Skip to main content

Symbolic Visibly Pushdown Automata

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 8559)


Nested words model data with both linear and hierarchical structure such as XML documents and program traces. A nested word is a sequence of positions together with a matching relation that connects open tags (calls) with the corresponding close tags (returns). Visibly Pushdown Automata are a restricted class of pushdown automata that process nested words, and have many appealing theoretical properties such as closure under Boolean operations and decidable equivalence. However, like any classical automata models, they are limited to finite alphabets. This limitation is restrictive for practical applications to both XML processing and program trace analysis, where values for individual symbols are usually drawn from an unbounded domain. With this motivation, we introduce Symbolic Visibly Pushdown Automata (SVPA) as an executable model for nested words over infinite alphabets. In this model, transitions are labeled with predicates over the input alphabet, analogous to symbolic automata processing strings over infinite alphabets. A key novelty of SVPAs is the use of binary predicates to model relations between open and close tags in a nested word. We show how SVPAs still enjoy the decidability and closure properties of Visibly Pushdown Automata. We use SVPAs to model XML validation policies and program properties that are not naturally expressible with previous formalisms and provide experimental results for our implementation.


  • visibly pushdown automata
  • symbolic automata
  • XML

This research was supported by NSF Expeditions in Computing award CCF 1138996.


  1. Alur, R., Madhusudan, P.: Visibly pushdown languages. In: Proceedings of the 36th ACM Symposium on Theory of Computing, pp. 202–211 (2004)

    Google Scholar 

  2. Alur, R., Madhusudan, P.: Adding nesting structure to words. Journal of the ACM 56(3) (2009)

    Google Scholar 

  3. Bès, A.: An application of the Feferman-Vaught theorem to automata and logics for words over an infinite alphabet. CoRR, abs/0801.2498 (2008)

    Google Scholar 

  4. Bojanczyk, M., David, C., Muscholl, A., Schwentick, T., Segoufin, L.: Two-variable logic on data words. ACM Trans. Comput. Log. 12(4), 27 (2011)

    CrossRef  MathSciNet  Google Scholar 

  5. Chaudhuri, S., Alur, R.: Instrumenting C programs with nested word monitors. In: Bošnački, D., Edelkamp, S. (eds.) SPIN 2007. LNCS, vol. 4595, pp. 279–283. Springer, Heidelberg (2007)

    CrossRef  Google Scholar 

  6. Cheng, E.Y., Kaminski, M.: Context-free languages over infinite alphabets. Acta Informatica 35(3), 245–267 (1998)

    CrossRef  MATH  MathSciNet  Google Scholar 

  7. D’Antoni, L., Veanes, M.: Equivalence of extended symbolic finite transducers. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 624–639. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

  8. D’Antoni, L., Veanes, M.: Static analysis of string encoders and decoders. In: Giacobazzi, R., Berdine, J., Mastroeni, I. (eds.) VMCAI 2013. LNCS, vol. 7737, pp. 209–228. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

  9. D’Antoni, L., Veanes, M.: Minimization of Symbolic Automata. In: Proceedings of the 41st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (2014)

    Google Scholar 

  10. de Moura, L., Bjørner, N.S.: Z3: An efficient SMT solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  11. Debarbieux, D., Gauwin, O., Niehren, J., Sebastian, T., Zergaoui, M.: Early nested word automata for xPath query answering on XML streams. In: Konstantinidis, S. (ed.) CIAA 2013. LNCS, vol. 7982, pp. 292–305. Springer, Heidelberg (2013)

    Google Scholar 

  12. Driscoll, E., Thakur, A., Reps, T.: Opennwa: A nested-word automaton library. In: Madhusudan, P., Seshia, S.A. (eds.) CAV 2012. LNCS, vol. 7358, pp. 665–671. Springer, Heidelberg (2012)

    CrossRef  Google Scholar 

  13. Gauwin, O., Niehren, J.: Streamable fragments of forward xPath. In: Bouchou-Markhoff, B., Caron, P., Champarnaud, J.-M., Maurel, D. (eds.) CIAA 2011. LNCS, vol. 6807, pp. 3–15. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  14. Kaminski, M., Tan, T.: Tree automata over infinite alphabets. In: Avron, A., Dershowitz, N., Rabinovich, A. (eds.) Pillars of Computer Science. LNCS, vol. 4800, pp. 386–423. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  15. Kieżun, A., Ganesh, V., Guo, P.J., Hooimeijer, P., Ernst, M.D.: HAMPI: A solver for string constraints. In: Proceedings of the 2009 International Symposium on Software Testing and Analysis, ISSTA 2009, Chicago, IL, USA, July 21-23 (2009)

    Google Scholar 

  16. Kumar, V., Madhusudan, P., Viswanathan, M.: Visibly pushdown automata for streaming XML. In: WWW 2007: Proceedings of the 16th International Conference on World Wide Web, pp. 1053–1062. ACM, New York (2007)

    Google Scholar 

  17. Martens, W., Neven, F., Schwentick, T.: Complexity of decision problems for XML schemas and chain regular expressions. SIAM J. Comput. 39(4), 1486–1530 (2009)

    CrossRef  MATH  MathSciNet  Google Scholar 

  18. Mozafari, B., Zeng, K., D’Antoni, L., Zaniolo, C.: High-Performance Complex Event Processing over Hierarchical Data. In: ACM TODS’s Special Issue on Best of SIGMOD (2013)

    Google Scholar 

  19. Veanes, M.: Applications of symbolic finite automata. In: Konstantinidis, S. (ed.) CIAA 2013. LNCS, vol. 7982, pp. 16–23. Springer, Heidelberg (2013)

    Google Scholar 

  20. Veanes, M., Bjørner, N.: Symbolic automata: The toolkit. In: Flanagan, C., König, B. (eds.) TACAS 2012. LNCS, vol. 7214, pp. 472–477. Springer, Heidelberg (2012)

    CrossRef  Google Scholar 

  21. Veanes, M., Hooimeijer, P., Livshits, B., Molnar, D., Bjorner, N.: Symbolic finite state transducers: Algorithms and applications. In: Proceedings of the Symposium on Principles of Programming Languages (POPL) (January 2012)

    Google Scholar 

  22. Zilio, S.D., Lugiez, D.: XML schema, tree logic and sheaves automata. In: Nieuwenhuis, R. (ed.) RTA 2003. LNCS, vol. 2706, pp. 246–263. Springer, Heidelberg (2003)

    CrossRef  Google Scholar 

Download references

Author information

Authors and Affiliations


Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

D’Antoni, L., Alur, R. (2014). Symbolic Visibly Pushdown Automata. In: Biere, A., Bloem, R. (eds) Computer Aided Verification. CAV 2014. Lecture Notes in Computer Science, vol 8559. Springer, Cham.

Download citation

  • DOI:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-08866-2

  • Online ISBN: 978-3-319-08867-9

  • eBook Packages: Computer ScienceComputer Science (R0)