Deciding Universality of ptNFAs is PSpace-Complete

  • Tomáš Masopust
  • Markus Krötzsch
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10706)


An automaton is partially ordered if the only cycles in its transition diagram are self-loops. We study the universality problem for ptNFAs, a class of partially ordered NFAs recognizing piecewise testable languages. The universality problem asks if an automaton accepts all words over its alphabet. Deciding universality for both NFAs and partially ordered NFAs is PSpace-complete. For ptNFAs, the complexity drops to coNP-complete if the alphabet is fixed but is open if the alphabet may grow. We show, using a novel and nontrivial construction, that the problem is PSpace-complete if the alphabet may grow polynomially.


  1. 1.
    Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms. Addison-Wesley, Boston (1974)zbMATHGoogle Scholar
  2. 2.
    Almeida, J., Costa, J.C., Zeitoun, M.: Pointlike sets with respect to R and J. J. Pure Appl. Algebra 212(3), 486–499 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barceló, P., Libkin, L., Reutter, J.L.: Querying regular graph patterns. J. ACM 61(1), 8:1–8:54 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bojanczyk, M., Segoufin, L., Straubing, H.: Piecewise testable tree languages. Logical Methods Comput. Sci. 8(3) (2012)Google Scholar
  5. 5.
    Bouajjani, A., Muscholl, A., Touili, T.: Permutation rewriting and algorithmic verification. Inf. Comput. 205(2), 199–224 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brzozowski, J.A., Fich, F.E.: Languages of \({R}\)-trivial monoids. J. Comput. Syst. Sci. 20(1), 32–49 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Calvanese, D., De Giacomo, G., Lenzerini, M., Vardi, M.Y.: Reasoning on regular path queries. ACM SIGMOD Rec. 32(4), 83–92 (2003)CrossRefGoogle Scholar
  8. 8.
    Czerwiński, W., Martens, W., Masopust, T.: Efficient separability of regular languages by subsequences and suffixes. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013. LNCS, vol. 7966, pp. 150–161. Springer, Heidelberg (2013). CrossRefGoogle Scholar
  9. 9.
    Diekert, V., Gastin, P., Kufleitner, M.: A survey on small fragments of first-order logic over finite words. Int. J. Found. Comput. Science 19(3), 513–548 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fu, J., Heinz, J., Tanner, H.G.: An algebraic characterization of strictly piecewise languages. In: Ogihara, M., Tarui, J. (eds.) TAMC 2011. LNCS, vol. 6648, pp. 252–263. Springer, Heidelberg (2011). CrossRefGoogle Scholar
  11. 11.
    García, P., Ruiz, J.: Learning \(k\)-testable and \(k\)-piecewise testable languages from positive data. Grammars 7, 125–140 (2004)Google Scholar
  12. 12.
    García, P., Vidal, E.: Inference of k-testable languages in the strict sense and application to syntactic pattern recognition. IEEE Trans. Pattern Anal. Mach. Intell. 12(9), 920–925 (1990)CrossRefGoogle Scholar
  13. 13.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)zbMATHGoogle Scholar
  14. 14.
    Hofman, P., Martens, W.: Separability by short subsequences and subwords. In: Arenas, M., Ugarte, M. (eds.) ICDT 2015. LIPIcs, vol. 31, pp. 230–246. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2015).
  15. 15.
    Holub, Š., Jirásková, G., Masopust, T.: On upper and lower bounds on the length of alternating towers. In: Csuhaj-Varjú, E., Dietzfelbinger, M., Ésik, Z. (eds.) MFCS 2014. LNCS, vol. 8634, pp. 315–326. Springer, Heidelberg (2014). Google Scholar
  16. 16.
    Jones, N.D.: Space-bounded reducibility among combinatorial problems. J. Comput. Syst. Sci. 11(1), 68–85 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Klíma, O., Polák, L.: Alternative automata characterization of piecewise testable languages. In: Béal, M.-P., Carton, O. (eds.) DLT 2013. LNCS, vol. 7907, pp. 289–300. Springer, Heidelberg (2013). CrossRefGoogle Scholar
