On Properties and State Complexity of Deterministic State-Partition Automata

  • Galina Jirásková
  • Tomáš Masopust
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7604)


A deterministic automaton accepting a regular language L is a state-partition automaton with respect to a projection P if the state set of the deterministic automaton accepting the projected language P(L), obtained by the standard subset construction, forms a partition of the state set of the automaton. In this paper, we study fundamental properties of state-partition automata. We provide a construction of the minimal state-partition automaton for a regular language and a projection, discuss closure properties of state-partition automata under the standard constructions of deterministic automata for regular operations, and show that almost all of them fail to preserve the property of being a state-partition automaton. Finally, we define the notion of a state-partition complexity, and prove the tight bound on the state-partition complexity of regular languages represented by incomplete deterministic automata.


Regular languages finite automata descriptional complexity projections state-partition automata 


  1. 1.
    Boutin, O., Komenda, J., Masopust, T., Schmidt, K., van Schuppen, J.H.: Hierarchical control with partial observations: Sufficient conditions. In: Proc. of IEEE Conference on Decision and Control and European Control Conference (CDC-ECC 2011), Orlando, Florida, USA, pp. 1817–1822 (2011)Google Scholar
  2. 2.
    Cassandras, C.G., Lafortune, S.: Introduction to discrete event systems, 2nd edn. Springer (2008)Google Scholar
  3. 3.
    Cho, H., Marcus, S.I.: On supremal languages of classes of sublanguages that arise in supervisor synthesis problems with partial observation. Mathematics of Control, Signals, and Systems 2, 47–69 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Cho, H., Marcus, S.I.: Supremal and maximal sublanguages arising in supervisor synthesis problems with partial observations. Theory of Computing Systems 22(1), 177–211 (1989)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Hopcroft, J.E., Motwani, R., Ullman, J.D.: Introduction to automata theory, languages, and computation. Addison-Wesley, Boston (2003)Google Scholar
  6. 6.
    Jirásková, G., Masopust, T.: On a structural property in the state complexity of projected regular languages. Theoretical Computer Science 449C, 93–105 (2012)CrossRefGoogle Scholar
  7. 7.
    Jirásková, G., Okhotin, A.: State complexity of cyclic shift. RAIRO – Theoretical Informatics and Applications 42(2), 335–360 (2008)zbMATHCrossRefGoogle Scholar
  8. 8.
    Komenda, J., Masopust, T., Schmidt, K., van Schuppen, J.H.: Personal communication (2011)Google Scholar
  9. 9.
    Komenda, J., Masopust, T., van Schuppen, J.H.: Synthesis of controllable and normal sublanguages for discrete-event systems using a coordinator. Systems & Control Letters 60(7), 492–502 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Komenda, J., Masopust, T., van Schuppen, J.H.: Supervisory control synthesis of discrete-event systems using a coordination scheme. Automatica 48(2), 247–254 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Komenda, J., van Schuppen, J.H.: Supremal normal sublanguages of large distributed discrete-event systems. In: Proc. of International Workshop on Discrete Event Systems (WODES 2004), Reims, France, pp. 73–78 (2004)Google Scholar
  12. 12.
    Komenda, J., van Schuppen, J.H.: Modular control of discrete-event systems with coalgebra. IEEE Transactions on Automatic Control 53(2), 447–460 (2008)CrossRefGoogle Scholar
  13. 13.
    Rabin, M.O., Scott, D.: Finite automata and their decision problems. IBM Journal of Research and Development 3(2), 114–125 (1959)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Rozenberg, G., Salomaa, A.: Handbook of Formal Languages, vol. 1–3. Springer (1997)Google Scholar
  15. 15.
    Sakarovitch, J.: A construction on finite automata that has remained hidden. Theoretical Computer Science 204(1-2), 205–231 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Salomaa, A.: Formal languages. Academic Press, New York (1973)zbMATHGoogle Scholar
  17. 17.
    Schmidt, K., Breindl, C.: Maximally permissive hierarchical control of decentralized discrete event systems. IEEE Transactions on Automatic Control 56(4), 723–737 (2011)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Schmidt, K., Moor, T., Perk, S.: Nonblocking hierarchical control of decentralized discrete event systems. IEEE Transactions on Automatic Control 53(10), 2252–2265 (2008)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Sipser, M.: Introduction to the theory of computation. PWS Publishing Company, Boston (1997)zbMATHGoogle Scholar
  20. 20.
    Wong, K.: On the complexity of projections of discrete-event systems. In: Proc. of Workshop on Discrete Event Systems (WODES 1998), Cagliari, Italy, pp. 201–206 (1998)Google Scholar
  21. 21.
    Wong, K.C., Wonham, W.M.: Hierarchical control of discrete-event systems. Discrete Event Dynamic Systems: Theory and Applications 6(3), 241–273 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Wonham, W.M.: Supervisory control of discrete-event systems, lecture notes, University of Toronto (2011) (Online),
  23. 23.
    Yu, S.: Regular languages. In: Handbook of Formal Languages, vol. I, pp. 41–110. Springer (1997)Google Scholar
  24. 24.
    Zhong, H., Wonham, W.M.: On the consistency of hierarchical supervision in discrete-event systems. IEEE Transactions on Automatic Control 35(10), 1125–1134 (1990)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2012

Authors and Affiliations

  • Galina Jirásková
    • 1
  • Tomáš Masopust
    • 2
  1. 1.Mathematical InstituteSlovak Academy of SciencesKošiceSlovak Republic
  2. 2.Institute of MathematicsAcademy of Sciences of the Czech RepublicBrnoCzech Republic

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