A Note on Monitors and Büchi Automata

  • Volker Diekert
  • Anca Muscholl
  • Igor Walukiewicz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9399)


When a property needs to be checked against an unknown or very complex system, classical exploration techniques like model-checking are not applicable anymore. Sometimes a monitor can be used, that checks a given property on the underlying system at runtime. A monitor for a property L is a deterministic finite automaton \(\mathcal {M}_L\) that after each finite execution tells whether (1) every possible extension of the execution is in L, or (2) every possible extension is in the complement of L, or neither (1) nor (2) holds. Moreover, L being monitorable means that it is always possible that in some future the monitor reaches (1) or (2). Classical examples for monitorable properties are safety and cosafety properties. On the other hand, deterministic liveness properties like “infinitely many a’s” are not monitorable.

We discuss various monitor constructions with a focus on deterministic \(\omega \)-regular languages. We locate a proper subclass of deterministic \(\omega \)-regular languages but also strictly larger than the subclass of languages which are deterministic and codeterministic; and for this subclass there exist canonical monitors which also accept the language itself.

We also address the problem to decide monitorability in comparison with deciding liveness. The state of the art is as follows. Given a Büchi automaton, it is PSPACE-complete to decide liveness or monitorability. Given an LTL formula, deciding liveness becomes EXPSPACE-complete, but the complexity to decide monitorability remains open.


Turing Machine Regular Language Linear Temporal Logic Safety Property Deterministic Finite Automaton 
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.



The work was done while the first author was visiting LaBRI in the framework of the IdEx Bordeaux Visiting Professors Programme in June 2015. The hospitality of LaBRI and their members is greatly acknowledged.

The authors thank Andreas Bauer who communicated to us (in June 2012) that the complexity of \(\mathrm {LTL}\)-liveness should be regarded as open because published proofs stating PSPACE-completeness were not convincing. We also thank Ludwig Staiger, Gal Vardi, and Mikhail Volkov for helpful comments.