  18. 18.
    Kontorovich, L., Cortes, C., Mohri, M.: Kernel methods for learning languages. Theor. Comput. Sci. 405(3), 223–236 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Krötzsch, M., Masopust, T., Thomazo, M.: Complexity of universality and related problems for partially ordered NFAs. Inf. Comput. Part 1 255, 177–192 (2017).
  20. 20.
    Martens, W., Neven, F., Niewerth, M., Schwentick, T.: BonXai: combining the simplicity of DTD with the expressiveness of XML schema. In: Milo, T., Calvanese, D. (eds.) PODS 2015, pp. 145–156. ACM (2015).
  21. 21.
    Masopust, T.: Piecewise testable languages and nondeterministic automata. In: Faliszewski, P., Muscholl, A., Niedermeier, R. (eds.) MFCS 2016. LIPIcs, vol. 58, pp. 67:1–67:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2016).
  22. 22.
    Masopust, T., Krötzsch, M.: Universality of confluent, self-loop deterministic partially ordered NFAs is hard (2017).
  23. 23.
    Masopust, T., Thomazo, M.: On Boolean combinations forming piecewise testable languages. Theor. Comput. Sci. 682, 165–179 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Meyer, A.R., Stockmeyer, L.J.: The equivalence problem for regular expressions with squaring requires exponential space. In: Symposium on Switching and Automata Theory, pp. 125–129. IEEE Computer Society (1972).
  25. 25.
    Perrin, D., Pin, J.E.: Infinite Words: Automata, Semigroups, Logic and Games, Pure and Applied Mathematics, vol. 141. Academic Press, Cambridge (2004)zbMATHGoogle Scholar
  26. 26.
    Place, T., van Rooijen, L., Zeitoun, M.: Separating regular languages by piecewise testable and unambiguous languages. In: Chatterjee, K., Sgall, J. (eds.) MFCS 2013. LNCS, vol. 8087, pp. 729–740. Springer, Heidelberg (2013). CrossRefGoogle Scholar
  27. 27.
    Rampersad, N., Shallit, J., Xu, Z.: The computational complexity of universality problems for prefixes, suffixes, factors, and subwords of regular languages. Fundamenta Informatica 116(1–4), 223–236 (2012)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Rogers, J., Heinz, J., Bailey, G., Edlefsen, M., Visscher, M., Wellcome, D., Wibel, S.: On languages piecewise testable in the strict sense. In: Ebert, C., Jäger, G., Michaelis, J. (eds.) MOL 2007/2009. LNCS (LNAI), vol. 6149, pp. 255–265. Springer, Heidelberg (2010). CrossRefGoogle Scholar
  29. 29.
    Rogers, J., Heinz, J., Fero, M., Hurst, J., Lambert, D., Wibel, S.: Cognitive and sub-regular complexity. In: Morrill, G., Nederhof, M.-J. (eds.) FG 2012-2013. LNCS, vol. 8036, pp. 90–108. Springer, Heidelberg (2013). CrossRefGoogle Scholar
  30. 30.
    Schwentick, T., Thérien, D., Vollmer, H.: Partially-ordered two-way automata: a new characterization of DA. In: Kuich, W., Rozenberg, G., Salomaa, A. (eds.) DLT 2001. LNCS, vol. 2295, pp. 239–250. Springer, Heidelberg (2002). CrossRefGoogle Scholar
  31. 31.
    Simon, I.: Hierarchies of events with dot-depth one. Ph.D. thesis, University of Waterloo, Canada (1972)Google Scholar
  32. 32.
    Stefanoni, G., Motik, B., Krötzsch, M., Rudolph, S.: The complexity of answering conjunctive and navigational queries over OWL 2 EL knowledge bases. J. Artif. Intell. Res. 51, 645–705 (2014)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Stockmeyer, L.J., Meyer, A.R.: Word problems requiring exponential time: preliminary report. In: Aho, A.V., Borodin, A., Constable, R.L., Floyd, R.W., Harrison, M.A., Karp, R.M., Strong, R. (eds.) ACM Symposium on the Theory of Computing, pp. 1–9. ACM (1973).

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Authors and Affiliations

  1. 1.Institute of MathematicsCzech Academy of SciencesBrnoCzech Republic
  2. 2.cfaedTU DresdenDresdenGermany

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