Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Patnaik, S., Immerman, N.: Dyn-FO: A parallel, dynamic complexity class. In: PODS, pp. 210–221. ACM Press (1994)
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)
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)
Dong, G., Libkin, L., Wong, L.: Incremental recomputation in local languages. Inf. Comput. 181(2), 88–98 (2003)
Hesse, W.: Dynamic Computational Complexity. PhD thesis, University of Massachusetts Amherst (2003)
Gelade, W., Marquardt, M., Schwentick, T.: The dynamic complexity of formal languages. In: STACS, pp. 481–492 (2009)
Grädel, E., Siebertz, S.: Dynamic definability. In: ICDT, 236–248 (2012)
Patrascu, M., Demaine, E.D.: Lower bounds for dynamic connectivity. In: Babai, L. (ed.) STOC, pp. 546–553. ACM (2004)
Zeume, T., Schwentick, T.: On the quantifier-free dynamic complexity of reachability. CoRR abs/1306.3056 (2013), http://arxiv.org/abs/1306.3056
Weber, V., Schwentick, T.: Dynamic complexity theory revisited. Theory Comput. Syst. 40(4), 355–377 (2007)
Gelade, W., Marquardt, M., Schwentick, T.: The dynamic complexity of formal languages. ACM Trans. Comput. Log. 13(3), 19 (2012)
Etessami, K.: Dynamic tree isomorphism via first-order updates. In: PODS, pp. 235–243. ACM Press (1998)
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)
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Zeume, T., Schwentick, T. (2013). On the Quantifier-Free Dynamic Complexity of Reachability. In: Chatterjee, K., Sgall, J. (eds) Mathematical Foundations of Computer Science 2013. MFCS 2013. Lecture Notes in Computer Science, vol 8087. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40313-2_73
Download citation
DOI: https://doi.org/10.1007/978-3-642-40313-2_73
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-40312-5
Online ISBN: 978-3-642-40313-2
eBook Packages: Computer ScienceComputer Science (R0)