Probabilistic Automata for Safety LTL Specifications

  • Dileep Kini
  • Mahesh Viswanathan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8318)


Automata constructions for logical properties play an important role in the formal analysis of the system both statically and dynamically. In this paper, we present constructions of finite-state probabilistic monitors (FPM) for safety properties expressed in LTL. FPMs are probabilistic automata on infinite words that have a special, absorbing reject state, and given a cut-point λ ∈ [0,1], accept all words whose probability of reaching the reject state is at most 1 − λ. We consider Safe-LTL, the collection of LTL formulas built using conjunction, disjunction, next, and release operators, and show that (a) for any formula ϕ, there is an FPM with cut-point 1 of exponential size that recognizes the models of ϕ, and (b) there is a family of Safe-LTL formulas, such that the smallest FPM with cut-point 0 for this family is of doubly exponential size. Next, we consider the fragment LTL(G) of Safe-LTL wherein always operator is used instead of release operator and show that for any formula ϕ ∈ LTL(G) (c) there is an FPM with cut-point 0 of exponential size for ϕ, and (d) there is a robust FPM of exponential size for ϕ, where a robust FPM is one in which the acceptance probability of any word is bounded away from the cut-point. We also show that these constructions are optimal.


Model Check Boolean Function Communication Complexity Linear Temporal Logic Safety Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Chadha, R., Sistla, A.P., Viswanathan, M.: On the expressiveness and complexity of randomization in finite state monitors. J. ACM 56(5), 26:1–26:44 (2009)Google Scholar
  2. 2.
    Baier, C., Gröβer, M.: Recognizing ω-regular languages with probabilistic automata. In: Proceedings of the IEEE Symposium on Logic in Computer Science, pp. 137–146 (2005)Google Scholar
  3. 3.
    Rabin, M.: Probabilitic automata. Information and Control 6(3), 230–245 (1963)CrossRefzbMATHGoogle Scholar
  4. 4.
    Paz, A.: Introduction to Probabilistic Automata. Academic Press (1971)Google Scholar
  5. 5.
    Baier, C., Bertrand, N., Größer, M.: On decision problems for probabilistic büchi automata. In: Amadio, R.M. (ed.) FOSSACS 2008. LNCS, vol. 4962, pp. 287–301. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Sistla, A.P.: Safety, liveness and fairness in temporal logic. Formal Aspect of Computing, 495–511 (1999)Google Scholar
  7. 7.
    Kupferman, O., Vardi, M.Y.: Model checking of safety properties. In: Halbwachs, N., Peled, D.A. (eds.) CAV 1999. LNCS, vol. 1633, pp. 172–183. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  8. 8.
    Manna, Z., Pnueli, A.: Temporal verification of reactive and concurrent systems: Specification. Springer (1992)Google Scholar
  9. 9.
    Alur, R., La Torre, S.: Deterministic generators and games for ltl fragments. ACM Trans. Comput. Logic 5(1), 1–25 (2004)CrossRefGoogle Scholar
  10. 10.
    Yao, A.: Some complexity questions related to distributed computing. In: Proceedings of the ACM Symposium on Theory of Computation, pp. 209–213 (1979)Google Scholar
  11. 11.
    Kushilevtiz, E., Nisan, N.: Communication Complexity. Cambridge University Press (1996)Google Scholar
  12. 12.
    Kremer, I., Nisan, N., Ron, D.: On randomized one-round communication complexity. In: Symposium on Theory of Computing (June 1995)Google Scholar
  13. 13.
    Motwani, R., Raghavan, P.: Randomized algorithms. Cambridge University Press, New York (1995)CrossRefzbMATHGoogle Scholar
  14. 14.
    Vardi, M.Y.: An automata-theoretic approach to linear temporal logic. In: Moller, F., Birtwistle, G. (eds.) Logics for Concurrency. LNCS, vol. 1043, pp. 238–266. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  15. 15.
    Vapnik, V.N., Chervonenkis, A.Y.: On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability & Its Applications 16(2), 264–280 (1971)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Kupferman, O., Rosenberg, A.: The blow-up in translating LTL to deterministic automata. In: van der Meyden, R., Smaus, J.-G. (eds.) MoChArt 2010. LNCS, vol. 6572, pp. 85–94. Springer, Heidelberg (2011)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Dileep Kini
    • 1
  • Mahesh Viswanathan
    • 1
  1. 1.Department of Computer ScienceUniversity of Illinois at Urbana-ChampaignUSA

Personalised recommendations