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Some results on complexity of μ-calculusevaluation in the black-box model∗

Published online by Cambridge University Press:  10 January 2013

Paweł Parys
Paweł Parys*
Affiliation:
Institute of Informatics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland.. [email protected]
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Abstract

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

Keywords

μ-calculusblack-box modellower boundexpression complexity
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 47 , Issue 1 , January 2013 , pp. 97 - 109
Copyright
© EDP Sciences 2013

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References

Arnold, A. and Niwiński, D., Rudiments of μ-Calculus, Studies in Logic and the Foundations of Mathematics. North Holland 146 (2001). Google Scholar
Dawar, A. and Kreutzer, S., Generalising automaticity to modal properties of finite structures. Theor. Comput. Sci. 379 (2007) 266–285. Google Scholar
S. Dziembowski, M. Jurdziński and D. Niwiński, On the expression complexity of the modal μ-calculus model checking, unpublished manuscript.
E.A. Emerson and C.-L. Lei, 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.
Jurdziński, M., Paterson, M. and Zwick, U., A deterministic subexponential algorithm for solving parity games. SIAM J. Comput. 38 (2008) 1519–1532. Google Scholar
D.E. Long, A. Browne, E.M. Clarke, S. Jha and W.R. Marrero, 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.
D. Niwiński, Computing flat vectorial Boolean fixed points, unpublished manuscript.
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.
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.