Finding All Minimum-Size DFA Consistent with Given Examples: SAT-Based Approach

  • Ilya ZakirzyanovEmail author
  • Anatoly Shalyto
  • Vladimir Ulyantsev
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10729)


Deterministic finite automaton (DFA) is a fundamental concept in the theory of computation. The NP-hard DFA identification problem can be efficiently solved by translation to the Boolean satisfiability problem (SAT). Previously we developed a technique to reduce the problem search space by enforcing DFA states to be enumerated in breadth-first search (BFS) order. We proposed symmetry breaking predicates, which can be added to Boolean formulae representing various automata identification problems. In this paper we continue the study of SAT-based approaches. First, we propose new predicates based on depth-first search order. Second, we present three methods to identify all non-isomorphic automata of the minimum size instead of just one—the Open image in new window P-complete problem which has not been solved before. Third, we revisited our implementation of the BFS-based approach and conducted new evaluation experiments. It occurs that BFS-based approach outperforms all other exact algorithms for DFA identification and can be effectively applied for finding all solutions of the problem.


Grammatical inference Automata identification Symmetry breaking Boolean satisfiability 



The authors would like to thank Igor Buzhinsky, Daniil Chivilikhin, Maxim Buzdalov for useful comments. This work was financially supported by the Government of Russian Federation, Grant 074-U01.


  1. 1.
    Hopcroft, J., Motwani, R., Ullman, J.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Boston (2006)zbMATHGoogle Scholar
  2. 2.
    De La Higuera, C.: A bibliographical study of grammatical inference. Pattern Recogn. 38(9), 1332–1348 (2005)CrossRefGoogle Scholar
  3. 3.
    Gold, E.M.: Complexity of automaton identification from given data. Inf. Control 37(3), 302–320 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dupont, P.: Regular grammatical inference from positive and negative samples by genetic search: the GIG method. In: Carrasco, R.C., Oncina, J. (eds.) ICGI 1994. LNCS, vol. 862, pp. 236–245. Springer, Heidelberg (1994). CrossRefGoogle Scholar
  5. 5.
    Luke, S., Hamahashi, S., Kitano, H.: Genetic programming. In: Proceedings of the genetic and evolutionary computation conference, vol. 2, pp. 1098–1105 (1999)Google Scholar
  6. 6.
    Lucas, S.M., Reynolds, T.J.: Learning DFA: evolution versus evidence driven state merging. In: The 2003 Congress on Evolutionary Computation, 2003. CEC 2003, vol. 1, pp. 351–358. IEEE (2003)Google Scholar
  7. 7.
    Lang, K.J., Pearlmutter, B.A., Price, R.A.: Results of the Abbadingo one DFA learning competition and a new evidence-driven state merging algorithm. In: Honavar, V., Slutzki, G. (eds.) ICGI 1998. LNCS, vol. 1433, pp. 1–12. Springer, Heidelberg (1998). CrossRefGoogle Scholar
  8. 8.
    Lang, K.J.: Faster algorithms for finding minimal consistent DFAs. Technical report (1999)Google Scholar
  9. 9.
    Bugalho, M., Oliveira, A.L.: Inference of regular languages using state merging algorithms with search. Pattern Recogn. 38(9), 1457–1467 (2005)CrossRefzbMATHGoogle Scholar
  10. 10.
    Heule, M.J.H., Verwer, S.: Exact DFA identification using SAT solvers. In: Sempere, J.M., García, P. (eds.) ICGI 2010. LNCS (LNAI), vol. 6339, pp. 66–79. Springer, Heidelberg (2010). CrossRefGoogle Scholar
  11. 11.
    Lohfert, R., Lu, J.J., Zhao, D.: Solving SQL constraints by incremental translation to SAT. In: Nguyen, N.T., Borzemski, L., Grzech, A., Ali, M. (eds.) IEA/AIE 2008. LNCS (LNAI), vol. 5027, pp. 669–676. Springer, Heidelberg (2008). CrossRefGoogle Scholar
  12. 12.
    Galeotti, J.P., Rosner, N., Pombo, C.G.L., Frias, M.F.: TACO: efficient SAT-based bounded verification using symmetry breaking and tight bounds. IEEE Trans. Softw. Eng. 39(9), 1283–1307 (2013)CrossRefGoogle Scholar
  13. 13.
    Ulyantsev, V., Tsarev, F.: Extended finite-state machine induction using SAT-solver. In: Proceedings of ICMLA 2011, vol. 2, pp. 346–349. IEEE (2011)Google Scholar
  14. 14.
    Zbrzezny, A.: A new translation from ECTL* to SAT. Fundamenta Informaticae 120(3–4), 375–395 (2012)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Walkinshaw, N., Lambeau, B., Damas, C., Bogdanov, K., Dupont, P.: STAMINA: a competition to encourage the development and assessment of software model inference techniques. Empirical Software Engineering 18(4), 791–824 (2013)CrossRefGoogle Scholar
  16. 16.
    Crawford, J., Ginsberg, M., Luks, E., Roy, A.: Symmetry-breaking predicates for search problems. KR 96, 148–159 (1996)Google Scholar
  17. 17.
    Ulyantsev, V., Zakirzyanov, I., Shalyto, A.: BFS-based symmetry breaking predicates for DFA identification. In: Dediu, A.-H., Formenti, E., Martín-Vide, C., Truthe, B. (eds.) LATA 2015. LNCS, vol. 8977, pp. 611–622. Springer, Cham (2015). Google Scholar
  18. 18.
    Ulyantsev, V., Buzhinsky, I., Shalyto, A.: Exact finite-state machine identification from scenarios and temporal properties. Int. J. Softw. Tools Technol. Transf. 1–21 (2016)Google Scholar
  19. 19.
    Eén, N., Sörensson, N.: An extensible SAT-solver. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 502–518. Springer, Heidelberg (2004). CrossRefGoogle Scholar
  20. 20.
    Biere, A.: Splatz, lingeling, plingeling, treengeling, YalSAT entering the SAT competition 2016. In: Proceedings of SAT Competition, pp. 44–45 (2016)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Ilya Zakirzyanov
    • 1
    • 2
    Email author
  • Anatoly Shalyto
    • 1
  • Vladimir Ulyantsev
    • 1
  1. 1.ITMO UniversitySaint PetersburgRussia
  2. 2.JetBrains ResearchSaint PetersburgRussia

Personalised recommendations