  1. 1.
    Angluin, D., Fisman, D.: Learning regular omega languages. In: Auer, P., Clark, A., Zeugmann, T., Zilles, S. (eds.) ALT 2014. LNCS, vol. 8776, pp. 125–139. Springer, Heidelberg (2014) Google Scholar
  2. 2.
    Basin, D., Klaedtke, F., Müller, S., Zalinescu, E.: Monitoring metric first-order temporal properties. J. ACM 62, 15:1–15:45 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bauer, A., Leucker, M., Schallhart, C.: Monitoring of real-time properties. In: Arun-Kumar, S., Garg, N. (eds.) FSTTCS 2006. LNCS, vol. 4337, pp. 260–272. Springer, Heidelberg (2006) CrossRefGoogle Scholar
  4. 4.
    Diekert, V., Gastin, P.: First-order definable languages. In: Flum, J., Grädel, E., Wilke, Th. (eds.) Logic and Automata: History and Perspectives, Texts in Logic and Games, pp. 261–306. Amsterdam University Press (2008)Google Scholar
  5. 5.
    Diekert, V., Leucker, M.: Topology, monitorable properties and runtime verification. Theor. Comput. Sci. 537, 29–41 (2014). Special Issue of ICTAC 2012MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Diekert, V., Muscholl, A.: On distributed monitoring of asynchronous systems. In: Ong, L., de Queiroz, R. (eds.) WoLLIC 2012. LNCS, vol. 7456, pp. 70–84. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  7. 7.
    Gondi, K., Patel, Y., Sistla, A.P.: Monitoring the full range of \(\omega \)-regular properties of stochastic systems. In: Jones, N.D., Müller-Olm, M. (eds.) VMCAI 2009. LNCS, vol. 5403, pp. 105–119. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  8. 8.
    Hopcroft, J.E., Ulman, J.D.: Introduction to Automata Theory, Languages and Computation. Addison-Wesley, Reading (1979) Google Scholar
  9. 9.
    Kamp, H.: Tense logic and the theory of linear order. Ph.D. thesis, University of California (1968)Google Scholar
  10. 10.
    Knuth, D., Morris, J.H., Pratt, V.: Fast pattern matching in strings. SIAM J. Comput. 6, 323–350 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kupferman, O., Vardi, G.: On relative and probabilistic finite counterabilty. In: Kreutzer, S. (ed.) Proceedings 24th EACSL Annual Conference on Computer Science Logic (CSL 2015). LIPIcs, vol. 41, Dagstuhl, Germany, pp. 175–192. Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)Google Scholar
  12. 12.
    Landweber, L.H.: Decision problems for \(\omega \)-automata. Math. Syst. Theory 3(4), 376–384 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Leucker, M., Schallhart, C.: A brief account of runtime verification. J. Logic Algebraic Program. 78(5), 293–303 (2009)CrossRefzbMATHGoogle Scholar
  14. 14.
    Maler, O., Pnueli, A.: On the learnability of infinitary regular sets. Inf. Comput. 118, 316–326 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Martugin, P.V.: A series of slowly synchronizing automata with a zero state over a small alphabet. Inf. Comput. 206, 1197–1203 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Matiyasevich, Y.: Real-time recognition of the inclusion relation. J. Sov. Math. 1, 64–70 (1973). Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, vol. 20, pp. 104–114 (1971)CrossRefzbMATHGoogle Scholar
  17. 17.
    Nitsche, U., Wolper, P.: Relative liveness and behavior abstraction (extended abstract). In: Burns, J.E., Attiya, H. (eds.) Proceedings of the Sixteenth Annual ACM Symposium on Principles of Distributed Computing (PODS 1997), Santa Barbara, California, USA, 21–24 August 1997, pp. 45–52. ACM (1997)Google Scholar
  18. 18.
    Pnueli, A., Zaks, A.: PSL model checking and run-time verification via testers. In: Misra, J., Nipkow, T., Sekerinski, E. (eds.) FM 2006. LNCS, vol. 4085, pp. 573–586. Springer, Heidelberg (2006) CrossRefGoogle Scholar
  19. 19.
    Rystsov, I.: Reset words for commutative and solvable automata. Theor. Comput. Sci. 172, 273–279 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Sistla, A.P., Žefran, M., Feng, Y.: Monitorability of stochastic dynamical systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 720–736. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  21. 21.
    Staiger, L.: Reguläre Nullmengen. Elektronische Informationsverarbeitung und Kybernetik 12(6), 307–311 (1976)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Staiger, L.: Finite-state \(\omega \)-languages. J. Comput. Syst. Sci. 27, 434–448 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Staiger, L., Wagner, K.W.: Automatentheoretische und automatenfreie Charakterisierungen topologischer Klassen regulärer Folgenmengen. Elektronische Informationsverarbeitung und Kybernetik 10, 379–392 (1974)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Tabakov, D., Rozier, K.Y., Vardi, M.Y.: Optimized temporal monitors for SystemC. Formal Methods Syst. Des. 41, 236–268 (2012)CrossRefzbMATHGoogle Scholar
  25. 25.
    Thomas, W.: Automata on infinite objects. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, chap. 4, pp. 133–191. Elsevier Science Publishers B.V., Amsterdam (1990)Google Scholar
  26. 26.
    Volkov, M.V.: Synchronizing automata and the Cerný conjecture. In: Martín-Vide, C., Otto, F., Fernau, H. (eds.) LATA 2008. LNCS, vol. 5196, pp. 11–27. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  27. 27.
    Wagner, K.W.: On omega-regular sets. Inf. Control 43, 123–177 (1979)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Volker Diekert
    • 1
  • Anca Muscholl
    • 2
  • Igor Walukiewicz
    • 2
  1. 1.FMIUniversität StuttgartStuttgartGermany
  2. 2.LaBRIUniversity of BordeauxTalenceFrance

Personalised recommendations