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Deciding Universality of ptNFAs is PSpace-Complete

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SOFSEM 2018: Theory and Practice of Computer Science (SOFSEM 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10706))

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Abstract

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.

Supported by DFG grants KR 4381/1-1 & CRC 912 (HAEC), and by RVO 67985840.

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References

  1. Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms. Addison-Wesley, Boston (1974)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barceló, P., Libkin, L., Reutter, J.L.: Querying regular graph patterns. J. ACM 61(1), 8:1–8:54 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bojanczyk, M., Segoufin, L., Straubing, H.: Piecewise testable tree languages. Logical Methods Comput. Sci. 8(3) (2012)

    Google Scholar 

  5. Bouajjani, A., Muscholl, A., Touili, T.: Permutation rewriting and algorithmic verification. Inf. Comput. 205(2), 199–224 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brzozowski, J.A., Fich, F.E.: Languages of \({R}\)-trivial monoids. J. Comput. Syst. Sci. 20(1), 32–49 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  7. Calvanese, D., De Giacomo, G., Lenzerini, M., Vardi, M.Y.: Reasoning on regular path queries. ACM SIGMOD Rec. 32(4), 83–92 (2003)

    Article  Google Scholar 

  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). https://doi.org/10.1007/978-3-642-39212-2_16

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1007/978-3-642-20877-5_26

    Chapter  Google Scholar 

  11. García, P., Ruiz, J.: Learning \(k\)-testable and \(k\)-piecewise testable languages from positive data. Grammars 7, 125–140 (2004)

    Google Scholar 

  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)

    Article  Google Scholar 

  13. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)

    MATH  Google Scholar 

  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). https://doi.org/10.4230/LIPIcs.ICDT.2015.230

  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). https://doi.org/10.1007/978-3-662-44522-8_27

    Google Scholar 

  16. Jones, N.D.: Space-bounded reducibility among combinatorial problems. J. Comput. Syst. Sci. 11(1), 68–85 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1007/978-3-642-38771-5_26

    Chapter  Google Scholar 

  18. Kontorovich, L., Cortes, C., Mohri, M.: Kernel methods for learning languages. Theor. Comput. Sci. 405(3), 223–236 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1016/j.ic.2017.06.004

  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). https://doi.org/10.1145/2745754.2745774

  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). https://doi.org/10.4230/LIPIcs.MFCS.2016.67

  22. Masopust, T., Krötzsch, M.: Universality of confluent, self-loop deterministic partially ordered NFAs is hard (2017). http://arxiv.org/abs/1704.07860

  23. Masopust, T., Thomazo, M.: On Boolean combinations forming piecewise testable languages. Theor. Comput. Sci. 682, 165–179 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1109/SWAT.1972.29

  25. Perrin, D., Pin, J.E.: Infinite Words: Automata, Semigroups, Logic and Games, Pure and Applied Mathematics, vol. 141. Academic Press, Cambridge (2004)

    MATH  Google Scholar 

  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). https://doi.org/10.1007/978-3-642-40313-2_64

    Chapter  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1007/978-3-642-14322-9_19

    Chapter  Google Scholar 

  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). https://doi.org/10.1007/978-3-642-39998-5_6

    Chapter  Google Scholar 

  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). https://doi.org/10.1007/3-540-46011-X_20

    Chapter  Google Scholar 

  31. Simon, I.: Hierarchies of events with dot-depth one. Ph.D. thesis, University of Waterloo, Canada (1972)

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1145/800125.804029

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Author information

Authors and Affiliations

  1. Institute of Mathematics, Czech Academy of Sciences, Brno, Czech Republic

    Tomáš Masopust

  2. cfaed, TU Dresden, Dresden, Germany

    Markus Krötzsch

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  1. Tomáš Masopust
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  2. Markus Krötzsch
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Correspondence to Tomáš Masopust .

Editor information

Editors and Affiliations

  1. Vienna University of Technology , Vienna, Austria

    A Min Tjoa

  2. ISAE-ENSMA, Chasseneuil-du-Poitou, France

    Ladjel Bellatreche

  3. Vienna University of Technology, Vienna, Austria

    Stefan Biffl

  4. Utrecht University, Utrecht, The Netherlands

    Jan van Leeuwen

  5. Academy of Sciences, Prague, Czech Republic

    Jiří Wiedermann

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Masopust, T., Krötzsch, M. (2018). Deciding Universality of ptNFAs is PSpace-Complete. In: Tjoa, A., Bellatreche, L., Biffl, S., van Leeuwen, J., Wiedermann, J. (eds) SOFSEM 2018: Theory and Practice of Computer Science. SOFSEM 2018. Lecture Notes in Computer Science(), vol 10706. Edizioni della Normale, Cham. https://doi.org/10.1007/978-3-319-73117-9_29

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  • DOI: https://doi.org/10.1007/978-3-319-73117-9_29

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  • Publisher Name: Edizioni della Normale, Cham

  • Print ISBN: 978-3-319-73116-2

  • Online ISBN: 978-3-319-73117-9

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