In data words, each position carries not only a letter form a finite alphabet, as the usual words do, but also a data value coming from an infinite domain. There has been a renewed interest in them due to applications in querying and reasoning about data models with complex structural properties, notably XML, and more recently, graph databases. Logical formalisms designed for querying such data often require concise and easily understandable presentations of regular languages over data words.

Our goal, therefore, is to define and study regular expressions for data words. As the automaton model, we take register automata, which are a natural analog of NFAs for data words. We first equip standard regular expressions with limited memory, and show that they capture the class of data words defined by register automata. The complexity of the main decision problems for these expressions (nonemptiness, membership) also turns out to be the same as for register automata. We then look at a subclass of these regular expressions that can define many properties of interest in applications of data words, and show that the main decision problems can be solved efficiently for it.


Regular Expression Regular Language Closure Property State Automaton Graph Database 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Angles, R., Gutiérrez, C.: Survey of graph database models. ACM Comput. Surv. 40(1) (2008)Google Scholar
  2. 2.
    Barceló, P., Hurtado, C., Libkin, L., Wood, P.: Expressive languages for path queries over graph-structured data. In: PODS 2010, pp. 3–14 (2010)Google Scholar
  3. 3.
    Benedikt, M., Ley, C., Puppis, G.: Automata vs. Logics on Data Words. In: Dawar, A., Veith, H. (eds.) CSL 2010. LNCS, vol. 6247, pp. 110–124. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  4. 4.
    Bojanczyk, M., Parys, P.: XPath evaluation in linear time. In: PODS 2008, pp. 241–250 (2008)Google Scholar
  5. 5.
    Bojanczyk, M., David, C., Muscholl, A., Schwentick, T., Segoufin, L.: Two-variable logic on words with data. ACM TOCL 12(4) (2011)Google Scholar
  6. 6.
    Bojanczyk, M., Lasota, S.: An extension of data automata that captures XPath. In: LICS 2010, pp. 243–252 (2010)Google Scholar
  7. 7.
    Calvanese, D., de Giacomo, G., Lenzerini, M., Vardi, M.Y.: Rewriting of regular expressions and regular path queries. JCSS 64(3), 443–465 (2002)zbMATHGoogle Scholar
  8. 8.
    Colcombet, T., Ley, C., Puppis, G.: On the Use of Guards for Logics with Data. In: Murlak, F., Sankowski, P. (eds.) MFCS 2011. LNCS, vol. 6907, pp. 243–255. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  9. 9.
    Demri, S., Lazic, R.: LTL with the freeze quantifier and register automata. ACM TOCL 10(3) (2009)Google Scholar
  10. 10.
    Figueira, D.: Satisfiability of downward XPath with data equality tests. In: PODS 2009, pp. 197–206 (2009)Google Scholar
  11. 11.
    Glaister, I., Shallit, J.: A lower bound technique for the size of nondeterministic finite automata. IPL 59, 75–77 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Grumberg, O., Kupferman, O., Sheinvald, S.: Variable Automata over Infinite Alphabets. In: Dediu, A.-H., Fernau, H., Martín-Vide, C. (eds.) LATA 2010. LNCS, vol. 6031, pp. 561–572. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  13. 13.
    Kaminski, M., Francez, N.: Finite memory automata. Theoretical Computer Science 134(2), 329–363 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Kaminski, M., Tan, T.: Regular expressions for languages over infinite alphabets. Fundam. Inform. 69(3), 301–318 (2006)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Libkin, L.: Logics for unranked trees: an overview. Logical Methods in Computer Science 2(3) (2006)Google Scholar
  16. 16.
    Libkin, L., Vrgoč, D.: Regular path queries on graphs with data. In: ICDT 2012 (to appear, 2012)Google Scholar
  17. 17.
    Marx, M.: Conditional XPath. ACM TODS 30, 929–959 (2005)CrossRefGoogle Scholar
  18. 18.
    Mendelzon, A.O., Wood, P.T.: Finding regular simple paths in graph databases. SIAM J. Comput. 24(6), 1235–1258 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Neven, F.: Automata theory for XML researchers. SIGMOD Record 31(3), 39–46 (2002)CrossRefGoogle Scholar
  20. 20.
    Neven, F., Schwentick, T., Vianu, V.: Finite state machines for strings over infinite alphabets. ACM TOCL 5(3), 403–435 (2004)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Sakamoto, H., Ikeda, D.: Intractability of decision problems for finite-memory automata. Theor. Comput. Sci. 231(2), 297–308 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Schwentick, T.: Automata for XML – A survey. JCSS 73(3), 289–315 (2007)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Segoufin, L.: Automata and Logics for Words and Trees over an Infinite Alphabet. In: Ésik, Z. (ed.) CSL 2006. LNCS, vol. 4207, pp. 41–57. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  24. 24.
    Sipser, M.: Introduction to the Theory of Computation. PWS Publishing (1997)Google Scholar
  25. 25.
    Tan, T.: Graph reachability and pebble automata over infinite alphabets. In: LICS 2009, pp. 157–166 (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Leonid Libkin
    • 1
  • Domagoj Vrgoč
    • 1
  1. 1.School of InformaticsUniversity of EdinburghUK

Personalised recommendations