Park, and Holger Thies

Published in: LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Abstract
We propose a framework for certified computation on hyperspaces by formalizing various higher-order data types and operations in a constructive dependent type theory. Our approach builds on our previous work on axiomatization of exact real computation where we formalize nondeterministic first-order partial computations over real and complex numbers. Based on the axiomatization, we first define open, closed, compact and overt subsets in an abstract topological way that allows short and elegant proofs with computational content coinciding with standard definitions in computable analysis. From these proofs we extract programs for testing inclusion, overlapping of sets, et cetera. To improve extracted programs, our framework specializes the Euclidean space ℝ^m making use of metric properties. To define interesting operations over hyperspaces of Euclidean space, we introduce a nondeterministic version of a continuity principle valid under the standard type-2 realizability interpretation. Instead of choosing one of the usual formulations, we define it in a way similar to an interval extension operator, which often is already available in exact real computation software. We prove that the operations on subsets preserve the encoding, and thereby define a small calculus to built new subsets from given ones, including limits of converging sequences with regards to the Hausdorff metric. From the proofs, we extract programs that generate drawings of subsets of ℝ^m with any given precision efficiently. As an application we provide a function that constructs fractals, such as the Sierpinski triangle, from iterated function systems using the limit operation, resulting in certified programs that errorlessly draw such fractals up to any desired resolution.
Cite as

Michal Konečný, Sewon Park, and Holger Thies. Formalizing Hyperspaces for Extracting Efficient Exact Real Computation. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 59:1-59:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{konecny_et_al:LIPIcs.MFCS.2023.59,
  author =	{Kone\v{c}n\'{y}, Michal and Park, Sewon and Thies, Holger},
  title =	{{Formalizing Hyperspaces for Extracting Efficient Exact Real Computation}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{59:1--59:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-292-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{272},
  editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.59},
  URN =		{urn:nbn:de:0030-drops-185935},
  doi =		{10.4230/LIPIcs.MFCS.2023.59},
  annote =	{Keywords: Computable analysis, type theory, program extraction}
}
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