Abstract
Attempts to manage the reasoning about systems with fairness properties are long running. The popular but restricted Computational Tree Logic (CTL) is amenable to automated reasoning but has difficulty expressing some fairness properties. More expressive languages such as CTL* and CTL+ are computationally complex. The main contribution of this paper is to show the usefulness and practicality of employing the bundled variants of these languages to handle fairness. In particular we present a tableau for a bundled variant of CTL that still has the similar computational complexity to the CTL tableau and a simpler implementation. We further show that the decision problem remains in EXPTIME even if a bounded number of CTL* fairness constraints are allowed in the input formulas. By abandoning limit closure the bundled logics can simultaneously be easier to automate and express many typical fairness constraints.
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Ben-Ari, M., Manna, Z., Pnueli, A.: The temporal logic of branching time. In: POPL ’81: Proceedings of the 8th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL), pp 164–176. ACM, New York, NY, USA (1981)
Bolotov, A., Basukoski, A.: A clausal resolution method for branching-time logic ECTL+. Ann. Math. Artif. Intell. 46(3), 235–263 (2006)
Browne, M.C., Clarke, E.M., Grumberg, O.: Characterizing finite kripke structures in propositional temporal logic. Theor. Comput. Sci. 59, 115–131 (1988)
Burch, J.R., Clarke, E.M., McMillan, K.L., Dill, D.L.: Sequential circuit verification using symbolic model checking. In: Design Automation Conference, 1990. Proceedings. 27th ACM/IEEE, pp. 46–51. IEEE (1990)
Courcoubetis, C., Vardi, M.Y., Wolper, P.: Reasoning about fair concurrent programs. In: Proceedings of the Eighteenth Annual ACM Symposium on Theory of Computing, pp. 283–294. ACM (1986)
Emerson, E.A., Halpern, J.Y.: Decision Procedures and Expressiveness in the Temporal Logic of Branching Time. In: STOC, pp. 169–180. ACM (1982)
Emerson, E.A., Halpern, J.Y.: Decision procedures and expressiveness in the temporal logic of branching time. J. Comput. Syst. Sci. 30(1), 1–24 (1985). doi:10.1016/0022-0000(85)90001-7
Emerson, E.A., Halpern, J.Y.: “Sometimes” and “not never” revisited: on branching versus linear time temporal logic. J. ACM 33(1), 151–178 (1986)
Emerson, E.A., Jutla, C.S.: The complexity of tree automata and logics of programs. SIAM J. Comput. 29(1), 132–158 (2000)
Emerson, E.A., Lei, C.-L.: Modalities for model checking: branching time logic strikes back. Sci. Comput. Program. 8(3), 275–306 (1987)
Emerson, E.A., Sistla, A.P.: Deciding Branching Time Logic: a Triple Exponential Decision Procedure for CTL*. In: E. M. Clarke and D. Kozen, editors, Logic of Programs, volume 164, pages 176–192. Springer-Verlag (1983)
Emerson, E.A., Sistla, A.P.: Deciding branching time logic. In: STOC ’84: Proceedings of the 16th Annual ACM Symposium on Theory of Computing, pp 14–24. ACM Press, New York, NY, USA (1984)
Friedmann, O., Latte, M., Lange, M.: A decision procedure for CTL* based on tableaux and automata. In: Giesl, J., Hähnle, R. (eds.) IJCAR, vol. 6173, pp. 331–345. Springer (2010)
Gabbay, D.M., Pnueli, A., Shelah, S., Stavi, J.: On the temporal analysis of fairness. In: POPL ’80: Proceedings of the 7th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pp 163–173. ACM, New York, NY, USA (1980)
Goré, R., Thomson, J., Widmann, F.: An experimental comparison of theorem provers for CTL. In: Combi, C., Leucker, M., Wolter, F. (eds.) Proceedings of the 18th International Symposium on Temporal Representation and Reasoning, pp. 49–56. IEEE (2011)
Johannsen, J., Lange, M.: CTL+ is complete for double exponential time. In: Automata, Languages and Programming, pp. 767–775. Springer (2003)
Lehmann, D.J., Shelah, S.: Reasoning with time and chance. Inf. Control. 53(3), 165–198 (1982)
Marrero, W.: Using BDDs to decide CTL. In: TACAS’05, pp 222–236. Springer, Berlin, Heidelberg (2005). See also third party implementation at http://rsise.anu.edu.au/~rpg/CTLProvers/bddctl.tgz
Masini, A., Viganó, L., Volpe, M.: Labelled natural deduction for a bundled branching temporal logic. J. Log. Comput. 21(6), 1093–1163 (2011)
McCabe-Dansted, J.C.: A rooted tableau for BCTL*. In: The International Methods for Modalities Workshop, vol. 278, pp 145–158. Amsterdam, The Netherlands (2011). Elsevier Science Publishers B. V. see also expanded version: http://staffhome.ecm.uwa.edu.au/~00061811/papers/Rooted_BCTL_Tableau.pdf
McCabe-Dansted, J.C.: A Tableau for the Combination of CTL and BCTL*. In: B. Moszkowski, M. Reynolds, and P. Terenziani, editors, Proceedings of the International Symposium on Temporal Representation and Reasoning, pages 29–36. IEEE Computer Society (2012)
Reynolds, M.: A tableau for Bundled CTL*. J. Log. Comput. 17(1), 117–132 (2007)
Reynolds, M.: A tableau-based decision procedure for CTL*. Journal of Formal Aspects of Computing 23, 1–41 (2011)
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), pp 240–251. ACM, New York, NY, USA (1985)
Wan, W., Bentahar, J., Yahyaoui, H., Ben Hamza, A.: Verifying concurrent probabilistic systems using probabilistic-epistemic logic specifications. Appl. Intell. 45(3), 747–776 (2016)
Wilke, T.: CTL+ is Exponentially More Succinct than CTL. In: Pandu Rangan, C., Raman, V., Ramanujam, R. (eds.) Foundations of Software Technology and Theoretical Computer Science, 19th Conference, Chennai, India, December 13–15, 1999, Proceedings, vol. 1738, pp. 110–121. Springer (1999)
Zhang, L., Hustadt, U., Dixon, C.: A resolution calculus for the branching-time temporal logic CTL. ACM Trans. Comput. Log. (TOCL) 15(1), 10 (2014)
Acknowledgements
This project is supported by the Australian Government’s International Science Linkages program (French-Australian Science and Technology) and the Australian Research Council (ARC). In particular the ARC Grants: ‘Combining modal logics for dynamic and multi-agent systems’ (2004–2008, DP0452628), ‘Automatic Strategic Reasoning’ (2014–2016, DP140103365), ‘Automation of metric temporal reasoning’ (2011–2013, DP110104669). The funding bodies had no input into the conclusions of this paper.
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McCabe-Dansted, J., Reynolds, M. To be fair, use bundles. Ann Math Artif Intell 80, 317–364 (2017). https://doi.org/10.1007/s10472-017-9546-x
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DOI: https://doi.org/10.1007/s10472-017-9546-x