# Some results on complexity of μ-calculus evaluation in the black-box model

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications (2013)

- Volume: 47, Issue: 1, page 97-109
- ISSN: 0988-3754

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topParys, Paweł. "Some results on complexity of μ-calculus evaluation in the black-box model." RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications 47.1 (2013): 97-109. <http://eudml.org/doc/273068>.

@article{Parys2013,

abstract = {We consider μ-calculus formulas in a normal form: after a prefix of fixed-point quantifiers follows a quantifier-free expression. We are interested in the problem of evaluating (model checking) such formulas in a powerset lattice. We assume that the quantifier-free part of the expression can be any monotone function given by a black-box – we may only ask for its value for given arguments. As a first result we prove that when the lattice is fixed, the problem becomes polynomial (the assumption about the quantifier-free part strengthens this result). As a second result we show that any algorithm solving the problem has to ask at least about n2 (namely Ω(n2/log n)) queries to the function, even when the expression consists of one μ and one ν (the assumption about the quantifier-free part weakens this result).},

author = {Parys, Paweł},

journal = {RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications},

keywords = {μ-calculus; black-box model; lower bound; expression complexity; -calculus},

language = {eng},

number = {1},

pages = {97-109},

publisher = {EDP-Sciences},

title = {Some results on complexity of μ-calculus evaluation in the black-box model},

url = {http://eudml.org/doc/273068},

volume = {47},

year = {2013},

}

TY - JOUR

AU - Parys, Paweł

TI - Some results on complexity of μ-calculus evaluation in the black-box model

JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

PY - 2013

PB - EDP-Sciences

VL - 47

IS - 1

SP - 97

EP - 109

AB - We consider μ-calculus formulas in a normal form: after a prefix of fixed-point quantifiers follows a quantifier-free expression. We are interested in the problem of evaluating (model checking) such formulas in a powerset lattice. We assume that the quantifier-free part of the expression can be any monotone function given by a black-box – we may only ask for its value for given arguments. As a first result we prove that when the lattice is fixed, the problem becomes polynomial (the assumption about the quantifier-free part strengthens this result). As a second result we show that any algorithm solving the problem has to ask at least about n2 (namely Ω(n2/log n)) queries to the function, even when the expression consists of one μ and one ν (the assumption about the quantifier-free part weakens this result).

LA - eng

KW - μ-calculus; black-box model; lower bound; expression complexity; -calculus

UR - http://eudml.org/doc/273068

ER -

## References

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- [8] S. Schewe, Solving parity games in big steps, in Proc. of 27th Int. Conf. on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2007 Kharagpur, Dec. 2007, edited by V. Arvind and S. Prasad, Springer. Lect. Notes Comput. Sci. 4855 (2007) 449–460. Zbl1135.68480MR2480222
- [9] S. Zhang, O. Sokolsky and S.A. Smolka, On the parallel complexity of model checking in the modal mu-calculus, in Proc. 9th Ann. IEEE Symp. on Logic in Computer Science, LICS ’94 Paris, July 1994. IEEE CS Press. (1994) 154–163.

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