Some results on complexity of μ-calculus evaluation in the black-box model
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 47 (2013) no. 1, p. 97-109

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).

DOI : https://doi.org/10.1051/ita/2012030
Classification:  68Q17,  03B70
Keywords: μ-calculus, black-box model, lower bound, expression complexity
@article{ITA_2013__47_1_97_0,
     author = {Parys, Pawe\l },
     title = {Some results on complexity of $\mu $-calculus evaluation in the black-box model},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {1},
     year = {2013},
     pages = {97-109},
     doi = {10.1051/ita/2012030},
     zbl = {1269.68056},
     language = {en},
     url = {http://www.numdam.org/item/ITA_2013__47_1_97_0}
}
Parys, Paweł. Some results on complexity of $\mu $-calculus evaluation in the black-box model. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 47 (2013) no. 1, pp. 97-109. doi : 10.1051/ita/2012030. http://www.numdam.org/item/ITA_2013__47_1_97_0/

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