Abstract
In the quantitative verification and synthesis of reactive systems, the states or transitions of a system are associated with payoffs, and a quantitative property of a behavior of the system is often characterized by the mean payoff for the behavior. This paper proposes an extension of LTL thatdescribes mean-payoff constraints. For each step of a behavior of a system, the payment depends on a system transition and a temporal property of the behavior. A mean-payoff constraint is a threshold condition for the limit supremum or limit infimum of the mean payoffs of a behavior. This extension allows us to describe specifications reflecting qualitative and quantitative requirements on long-run average of costs and the frequencies of satisfaction of temporal properties. Moreover, we develop an algorithm for the emptiness problems of multi-dimensional payoff automata with Büchi acceptance conditions and multi-threshold mean-payoff acceptance conditions. The emptiness problems are decided by solving linear constraint satisfaction problems, and the decision problems of our logic are reduced to the emptiness problems. Consequently, we obtain exponential-time algorithms for the model- and satisfiability-checking of the logic. Some optimization problems of the logic can also be reduced to linear programming problems.
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
Acacia+, http://lit2.ulb.ac.be/acaciaplus/
Alur, R., Degorre, A., Maler, O., Weiss, G.: On Omega-Languages Defined by Mean-Payoff Conditions. In: de Alfaro, L. (ed.) FOSSACS 2009. LNCS, vol. 5504, pp. 333–347. Springer, Heidelberg (2009)
Andova, S., Hermanns, H., Katoen, J.P.: Discrete-Time Rewards Model-Checked. In: Larsen, K.G., Niebert, P. (eds.) FORMATS 2003. LNCS, vol. 2791, pp. 88–104. Springer, Heidelberg (2004)
Aziz, A., Sanwal, K., Singhal, V., Brayton, R.: Verifying Continuous Time Markov Chains. In: Alur, R., Henzinger, T.A. (eds.) CAV 1996. LNCS, vol. 1102, pp. 269–276. Springer, Heidelberg (1996)
Baier, C., Haverkort, B., Hermanns, H., Katoen, J.P.: Model-checking algorithms for continuous-time markov chains. IEEE Transactions on Software Engineering 29(6), 524–541 (2003)
Baier, C., Haverkort, B., Hermanns, H., Katoen, J.-P.: On the Logical Characterisation of Performability Properties. In: Welzl, E., Montanari, U., Rolim, J.D.P. (eds.) ICALP 2000. LNCS, vol. 1853, pp. 780–792. Springer, Heidelberg (2000)
Bloem, R., Chatterjee, K., Henzinger, T.A., Jobstmann, B.: Better Quality in Synthesis through Quantitative Objectives. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 140–156. Springer, Heidelberg (2009)
Boker, U., Chatterjee, K., Henzinger, T., Kupferman, O.: Temporal specifications with accumulative values. In: LICS 2011, pp. 43–52 (2011)
Brázdil, T., Forejt, V., Kretínský, J., Kucera, A.: The satisfiability problem for probabilistic ctl. In: LICS 2008, pp. 391–402 (2008)
Černý, P., Chatterjee, K., Henzinger, T.A., Radhakrishna, A., Singh, R.: Quantitative Synthesis for Concurrent Programs. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 243–259. Springer, Heidelberg (2011)
Chatterjee, K., Doyen, L., Edelsbrunner, H., Henzinger, T.A., Rannou, P.: Mean-Payoff Automaton Expressions. In: Gastin, P., Laroussinie, F. (eds.) CONCUR 2010. LNCS, vol. 6269, pp. 269–283. Springer, Heidelberg (2010)
Chatterjee, K., Doyen, L., Henzinger, T.A.: Quantitative Languages. In: Kaminski, M., Martini, S. (eds.) CSL 2008. LNCS, vol. 5213, pp. 385–400. Springer, Heidelberg (2008)
Chatterjee, K., Henzinger, T.A., Jobstmann, B., Singh, R.: Measuring and Synthesizing Systems in Probabilistic Environments. In: Touili, T., Cook, B., Jackson, P. (eds.) CAV 2010. LNCS, vol. 6174, pp. 380–395. Springer, Heidelberg (2010)
Chatterjee, K., Majumdar, R., Henzinger, T.A.: Markov Decision Processes with Multiple Objectives. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 325–336. Springer, Heidelberg (2006)
Droste, M., Gastin, P.: Weighted automata and weighted logics. Theoretical Computer Science 380(1-2), 69–86 (2007)
Hansson, H., Jonsson, B.: A logic for reasoning about time and reliability. Formal Aspects of Computing 6(5), 512–535 (1994)
Kupferman, O., Lustig, Y.: Lattice Automata. In: Cook, B., Podelski, A. (eds.) VMCAI 2007. LNCS, vol. 4349, pp. 199–213. Springer, Heidelberg (2007)
Pnueli, A., Rosner, R.: On the synthesis of a reactive module. In: POPL 1989, pp. 179–190 (1989)
Pnueli, A.: The temporal logic of programs. In: FOCS 1977, pp. 46–57 (1977)
Tomita, T., Hagihara, S., Yonezaki, N.: A probabilistic temporal logic with frequency operators and its model checking. In: INFINITY 2011. EPTCS, vol. 73, pp. 79–93 (2011)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Tomita, T., Hiura, S., Hagihara, S., Yonezaki, N. (2012). A Temporal Logic with Mean-Payoff Constraints. In: Aoki, T., Taguchi, K. (eds) Formal Methods and Software Engineering. ICFEM 2012. Lecture Notes in Computer Science, vol 7635. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34281-3_19
Download citation
DOI: https://doi.org/10.1007/978-3-642-34281-3_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-34280-6
Online ISBN: 978-3-642-34281-3
eBook Packages: Computer ScienceComputer Science (R0)