Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Konečný, Michal; Steinberg, Florian; Thies, Holger https://www.dagstuhl.de/lipics License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
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Continuous and Monotone Machines

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Abstract

We investigate a variant of the fuel-based approach to modeling diverging computation in type theories and use it to abstractly capture the essence of oracle Turing machines. The resulting objects we call continuous machines. We prove that it is possible to translate back and forth between such machines and names in the standard function encoding used in computable analysis. Put differently, among the operators on Baire space, exactly the partial continuous ones are implementable by continuous machines and the data that such a machine provides is a description of the operator as a sequentially realizable functional. Continuous machines are naturally formulated in type theories and we have formalized our findings in Coq as part of Incone, a Coq library for computable analysis. The correctness proofs use a classical meta-theory with countable choice. Along the way we formally prove some known results such as the existence of a self-modulating modulus of continuity for partial continuous operators on Baire space. To illustrate their versatility we use continuous machines to specify some algorithms that operate on objects that cannot be fully described by finite means, such as real numbers and functions. We present particularly simple algorithms for finding the multiplicative inverse of a real number and for composition of partial continuous operators on Baire space. Some of the simplicity is achieved by utilizing the fact that continuous machines are compatible with multivalued semantics.

BibTeX - Entry

@InProceedings{konen_et_al:LIPIcs:2020:12723,
  author =	{Michal Konečn{\'y} and Florian Steinberg and Holger Thies},
  title =	{{Continuous and Monotone Machines}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{56:1--56:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Javier Esparza and Daniel Kr{\'a}ľ},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12723},
  URN =		{urn:nbn:de:0030-drops-127230},
  doi =		{10.4230/LIPIcs.MFCS.2020.56},
  annote =	{Keywords: Computable Analysis, exact real computation, formal proofs, proof assistant, Coq}
}

Keywords: Computable Analysis, exact real computation, formal proofs, proof assistant, Coq
Seminar: 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)
Issue date: 2020
Date of publication: 18.08.2020
Supplementary Material: Project Page https://holgerthies.github.io/continuous-machines/


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