Skip to main content
SpringerLink
Log in

Bounded Variable Logic, Parameterized Logarithmic Space, and Savitch’s Theorem

  • Conference paper
Mathematical Foundations of Computer Science 2014 (MFCS 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8634))

Abstract

We study the parameterized space complexity of model-checking first-order logic with a bounded number of variables. By restricting the number of the quantifier alternations we obtain problems complete for a natural hierarchy between parameterized logarithmic space and FPT. We call this hierarchy the tree hierarchy, provide a machine characterization, and link it to the recently introduced classes PATH and TREE. We show that the lowest class PATH collapses to parameterized logarithmic space only if Savitch’s theorem can be improved. Finally, we settle the complexity with respect to the tree-hierarchy of finding short undirected paths and small undirected trees.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. of Comp. and Syst. Sciences 75(8), 423–434 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  2. Borodin, A., Cook, S.A., Dymond, P.W., Ruzzo, W.L., Tompa, M.: Two applications of inductive counting for complementation problems. SIAM J. on Computing 18(3), 559–578 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cai, L., Chen, J., Downey, R.G., Fellows, M.R.: On the structure of parameterized problems in NP. Information and Computation 123, 38–49 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cai, L., Chen, J., Downey, R.G., Fellows, M.R.: Advice classes of parameterized tractability. Annals of Pure and Applied Logic 84(1), 119–138 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chen, H., Müller, M.: The fine classification of conjunctive queries and parameterized logarithmic space complexity. In: Proc. of PODS 2013, pp. 309–320 (2013); full version arXiv:1306.5424 [cs.CC]

    Google Scholar 

  6. Chen, J., Kneis, J., Lu, S., Mölle, D., Richter, S., Rossmanith, P., Sze, S.-H., Zhang, F.: Randomized divide-and-conquer: Improved path, matching, and packing algorithms. SIAM J. on Computing 38(6), 2526–2547 (2009)

    Article  MATH  Google Scholar 

  7. Chen, Y., Flum, J.: On parameterized path and chordless path problems. In: Proc. of CCC 2007, pp. 250–263 (2007)

    Google Scholar 

  8. Chen, Y., Flum, J.: On optimal inverters. Bull. Symb. Logic (to appear, 2014)

    Google Scholar 

  9. Elberfeld, M., Stockhusen, C., Tantau, T.: On the space complexity of parameterized problems. In: Thilikos, D.M., Woeginger, G.J. (eds.) IPEC 2012. LNCS, vol. 7535, pp. 206–217. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  10. Flum, J., Grohe, M.: Describing parameterized complexity classes. Information and Computation 187(2), 291–319 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer (2006)

    Google Scholar 

  12. M. Frick and M. Grohe. Deciding first-order properties of locally tree-decomposable structures. J. of the ACM, 48(6):1184–1206, 2001.

    Google Scholar 

  13. Grohe, M.: The complexity of homomorphism and constraint satisfaction problems seen from the other side. J. of the ACM 54(1), 1:1–1:24 (2007)

    Google Scholar 

  14. Grohe, M., Kreutzer, S.: Methods for algorithmic meta theorems. In: Model Theoretic Meth. in Finite Combinatorics, Cont. Math., vol. 558, pp. 181–206. AMS (2011)

    Google Scholar 

  15. Grohe, M., Schwentick, T., Segoufin, L.: When is the evaluation of conjunctive queries tractable? In: Proc. of STOC 2001 (2001)

    Google Scholar 

  16. Hemaspaandra, L.A., Ogihara, M., Toda, S.: Space-efficient recognition of sparse self-reducible languages. Computational Complexity 4, 262–296 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  17. Levin, L.: Universal sequential search problems. Problems of Information Transmission 9(3), 265–266 (1973)

    Google Scholar 

  18. Lipton, J.R.: The P = NP Question and Gödel’s Lost Letter. Springer (2010)

    Google Scholar 

  19. Montoya, J.-A., Müller, M.: Parameterized random complexity. Theory of Computing Systems 52(2), 221–270 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  20. Potechin, A.: Bounds on monotone switching networks for directed connectivity. In: Proc. of FOCS 2010, pp. 553–562 (2010)

    Google Scholar 

  21. Reingold, O.: Undirected connectivity in log-space. J. of the ACM 55(4) (2008)

    Google Scholar 

  22. Savitch, W.J.: Relationships between nondeterministic and deterministic tape complexities. J. of Comp. and Syst. Sciences 4(2), 177–192 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  23. Schweikardt, N., Schwentick, T., Segoufin, L.: Database theory: Query languages. In: Algorithms and Theory of Computation, pp. 19.1–19.34. Chapman & Hall/CRC (2010)

    Google Scholar 

  24. Vardi, M.Y.: On the complexity of bounded-variable queries. In: Proc. of PODS 1995, pp. 266–276. ACM Press (1995)

    Google Scholar 

  25. Venkateswaran, H.: Properties that characterize LOGCFL. J. of Comp. and Syst. Sciences 43(2), 380–404 (1991)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Shanghai Jiaotong University, China

    Yijia Chen

  2. Kurt Gödel Research Center, University of Vienna, Austria

    Moritz Müller

Authors
  1. Yijia Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Moritz Müller
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Faculty of Informatics, Eötvös Loránd University, Pázmány Péter sétány 1/C, 1117, Budapest, Hungary

    Erzsébet Csuhaj-Varjú

  2. Fakultät für Informatik und Automatisierung, Technische Universität Ilmenau, Helmholtzplatz, 598693, Ilmenau, Germany

    Martin Dietzfelbinger

  3. Institute of Informatics, Szeged University, Tisza Lajos krt. 103, 6720, Szeged, Hungary

    Zoltán Ésik

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag GmbH Berlin Heidelberg

About this paper

Cite this paper

Chen, Y., Müller, M. (2014). Bounded Variable Logic, Parameterized Logarithmic Space, and Savitch’s Theorem. In: Csuhaj-Varjú, E., Dietzfelbinger, M., Ésik, Z. (eds) Mathematical Foundations of Computer Science 2014. MFCS 2014. Lecture Notes in Computer Science, vol 8634. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44522-8_16

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-44522-8_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-44521-1

  • Online ISBN: 978-3-662-44522-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation