Three notes on the complexity of model checking fixpoint logic with chop

Lange, Martin

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 41 (2007), p. 177-190 / Harvested from Numdam

Résumé

This paper analyses the complexity of model checking fixpoint logic with Chop - an extension of the modal $\mu $-calculus with a sequential composition operator. It uses two known game-based characterisations to derive the following results: the combined model checking complexity as well as the data complexity of FLC are EXPTIME-complete. This is already the case for its alternation-free fragment. The expression complexity of FLC is trivially P-hard and limited from above by the complexity of solving a parity game, i.e. in UP$\cap $co-UP. For any fragment of fixed alternation depth, in particular alternation- free formulas it is P-complete.

Détail

Publié le : 2007-01-01

MR 2350643 | Zbl 1133.68046

DOI : https://doi.org/10.1051/ita:2007011
Classification:  03B44,  68Q17,  68Q60

Bibliographie

[1] H. Békič, Programming Languages and Their Definition, Selected Papers. Lect. Notes Comput. Sci. 177 (1984). | MR 760077 | Zbl 0548.68004

[2] A. Browne, E.M. Clarke, S. Jha, D.E. Long and W. Marrero, An improved algorithm for the evaluation of fixpoint expressions. TCS 178 (1997) 237-255. | Zbl 0901.68118

[3] R. Cleaveland, Tableau-based model checking in the propositional $\mu $-calculus. Acta Informatica 27 (1990) 725-748. | Zbl 0676.03033

[4] A. Dawar, E. Grädel and S. Kreutzer, Inflationary fixed points in modal logic, in Proc. 15th Workshop on Computer Science Logic, CSL'01, edited by L. Fribourg, Paris, France. Lect. Notes Comput. Sci. 277-291 (2001). | Zbl 0999.03019

[5] S. Dziembowski, M. Jurdziński and D. Niwiński, On the expression complexity of the modal $\mu $-calculus model checking. Unpublished manuscript (1996).

[6] E.A. Emerson and C.S. Jutla, Tree automata, $\mu $-calculus and determinacy, in Proc. 32nd Symp. on Foundations of Computer Science, San Juan, Puerto Rico 368-377 (1991). IEEE.

[7] E.A. Emerson, C.S. Jutla and A.P. Sistla, On model-checking for fragments of $\mu $-calculus, in Proc. 5th Conf. on Computer Aided Verification, CAV'93. Lect. Notes Comput. Sci. 697 (1993) 385-396.

[8] E. Grädel, W. Thomas and Th. Wilke, Eds. Automata, Logics, and Infinite Games. Lect. Notes Comput. Sci. (2002). | Zbl 1011.00037

[9] D. Janin and I. Walukiewicz, On the expressive completeness of the propositional $\mu $-calculus with respect to monadic second order logic, in Proc. 7th Conf. on Concurrency Theory, CONCUR'96, edited by U. Montanari and V. Sassone, Pisa, Italy. Lect. Notes Comput. Sci. 1119 (1996) 263-277.

[10] M. Jurdziński, Deciding the winner in parity games is in ${UP}\cap $co-${UP}$. Inform. Process. Lett. 68 (1998) 119-124.

[11] M. Jurdziński, Small progress measures for solving parity games, in Proc. 17th Ann. Symp. on Theoretical Aspects of Computer Science, STACS'00, edited by H. Reichel and S. Tison. Lect. Notes Comput. Sci. 1770 (2000) 290-301. | Zbl 0962.68111

[12] D. Kozen, Results on the propositional $\mu $-calculus. TCS 27 (1983) 333-354. | Zbl 0553.03007

[13] M. Lange, Alternating context-free languages and linear time $\mu $-calculus with sequential composition, in Proc. 9th Workshop on Expressiveness in Concurrency, EXPRESS'02, edited by P. Panangaden and U. Nestmann, Brno, Czech Republic, Elsevier. ENTCS 68.2 (2002) 71-87.

[14] M. Lange, Local model checking games for fixed point logic with chop, in Proc. 13th Conf. on Concurrency Theory, CONCUR'02, edited by L. Brim, P. Jančar, M. Křetínský and A. Kučera, Brno, Czech Republic. Lect. Notes Comput. Sci. 2421 (2002) 240-254. | Zbl 1012.68130

[15] M. Lange and R. Somla, The complexity of model checking higher order fixpoint logic, in Proc. 30th Int. Symp. on Math. Foundations of Computer Science, MFCS'05, edited by J. Jedrzejowicz and A. Szepietowski, Gdansk, Poland. Lect. Notes Comput. Sci. 3618 (2005) 640-651. | Zbl pre05065423

[16] M. Lange and C. Stirling, Model checking fixed point logic with chop, in Proc. 5th Conf. on Foundations of Software Science and Computation Structures, FOSSACS'02, edited by M. Nielsen and U. H. Engberg, Grenoble, France. Lect. Notes Comput. Sci. 2303 (2002) 250-263. | Zbl 1077.68690

[17] M. Müller-Olm, A modal fixpoint logic with chop, in Proc. 16th Symp. on Theoretical Aspects of Computer Science, STACS'99, edited by C. Meinel and S. Tison, Trier, Germany. Lect. Notes Comput. Sci. 1563 (1999) 510-520. | Zbl 0924.03046

[18] H. Seidl, Fast and simple nested fixpoint. Inform. Process. Lett. 59 (1996) 303-308. | Zbl 0900.68458

[19] C. Stirling, Local model checking games, in Proc. 6th Conf. on Concurrency Theory, CONCUR'95, edited by I. Lee and S. A. Smolka, Berlin, Germany. Lect. Notes Comput. Sci. 962 (1995) 1-11.

[20] M. Viswanathan and R. Viswanathan, A higher order modal fixed point logic, in Proc. 15th Int. Conf. on Concurrency Theory, CONCUR'04, edited by Ph. Gardner and N. Yoshida, London, UK. Lect. Notes Comput. Sci. 3170 (2004) 512-528. | Zbl 1099.03022

[21] I. Walukiewicz, Pushdown processes: Games and model-checking. Inform. Comput. 164 (2001) 234-263. | Zbl 1003.68072