Advertisement

Operational State Complexity and Decidability of Jumping Finite Automata

  • Simon BeierEmail author
  • Markus Holzer
  • Martin Kutrib
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10396)

Abstract

We consider jumping finite automata and their operational state complexity and decidability status. Roughly speaking, a jumping automaton is a finite automaton with a non-continuous input. This device has nice relations to semilinear sets and thus to Parikh images of regular sets, which will be exhaustively used in our proofs. In particular, we prove upper bounds on the intersection and complementation. The latter result on the complementation upper bound answers an open problem from G.J. Lavado, G. Pighizzini, S. Seki: Operational State Complexity of Parikh Equivalence [2014]. Moreover, we correct an erroneous result on the inverse homomorphism closure. Finally, we also consider the decidability status of standard problems as regularity, disjointness, universality, inclusion, etc. for jumping finite automata.

References

  1. 1.
    Baeza-Yates, R., Ribeiro-Neto, B.: Modern Information Retrieval: The Concepts and Technology Behind Search. Addison-Wesley, New York (2011)Google Scholar
  2. 2.
    Beier, S., Holzer, M., Kutrib, M.: On the descriptional complexity of operations on semilinear sets. IFIG Research Report 1701, Institut für Informatik, Universität Giessen (2017). http://www.informatik.uni-giessen.de/reports/Report1701.pdf
  3. 3.
    Dickson, L.E.: Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors. Am. J. Math. 35, 413–422 (1913)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Fernau, H., Paramasivan, M., Schmid, M.L.: Jumping finite automata: characterizations and complexity. In: Drewes, F. (ed.) CIAA 2015. LNCS, vol. 9223, pp. 89–101. Springer, Cham (2015). doi: 10.1007/978-3-319-22360-5_8 CrossRefGoogle Scholar
  5. 5.
    Fernau, H., Paramasivan, M., Schmid, M.L., Vorel, V.: Characterization and complexity results on jumping finite automata. Theor. Comput. Sci. 679, 31–52 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ginsburg, S., Spanier, E.H.: Bounded ALGOL-like languages. Trans. AMS 113, 333–368 (1964)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages and Computation. Addison-Wesley, Reading (1979)zbMATHGoogle Scholar
  8. 8.
    Huynh, D.T.: Deciding the inequivalence of context-free grammars with 1-letter terminal alphabet is \(\Sigma _2^{p}\)-complete. Theor. Comput. Sci. 33, 305–326 (1984)CrossRefzbMATHGoogle Scholar
  9. 9.
    Huynh, D.T.: The complexity of equivalence problems for commutative grammars. Inf. Control 66, 103–121 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Huynh, D.T.: A simple proof for the \(\Sigma _2^P\) upper bound of the inequivalence problem for semilinear sets. Elektr. Informationsverarb. Kybernet. 22, 147–156 (1986)zbMATHGoogle Scholar
  11. 11.
    Huynh, T.-D.: The complexity of semilinear sets. In: Bakker, J., Leeuwen, J. (eds.) ICALP 1980. LNCS, vol. 85, pp. 324–337. Springer, Heidelberg (1980). doi: 10.1007/3-540-10003-2_81 CrossRefGoogle Scholar
  12. 12.
    Kopczyńki, E., To, A.W.: Parikh images of grammar: Complexity and applications. In: Joiannaud, J.P. (ed.) Proceedings of the 25th Annual IEEE Symposium on Logic in Computer Science, pp. 80–89. IEEE (2010)Google Scholar
  13. 13.
    Kopczyński, E.: Complexity of problems of commutative grammars. Log. Methods Comput. Sci. 11(1) (2015). Paper 9Google Scholar
  14. 14.
    Lavado, G.J., Pighizzini, G., Seki, S.: Converting nondeterministic automata and context-free grammars into Parikh equivalent one-way and two-way deterministic automata. Inf. Comput. 228–229, 1–15 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lavado, G.J., Pighizzini, G., Seki, S.: Operational state complexity under parikh equivalence. In: Jürgensen, H., Karhumäki, J., Okhotin, A. (eds.) DCFS 2014. LNCS, vol. 8614, pp. 294–305. Springer, Cham (2014). doi: 10.1007/978-3-319-09704-6_26 Google Scholar
  16. 16.
    Meduna, A., Zemek, P.: Jumping finite automata. Int. J. Found. Comput. Sci. 23, 1555–1578 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    To, A.W.: Parikh images of regular languages: complexity and applications (2010). http://arxiv.org/abs/1002.1464v2
  18. 18.
    Vorel, V.: On basic properties of jumping finite automata (2015). http://arxiv.org/abs/1511.08396v2
  19. 19.
    Vorel, V.: Two results on discontinuous input processing. In: Câmpeanu, C., Manea, F., Shallit, J. (eds.) DCFS 2016. LNCS, vol. 9777, pp. 205–216. Springer, Cham (2016). doi: 10.1007/978-3-319-41114-9_16 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institut für InformatikUniversität GiessenGiessenGermany

Personalised recommendations