On the Quantifier-Free Dynamic Complexity of Reachability

  • Thomas Zeume
  • Thomas Schwentick
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8087)


The dynamic complexity of the reachability query is studied in the dynamic complexity framework of Patnaik and Immerman, restricted to quantifier-free update formulas.

It is shown that, with this restriction, the reachability query cannot be dynamically maintained, neither with binary auxiliary relations nor with unary auxiliary functions, and that ternary auxiliary relations are more powerful with respect to graph queries than binary auxiliary relations. Further results are obtained including more inexpressibility results for reachability in a different setting, inexpressibility results for some other queries and normal forms for quantifier-free update programs.


Normal Form Dynamic Program Dynamic Complexity Relation Symbol Auxiliary Data 
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.
    Patnaik, S., Immerman, N.: Dyn-FO: A parallel, dynamic complexity class. In: PODS, pp. 210–221. ACM Press (1994)Google Scholar
  2. 2.
    Hesse, W.: The dynamic complexity of transitive closure is in DynTC0. In: Van den Bussche, J., Vianu, V. (eds.) ICDT 2001. LNCS, vol. 1973, pp. 234–247. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  3. 3.
    Dong, G., Su, J.: Arity bounds in first-order incremental evaluation and definition of polynomial time database queries. J. Comput. Syst. Sci. 57(3), 289–308 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dong, G., Libkin, L., Wong, L.: Incremental recomputation in local languages. Inf. Comput. 181(2), 88–98 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hesse, W.: Dynamic Computational Complexity. PhD thesis, University of Massachusetts Amherst (2003)Google Scholar
  6. 6.
    Gelade, W., Marquardt, M., Schwentick, T.: The dynamic complexity of formal languages. In: STACS, pp. 481–492 (2009)Google Scholar
  7. 7.
    Grädel, E., Siebertz, S.: Dynamic definability. In: ICDT, 236–248 (2012)Google Scholar
  8. 8.
    Patrascu, M., Demaine, E.D.: Lower bounds for dynamic connectivity. In: Babai, L. (ed.) STOC, pp. 546–553. ACM (2004)Google Scholar
  9. 9.
    Zeume, T., Schwentick, T.: On the quantifier-free dynamic complexity of reachability. CoRR abs/1306.3056 (2013),
  10. 10.
    Weber, V., Schwentick, T.: Dynamic complexity theory revisited. Theory Comput. Syst. 40(4), 355–377 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gelade, W., Marquardt, M., Schwentick, T.: The dynamic complexity of formal languages. ACM Trans. Comput. Log. 13(3), 19 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Etessami, K.: Dynamic tree isomorphism via first-order updates. In: PODS, pp. 235–243. ACM Press (1998)Google Scholar
  13. 13.
    Schmitz, S., Schnoebelen, P.: Multiply-recursive upper bounds with Higman’s lemma. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part II. LNCS, vol. 6756, pp. 441–452. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  14. 14.
    Dong, G., Zhang, L.: Separating auxiliary arity hierarchy of first-order incremental evaluation systems using (3k+1)-ary input relations. Int. J. Found. Comput. Sci. 11(4), 573–578 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Thomas Zeume
    • 1
  • Thomas Schwentick
    • 1
  1. 1.TU Dortmund UniversityGermany

Personalised recommendations