Limit Deterministic and Probabilistic Automata for LTL ∖ GU

  • Dileep Kini
  • Mahesh Viswanathan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9035)


LTL ∖ GU is a fragment of linear temporal logic (LTL), where negations appear only on propositions, and formulas are built using the temporal operators X (next), F (eventually), G (always), and U (until, with the restriction that no until operator occurs in the scope of an always operator. Our main result is the construction of Limit Deterministic Büchi automata for this logic that are exponential in the size of the formula. One consequence of our construction is a new, improved EXPTIME model checking algorithm (as opposed to the previously known doubly exponential time) for Markov Decision Processes and LTL ∖ GU formulae. Another consequence is that it gives us a way to construct exponential sized Probabilistic Büchi Automata for LTL ∖ GU.


  1. 1.
    Alur, R., Torre, S.L.: Deterministic generators and games for ltl fragments. ACM Trans. Comput. Logic 5(1), 1–25 (2004)CrossRefGoogle Scholar
  2. 2.
    Babiak, T., Blahoudek, F., Křetínský, M., Strejček, J.: Effective translation of LTL to deterministic Rabin automata: Beyond the (F,G)-fragment. In: Van Hung, D., Ogawa, M. (eds.) ATVA 2013. LNCS, vol. 8172, pp. 24–39. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  3. 3.
    Baier, C., Größer, M.: Recognizing omega-regular languages with probabilistic automata. In: LICS, pp. 137–146 (2005)Google Scholar
  4. 4.
    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
  5. 5.
    Courcoubetis, C., Yannakakis, M.: The complexity of probabilistic verification. J. ACM 42(4), 857–907 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Esparza, J., Křetínský, J.: From LTL to deterministic automata: A safraless compositional approach. In: Biere, A., Bloem, R. (eds.) CAV 2014. LNCS, vol. 8559, pp. 192–208. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  7. 7.
    Gastin, P., Oddoux, D.: Fast LTL to büchi automata translation. In: Berry, G., Comon, H., Finkel, A. (eds.) CAV 2001. LNCS, vol. 2102, pp. 53–65. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  8. 8.
    Kini, D., Viswanathan, M.: Probabilistic automata for safety LTL specifications. In: McMillan, K.L., Rival, X. (eds.) VMCAI 2014. LNCS, vol. 8318, pp. 118–136. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  9. 9.
    Kini, D., Viswanathan, M.: Probabilistic büchi automata for LTL∖GU. Technical Report University of Illinois at Urbana-Champaign (2015),
  10. 10.
    Klein, J., Baier, C.: Experiments with deterministic ω-automata for formulas of linear temporal logic. Theoretical Computer Science 363(2), 182–195 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Křetínský, J., Esparza, J.: Deterministic automata for the (F,G)-fragment of LTL. In: Madhusudan, P., Seshia, S.A. (eds.) CAV 2012. LNCS, vol. 7358, pp. 7–22. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  12. 12.
    Křetínský, J., Garza, R.L.: Rabinizer 2: Small deterministic automata for LTL∖ GU. In: Van Hung, D., Ogawa, M. (eds.) ATVA 2013. LNCS, vol. 8172, pp. 446–450. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  13. 13.
    Morgenstern, A., Schneider, K.: From LTL to symbolically represented deterministic automata. In: Logozzo, F., Peled, D.A., Zuck, L.D. (eds.) VMCAI 2008. LNCS, vol. 4905, pp. 279–293. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  14. 14.
    Piterman, N., Pnueli, A., Sa’ar, Y.: Synthesis of reactive(1) designs. In: Emerson, E.A., Namjoshi, K.S. (eds.) VMCAI 2006. LNCS, vol. 3855, pp. 364–380. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  15. 15.
    Pnueli, A., Zaks, A.: On the merits of temporal testers. In: 25 Years of Model Checking, pp. 172–195 (2008)Google Scholar
  16. 16.
    Somenzi, F., Bloem, R.: Efficient Büchi automata from LTL formulae. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 248–263. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  17. 17.
    Vardi, M., Wolper, P., Sistla, A.P.: Reasoning about infinite computation paths. In: FOCS (1983)Google Scholar
  18. 18.
    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

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

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

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