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DOI: 10.4230/LIPIcs.FSTTCS.2011.339

A Tight Lower Bound for Streett Complementation

Authors: Yang Cai and Ting Zhang

Published in: LIPIcs, Volume 13, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)

Abstract

Finite automata on infinite words (omega-automata) proved to be a powerful weapon for modeling and reasoning infinite behaviors of reactive systems. Complementation of omega-automata is crucial in many of these applications. But the problem is non-trivial; even after extensive study during the past two decades, we still have an important type of omega-automata, namely Streett automata, for which the gap between the current best lower bound 2^(Omega(n lg nk)) and upper bound 2^(O (nk lg nk)) is substantial, for the Streett index size k can be exponential in the number of states n. In a previous work we showed a construction for complementing Streett automata with the upper bound 2^(O(n lg n+nk lg k)) for k = O(n) and 2^(O(n^2 lg n)) for k = omega(n). In this paper we establish a matching lower bound 2^(Omega (n lg n+nk lg k)) for k = O(n) and 2^(Omega (n^2 lg n)) for k = omega(n), and therefore showing that the construction is asymptotically optimal with respect to the ^(Theta(.)) notation.

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Yang Cai and Ting Zhang. A Tight Lower Bound for Streett Complementation. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 339-350, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{cai_et_al:LIPIcs.FSTTCS.2011.339,
  author =	{Cai, Yang and Zhang, Ting},
  title =	{{A Tight Lower Bound for Streett Complementation}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)},
  pages =	{339--350},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-34-7},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{13},
  editor =	{Chakraborty, Supratik and Kumar, Amit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2011.339},
  URN =		{urn:nbn:de:0030-drops-33474},
  doi =		{10.4230/LIPIcs.FSTTCS.2011.339},
  annote =	{Keywords: omega-automata, Streett automata, complementation, lower bounds}
}

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DOI: 10.4230/LIPIcs.CSL.2011.112

Tight Upper Bounds for Streett and Parity Complementation

Authors: Yang Cai and Ting Zhang

Published in: LIPIcs, Volume 12, Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL (2011)

Abstract

Complementation of finite automata on infinite words is not only a fundamental problem in automata theory, but also serves as a cornerstone for solving numerous decision problems in mathematical logic, model-checking, program analysis and verification. For Streett complementation, a significant gap exists between the current lower bound 2^{Omega(n*log(n*k))} and upper bound 2^{O(n*k*log(n*k))}, where n is the state size, k is the number of Streett pairs, and k can be as large as 2^{n}. Determining the complexity of Streett complementation has been an open question since the late 80's. In this paper we show a complementation construction with upper bound 2^{O(n*log(n)+n*k*log(k))} for k=O(n) and 2^{O(n^{2}*log(n))} for k=Omega(n), which matches well the lower bound obtained in the paper arXiv:1102.2963. We also obtain a tight upper bound 2^{O(n*log(n))} for parity complementation.

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Yang Cai and Ting Zhang. Tight Upper Bounds for Streett and Parity Complementation. In Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 12, pp. 112-128, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{cai_et_al:LIPIcs.CSL.2011.112,
  author =	{Cai, Yang and Zhang, Ting},
  title =	{{Tight Upper Bounds for Streett and Parity Complementation}},
  booktitle =	{Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL},
  pages =	{112--128},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-32-3},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{12},
  editor =	{Bezem, Marc},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2011.112},
  URN =		{urn:nbn:de:0030-drops-32269},
  doi =		{10.4230/LIPIcs.CSL.2011.112},
  annote =	{Keywords: Streett automata, omega-automata, parity automata, complementation, upper bounds}
}

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A Tight Lower Bound for Streett Complementation

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Tight Upper Bounds for Streett and Parity Complementation

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