Minimal Separating Sequences for All Pairs of States

  • Rick SmetsersEmail author
  • Joshua Moerman
  • David N. Jansen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9618)


Finding minimal separating sequences for all pairs of inequivalent states in a finite state machine is a classic problem in automata theory. Sets of minimal separating sequences, for instance, play a central role in many conformance testing methods. Moore has already outlined a partition refinement algorithm that constructs such a set of sequences in \(\mathcal {O}(mn)\) time, where m is the number of transitions and n is the number of states. In this paper, we present an improved algorithm based on the minimization algorithm of Hopcroft that runs in \(\mathcal {O}(m \log n)\) time. The efficiency of our algorithm is empirically verified and compared to the traditional algorithm.


Algorithms on automata and words Partition refinement 


  1. 1.
    Bonchi, F., Pous, D.: Checking NFA equivalence with bisimulations up to congruence. In: POPL, pp. 457–468 (2013)Google Scholar
  2. 2.
    Dorofeeva, R., El-Fakih, K., Maag, S., Cavalli, A., Yevtushenko, N.: FSM-based conformance testing methods: a survey annotated with experimental evaluation. Inf. Softw. Technol. 52(12), 1286–1297 (2010)CrossRefGoogle Scholar
  3. 3.
    Gill, A.: Introduction to the Theory of Finite-state Machines. McGraw-Hill, New York (1962)zbMATHGoogle Scholar
  4. 4.
    Gries, D.: Describing an algorithm by Hopcroft. Acta Informatica 2(2), 97–109 (1973)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Hierons, R.M., Türker, U.C.: Incomplete distinguishing sequences for finite state machines. Comput. J. 58, 1–25 (2015)CrossRefGoogle Scholar
  6. 6.
    Hopcroft, J.E.: An n log n algorithm for minimizing states in a finite automaton. In: Theory of Machines and Computations, pp. 189–196 (1971)Google Scholar
  7. 7.
    Knuutila, T.: Re-describing an algorithm by Hopcroft. Theoret. Comput. Sci. 250(1–2), 333–363 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Lee, D., Yannakakis, M.: Testing finite-state machines: state identification and verification. Computers 43(3), 306–320 (1994)MathSciNetGoogle Scholar
  9. 9.
    Moore, E.F.: Gedanken-experiments on sequential machines. Automata Stud. 34, 129–153 (1956)Google Scholar
  10. 10.
    Smeenk, W., Moerman, J., Vaandrager, F., Jansen, D.N.: Applying automata learning to embedded control software. In: Butler, M., et al. (eds.) ICFEM 2015. LNCS, vol. 9407, pp. 67–83. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-25423-4_5 CrossRefGoogle Scholar
  11. 11.
    Valmari, A., Lehtinen, P.: Efficient minimization of DFAs with partial transition functions. In: STACS, pp. 645–656 (2008)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Rick Smetsers
    • 1
    Email author
  • Joshua Moerman
    • 1
  • David N. Jansen
    • 1
  1. 1.Institute for Computing and Information SciencesRadboud UniversityNijmegenThe Netherlands

Personalised recommendations