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Computing Haar Measures

Authors: Arno Pauly, Dongseong Seon, and Martin Ziegler

Published in: LIPIcs, Volume 152, 28th EACSL Annual Conference on Computer Science Logic (CSL 2020)

According to Haar’s Theorem, every compact topological group G admits a unique (regular, right and) left-invariant Borel probability measure μ_G. Let the Haar integral (of G) denote the functional ∫_G:?(G)∋ f ↦ ∫ f d μ_G integrating any continuous function f:G → ℝ with respect to μ_G. This generalizes, and recovers for the additive group G=[0;1)mod 1, the usual Riemann integral: computable (cmp. Weihrauch 2000, Theorem 6.4.1), and of computational cost characterizing complexity class #P_1 (cmp. Ko 1991, Theorem 5.32). We establish that in fact, every computably compact computable metric group renders the Haar measure/integral computable: once using an elegant synthetic argument, exploiting uniqueness in a computably compact space of probability measures; and once presenting and analyzing an explicit, imperative algorithm based on "maximum packings" with rigorous error bounds and guaranteed convergence. Regarding computational complexity, for the groups SO(3) and SU(2), we reduce the Haar integral to and from Euclidean/Riemann integration. In particular both also characterize #P_1.

Cite as

Arno Pauly, Dongseong Seon, and Martin Ziegler. Computing Haar Measures. In 28th EACSL Annual Conference on Computer Science Logic (CSL 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 152, pp. 34:1-34:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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  author =	{Pauly, Arno and Seon, Dongseong and Ziegler, Martin},
  title =	{{Computing Haar Measures}},
  booktitle =	{28th EACSL Annual Conference on Computer Science Logic (CSL 2020)},
  pages =	{34:1--34:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-132-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{152},
  editor =	{Fern\'{a}ndez, Maribel and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2020.34},
  URN =		{urn:nbn:de:0030-drops-116779},
  doi =		{10.4230/LIPIcs.CSL.2020.34},
  annote =	{Keywords: Computable analysis, topological groups, exact real arithmetic, Haar measure}
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