Finding Shortest Witnesses to the Nonemptiness of Automata on Infinite Words

  • Orna Kupferman
  • Sarai Sheinvald-Faragy
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4137)


In the automata-theoretic approach to formal verification, the satisfiability and the model-checking problems for linear temporal logics are reduced to the nonemptiness problem of automata on infinite words. Modifying the nonemptiness algorithm to return a shortest witness to the nonemptiness (that is, a word of the form uv ω that is accepted by the automaton and for which |uv| is minimal) has applications in synthesis and counterexample analysis. Unlike shortest accepting runs, which have been studied in the literature, the definition of shortest witnesses is semantic and is independent on the specification formalism of the property or the system. In particular, its robustness makes it appropriate for analyzing counterexamples of concurrent systems.

We study the problem of finding shortest witnesses in automata with various types of concurrency. We show that while finding shortest witnesses is more complex than just checking nonemptiness in the nondeterministic and in the concurrent models of computation, it is not more complex in the alternating model. It follows that when the system is the composition of concurrent components, finding a shortest counterexample to its correctness is not harder than finding some counterexample. Our results give a computational motivation to translating temporal logic formulas to alternating automata, rather than going all the way to nondeterministic automata.


Model Check Hamiltonian Cycle Linear Temporal Logic Acceptance Condition Input Word 
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.


