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).

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/

[1] Rudiments of μ-Calculus, Studies in Logic and the Foundations of Mathematics. North Holland 146 (2001). | Zbl 0968.03002

and ,[2] Generalising automaticity to modal properties of finite structures. Theor. Comput. Sci. 379 (2007) 266-285. | Zbl 1121.03046

and ,[3] On the expression complexity of the modal μ-calculus model checking, unpublished manuscript.

, and ,[4] Efficient model checking in fragments of the propositional mu-calculus (extended abstract), in Proc. of 1st Ann. IEEE Symp. on Logic in Computer Science, LICS '86 Cambridge, MA, June 1986. IEEE CS Press. (1986) 267-278.

and ,[5] A deterministic subexponential algorithm for solving parity games. SIAM J. Comput. 38 (2008) 1519-1532. | Zbl 1173.91326

, and ,[6] An improved algorithm for the evaluation of fixpoint expressions. in Proc. of 6th Int. Conf. on Computer Aided Verification, CAV '94 Stanford, CA, June 1994, edited by D. L. Dill, Springer, Lect. Notes Comput. Sci. 818 (1994) 338-350. | Zbl 0901.68118

, , , and ,[7] Computing flat vectorial Boolean fixed points, unpublished manuscript.

,[8] 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. | MR 2480222 | Zbl 1135.68480

,[9] 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.

, and ,