Abstract
Graded path modalities count the number of paths satisfying a property, and generalize the existential (\(\mathsf {E}\)) and universal \((\mathsf {A})\) path modalities of \(\textsc {CTL}^{*}\). The resulting logic is denoted \(\textsc {G}\textsc {CTL}^{*}\), and is a very powerful logic since (as we show) it is equivalent, over trees, to monadic path logic. We settle the complexity of the satisfiability problem of \(\textsc {G}\textsc {CTL}^{*}\), i.e., 2ExpTime-Complete, and the complexity of the model checking problem of \(\textsc {G}\textsc {CTL}^{*}\), i.e., PSpace-Complete. The lower bounds already hold for \(\textsc {CTL}^{*}\), and so we supply the upper bounds. The significance of this work is two-fold: \(\textsc {G}\textsc {CTL}^{*}\) is much more expressive than \(\textsc {CTL}^{*}\) as it adds to it a form of quantitative reasoning, and this is done at no extra cost in computational complexity.
Benjamin Aminof is supported by the Austrian National Research Network S11403-N23 (RiSE) of the Austrian Science Fund (FWF) and by the Vienna Science and Technology Fund (WWTF) through grant ICT12-059. Aniello Murano is partially supported by the FP7 EU project 600958-SHERPA. Sasha Rubin is a Marie Curie fellow of the Istituto Nazionale di Alta Matematica.
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Notes
- 1.
Strictly speaking, GHTA generalise the symmetric variant of AHTA. That is, for every language accepted by an AHTA and that is closed under the operation of permuting siblings, there is a GHTA that accepts the same language.
- 2.
The combination of a Büchi and a co-Büchi condition that hesitant automata use can be thought of as a special case of the parity condition with 3 colors. Thus, we could have defined Graded Parity Tree Automata instead (using the parity condition, our automata strictly generalise the ones in [5, 19]) However, we do not need the full power of the parity condition, and in order to achieve optimal complexity for model checking of \(\textsc {G}\textsc {CTL}^{*}\) we need to be able to decide membership of our automata in a space efficient way, which cannot be done with the parity acceptance condition.
- 3.
For example, when building an automaton for \(\phi = \varphi _0 \vee \varphi _1\), in the degenerate case that \(\varphi _0 = \varphi _1\) then \(\mathsf {A}_{\varphi _1}\) is taken to be a copy of \(\mathsf {A}_{\varphi _0}\) with its states renamed to be disjoint from those of \(\mathsf {A}_{\varphi _0}\). Also, the new state \(q_0\) may be renamed to avoid a collision with any of the other states.
References
Almagor, S., Boker, U., Kupferman, O.: What’s decidable about weighted automata? In: Bultan, T., Hsiung, P.-A. (eds.) ATVA 2011. LNCS, vol. 6996, pp. 482–491. Springer, Heidelberg (2011)
Aminof, B., Kupferman, O., Lampert, R.: Rigorous approximated determinization of weighted automata. In: Symposium on Logic in Computer Science, pp. 345–354, IEEE (2011)
Aminof, B., Kupferman, O., Murano, A.: Improved model checking of hierarchical systems. Inf. Comput. 210, 68–86 (2012)
Arenas, M., Barceló, P., Libkin, L.: Combining temporal logics for querying XML documents. In: Schwentick, T., Suciu, D. (eds.) ICDT 2007. LNCS, vol. 4353, pp. 359–373. Springer, Heidelberg (2006)
Bianco, A., Mogavero, F., Murano, A.: Graded computation tree logic. In: Symposium on Logic in Computer Science, pp. 342–351, IEEE (2009)
Bianco, A., Mogavero, F., Murano, A.: Graded computation tree logic. ACM Trans. Comput. Log. 13(3), 25 (2012)
Bonatti, P.A., Lutz, C., Murano, A., Vardi, M.Y.: The complexity of enriched Mu-Calculi. Log. Methods Comput. Sci. 4(3), 1–27 (2008)
Calvanese, D., De Giacomo, G., Lenzerini, M.: Reasoning in expressive description logics with fixpoints based on automata on infinite trees. In: International Joint Conference on Artificial Intelligence, pp. 84–89 (1999)
de Rijke, M.: A note on graded modal logic. Studia Logica 64(2), 271–283 (2000)
Droste, M., Kuich, W., Vogler, H.: Handbook of Weighted Automata. Monographs in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg (2009)
Eisner, C., Fisman, D., Havlicek, J., Lustig, Y., McIsaac, A., Van Campenhout, D.: Reasoning with temporal logic on truncated paths. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 27–39. Springer, Heidelberg (2003)
Emerson, E.A., Jutla, C.S.: The complexity of tree automata and logics of programs. SIAM J. Comput. 29(1), 132–158 (1999)
Emerson, E.A., Sistla, A.P.: Deciding branching time logic, pp. 14–24. In: Symposium on Theory of Computing (1984)
Ferrante, A., Murano, A., Parente, M.: Enriched \(\mu \)-calculi module checking. Log. Methods Comput. Sci. 4(3), 1–21 (2008)
Ferrante, A., Napoli, M., Parente, M.: Model checking for graded CTL. Fundamenta Informaticae 96(3), 323–339 (2009)
Fine, K.: In so many possible worlds. Notre Dame J. Formal Log. 13, 516–520 (1972)
Gutiérrez-Basulto, V., Jung, J.C., Lutz, C.: Complexity of branching temporal description logics. In: European Conference on Artificial Intelligence, pp. 390–395 (2012)
Henzinger, T.A.: Quantitative reactive modeling and verification. Comput. Sci. 28(4), 331–344 (2013)
Kupferman, O., Sattler, U., Vardi, M.Y.: The complexity of the graded \(\mu \)-calculus. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392, pp. 423–437. Springer, Heidelberg (2002)
Kupferman, O., Vardi, M.Y., Wolper, P.: An automata theoretic approach to branching-time model checking. J. ACM 47(2), 312–360 (2000)
Malvone, V., Mogavero, F., Murano, A., Sorrentino, L.: On the counting of strategies. In: International Symposium on Temporal Representation and Reasoning, IEEE (2015, to appear)
Moller, F., Rabinovich, A.: Counting on CTL*: on the expressive power of monadic path logic. Inf. Comput. 184(1), 147–159 (2003)
Tobies, S.: PSPACE reasoning for graded modal logics. J. Log. Comput. 11(1), 85–106 (2001)
van der Hoek, W., Meyer, JJ.Ch.: Graded modalities in epistemic logic. In: Symposium on Logical Foundations of Computer Science, pp. 503–514 (1992)
Vardi, M.Y., Wolper, P.: Reasoning about infinite computations. Inf. Comput. 115(1), 1–37 (1994)
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Aminof, B., Murano, A., Rubin, S. (2015). On CTL* with Graded Path Modalities. In: Davis, M., Fehnker, A., McIver, A., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2015. Lecture Notes in Computer Science(), vol 9450. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48899-7_20
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