Abstract
There is a bundled variant, BCTL*, of the branching time logic CTL* which allows reasoning about models with fairness constraints on the possible futures. However, the stronger branching logic ATL*, which is well suited to reasoning about multi-agent systems, has no bundled variant. Schedulers, humans and so on may also exhibit “fair” behaviour that only manifests in the limit, motivating a similar variant of ATL*. In this paper we (1) show how to define a non-trivial Bundled variant of ATL* (BATL*); (2) Present a 2EXPTIME tableau for BATL* (so showing BATL* is 2EXPTIME-complete); (3) prove the correctness of the tableau; and (4) provide an implementation that can decide simple formulas for BATL* and another “non-local” variant NL-BCTL* that is well suited to verifying rewrite rules for ATL*.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alur, R., Henzinger, T.A., Kupferman, O.: Alternating-time temporal logic. Journal of the ACM (JACM) 49(5), 672–713 (2002)
Cerrito, S., David, A., Goranko, V.: Optimal tableaux-based decision procedure for testing satisfiability in the alternating-time temporal logic ATL+. In: Demri, S., Kapur, D., Weidenbach, C. (eds.) IJCAR 2014. LNCS, vol. 8562, pp. 277–291. Springer, Heidelberg (2014)
David, A.: Deciding ATL* satisfiability by tableaux. Technical report, Laboratoire IBISC – Université d’Évry Val-d’Essonne (2015), https://www.ibisc.univ-evry.fr/~adavid/fichiers/cade15_tableaux_atl_star_long.pdf (retrieved)
McCabe-Dansted, J.C., Reynolds, M.: EXPTIME fairness with bundled CTL. In: Cesta, A., Combi, C., Laroussinie, F. (eds.) Proceedings of the International Symposium on Temporal Representation and Reasoning, TIME 2014, pp. 164–173 (2014), http://www.csse.uwa.edu.au/~john/papers/ExpFair.pdf
Reynolds, M.: A tableau for Bundled CTL*. Journal of Logic and Computation 17(1), 117–132 (2007)
Reynolds, M.: A tableau-based decision procedure for CTL*. Formal Aspects of Computing 23(6), 739–779 (2011)
Schewe, S.: ATL* satisfiability is 2EXPTIME-complete. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 373–385. Springer, Heidelberg (2008)
Friedmann, O., Latte, M., Lange, M.: A decision procedure for CTL* based on tableaux and automata. In: Giesl, J., Hähnle, R. (eds.) IJCAR 2010. LNCS, vol. 6173, pp. 331–345. Springer, Heidelberg (2010)
Reynolds, M.: Axioms for branching time. Journal of Logic and Computation 12(4), 679–697 (2002)
Bauer, S., Hodkinson, I.M., Wolter, F., Zakharyaschev, M.: On non-local propositional and weak monodic quantified CTL*. Journal of Logic and Computation 14(1), 3–22 (2004)
McCabe-Dansted, J.C., Reynolds, M.: Verification of rewrite rules for computation tree logics. In: Cesta, A., Combi, C., Laroussinie, F. (eds.) Proceedings of the International Symposium on Temporal Representation and Reasoning, TIME 2014, pp. 142–151. IEEE Computer Society (2014), http://www.csse.uwa.edu.au/~john/papers/Rewrite-Long.pdf
McCabe-Dansted, J.C., Reynolds, M.: A tableau for bundled strategies (expanded version) (2015) http://www.csse.uwa.edu.au/~john/papers/BATL-Long.pdf
McCabe-Dansted, J.C.: Improved BCTL* tableau applet (2011) http://www.csse.uwa.edu.au/~john/BCTL2/
McCabe-Dansted, J.C.: A rooted tableau for BCTL*. In: The International Methods for Modalities Workshop, vol. 278, pp. 145–158. Elsevier Science Publishers B. V., Amsterdam, November 2011, http://www.csse.uwa.edu.au/~john/papers/Rooted_BCTL_Tableau.pdf
Emerson, E.A., Jutla, C.S.: The complexity of tree automata and logics of programs. SIAM Journal on Computing 29(1), 132–158 (2000)
Emerson, E.A., Sistla, A.P.: Deciding branching time logic: A triple exponential decision procedure for CTL*. In: Clarke, E., Kozen, D. (eds.) Logic of Programs 1983. LNCS, vol. 164, pp. 176–192. Springer, Heidelberg (1984)
Emerson, E.A., Sistla, A.P.: Deciding branching time logic. In: STOC 1984: Proceedings of the 16th Annual ACM Symposium on Theory of Computing, pp. 14–24. ACM Press, New York (1984)
Vardi, M.Y., Stockmeyer, L.J.: Improved upper and lower bounds for modal logics of programs. In: Proceedings of the 17th Annual ACM Symposium on Theory of Computing, STOC 1985, pp. 240–251. ACM, New York (1985)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
McCabe-Dansted, J., Reynolds, M. (2015). A Tableau for Bundled Strategies. In: De Nivelle, H. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2015. Lecture Notes in Computer Science(), vol 9323. Springer, Cham. https://doi.org/10.1007/978-3-319-24312-2_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-24312-2_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-24311-5
Online ISBN: 978-3-319-24312-2
eBook Packages: Computer ScienceComputer Science (R0)