License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CSL.2021.9
URN: urn:nbn:de:0030-drops-134430
URL: https://drops.dagstuhl.de/opus/volltexte/2021/13443/
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Arnold, André ; Niwiński, Damian ; Parys, Paweł

A Quasi-Polynomial Black-Box Algorithm for Fixed Point Evaluation

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LIPIcs-CSL-2021-9.pdf (0.6 MB)


Abstract

We consider nested fixed-point expressions like μ z. ν y. μ x. f(x,y,z) evaluated over a finite lattice, and ask how many queries to a function f are needed to find the value. The previous upper bounds for a monotone function f of arity d over the lattice {0,1}ⁿ were of the order n^{𝒪(d)}, whereas a lower bound of Ω(n²/(lg n)) is known in case when at least one alternation between the least (μ) and the greatest (ν) fixed point occurs in the expression. Following a recent development for parity games, we show here that a quasi-polynomial number of queries is sufficient, namely n^{lg(d/lg n)+𝒪(1)}. The algorithm is an abstract version of several algorithms proposed recently by a number of authors, which involve (implicitly or explicitly) the structure of a universal tree. We then show a quasi-polynomial lower bound for the number of queries used by the algorithms in consideration.

BibTeX - Entry

@InProceedings{arnold_et_al:LIPIcs:2021:13443,
  author =	{Andr{\'e} Arnold and Damian Niwiński and Pawe{\l} Parys},
  title =	{{A Quasi-Polynomial Black-Box Algorithm for Fixed Point Evaluation}},
  booktitle =	{29th EACSL Annual Conference on Computer Science Logic (CSL 2021)},
  pages =	{9:1--9:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-175-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{183},
  editor =	{Christel Baier and Jean Goubault-Larrecq},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/13443},
  URN =		{urn:nbn:de:0030-drops-134430},
  doi =		{10.4230/LIPIcs.CSL.2021.9},
  annote =	{Keywords: Mu-calculus, Parity games, Quasi-polynomial time, Black-box algorithm}
}

Keywords: Mu-calculus, Parity games, Quasi-polynomial time, Black-box algorithm
Collection: 29th EACSL Annual Conference on Computer Science Logic (CSL 2021)
Issue Date: 2021
Date of publication: 13.01.2021


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