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  1. 1.
    Alur, R., Henzinger, T.A., Kupferman, O., Vardi, M.Y.: Alternating refinement relations. In: Sangiorgi, D., de Simone, R. (eds.) CONCUR 1998. LNCS, vol. 1466, pp. 163–178. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  2. 2.
    Bouajjani, A., Fernandez, J.-C., Halbwachs, N.: Minimal model genration. In: Clarke, E., Kurshan, R.P. (eds.) CAV 1990. LNCS, vol. 531, pp. 197–203. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  3. 3.
    Birget, J.C.: State-complexity of finite-state devices, state compressibility and incompressibility. Mathematical Systems Theory 26(3), 237–269 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Brzozowski, J.A., Leiss, E.: Finite automata and sequential networks. TCS 10, 19–35 (1980)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Büchi, J.R.: On a decision method in restricted second order arithmetic. In: Proc. International Congress on Logic, Method, and Philosophy of Science, 1960, pp. 1–12. Stanford University Press, Stanford (1962)Google Scholar
  6. 6.
    Clarke, E.M., Emerson, E.A.: Design and synthesis of synchronization skeletons using branching time temporal logic. In: Kozen, D. (ed.) Logic of Programs 1981. LNCS, vol. 131, pp. 52–71. Springer, Heidelberg (1982)CrossRefGoogle Scholar
  7. 7.
    Clarke, E.M., Grumberg, O., Jha, S., Lu, Y., Veith, H.: Counterexample-guided abstraction refinement for symbolic model checking. J. ACM 50(5), 752–794 (2003)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Clarke, E.M., Grumberg, O., McMillan, K.L., Zhao, X.: Efficient generation of counterexamples and witnesses in symbolic model checking. In: Proc. 32nd DAC, pp. 427–432. IEEE Computer Society, Los Alamitos (1995)Google Scholar
  9. 9.
    Chandra, A.K., Kozen, D.C., Stockmeyer, L.J.: Alternation. J. ACM 28(1), 114–133 (1981)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Drusinsky, D., Harel, D.: On the power of bounded concurrency I: Finite automata. J. ACM 41(3), 517–539 (1994)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Emerson, E.A., Jutla, C.: Tree automata, μ-calculus and determinacy. In: Proc. 32nd FOCS, pp. 368–377 (1991)Google Scholar
  12. 12.
    Gurumurthy, S., Kupferman, O., Somenzi, F., Vardi, M.Y.: On Complementing Nondeterministic Büchi Automata. In: Geist, D., Tronci, E. (eds.) CHARME 2003. LNCS, vol. 2860, pp. 96–110. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  13. 13.
    Grumberg, O., Long, D.E.: Model checking and modular verification. ACM TOPLAS 16(3), 843–871 (1994)CrossRefGoogle Scholar
  14. 14.
    Gastin, P., Moro, P., Zeitoun, M.: Minimization of counterexamples in spin. In: Graf, S., Mounier, L. (eds.) SPIN 2004. LNCS, vol. 2989, pp. 92–108. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  15. 15.
    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
  16. 16.
    Galperin, H., Wigderson, A.: Succinct representations of graphs. I& C 56(3), 183–198 (1983)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Harel, D.: Statecharts: A visual formalism for complex systems. Sci. Computer Prog. 8, 231–274 (1987)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Harel, D., Kupferman, O., Vardi, M.Y.: On the complexity of verifying concurrent transition systems. I & C 173, 1–19 (2002)MathSciNetGoogle Scholar
  19. 19.
    Harel, D., Rosner, R., Vardi, M.Y.: On the power of bounded concurrency iii: Reasoning about programs. In: Proc. 5th LICS, pp. 478–488 (1990)Google Scholar
  20. 20.
    Jones, G.A., Jones, J.M., Tyrer-Jones, J.M.: Elementary Number Theory. Undergraduate Mathematics Series. Springer, Berlin (1998)zbMATHGoogle Scholar
  21. 21.
    Kozen, D.: Lower bounds for natural proof systems. In: Proc. 18th FOCS, pp. 254–266 (1977)Google Scholar
  22. 22.
    Kupferman, O., Vardi, M.Y.: Verification of fair transition systems. CJTCS 1998(2) (1998)Google Scholar
  23. 23.
    Kupferman, O., Vardi, M.Y.: Weak alternating automata are not that weak. ACM TOCL 2(2), 408–429 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Kupferman, O., Vardi, M.Y.: Vacuity detection in temporal model checking. J. STTT 4(2), 224–233 (2003)Google Scholar
  25. 25.
    Kupferman, O., Vardi, M.Y.: Safraless decision procedures. In: Proc. 46th FOCS, pp. 531–540 (2005)Google Scholar
  26. 26.
    Kupferman, O., Vardi, M.Y., Wolper, P.: An automata-theoretic approach to branching-time model checking. J. ACM 47(2), 312–360 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  27. 27.
    Miyano, S., Hayashi, T.: Alternating finite automata on ω-words. TCS 32, 321–330 (1984)CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Milner, R.: A Calculus of Communication Systems. LNCS, vol. 92. Springer, Heidelberg (1980)Google Scholar
  29. 29.
    Muller, D.E., Saoudi, A., Schupp, P.E.: Alternating automata, the weak monadic theory of the tree and its complexity. In: Kott, L. (ed.) ICALP 1986. LNCS, vol. 226, pp. 275–283. Springer, Heidelberg (1986)Google Scholar
  30. 30.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)zbMATHGoogle Scholar
  31. 31.
    Savitch, W.J.: Relationship between nondeterministic and deterministic tape complexities. Journal on Computer and System Sciences 4, 177–192 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Schuppan, V., Biere, A.: Shortest counterexamples for symbolic model checking of LTL with past. In: Halbwachs, N., Zuck, L.D. (eds.) TACAS 2005. LNCS, vol. 3440, pp. 493–509. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  33. 33.
    Sistla, A.P., Vardi, M.Y., Wolper, P.: The complementation problem for Büchi automata with applications to temporal logic. TCS 49, 217–237 (1987)CrossRefMathSciNetzbMATHGoogle Scholar
  34. 34.
    Vardi, M.Y.: Alternating automata and program verification. In: van Leeuwen, J. (ed.) Computer Science Today. LNCS, vol. 1000, pp. 471–485. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  35. 35.
    Vardi, M.Y., Wolper, P.: Reasoning about infinite computations. I& C 115(1), 1–37 (1994)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Orna Kupferman
    • 1
  • Sarai Sheinvald-Faragy
    • 1
  1. 1.School of Engineering and Computer ScienceHebrew UniversityJerusalemIsrael

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