Slicewise Definability in First-Order Logic with Bounded Quantifier Rank

Authors Yijia Chen, Jörg Flum, Xuangui Huang

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Yijia Chen
Jörg Flum
Xuangui Huang

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Yijia Chen, Jörg Flum, and Xuangui Huang. Slicewise Definability in First-Order Logic with Bounded Quantifier Rank. In 26th EACSL Annual Conference on Computer Science Logic (CSL 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 82, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


For every natural number q let FO_q denote the class of sentences of first-order logic FO of quantifier rank at most q. If a graph property can be defined in FO_q, then it can be decided in time O(n^q). Thus, minimizing q has favorable algorithmic consequences. Many graph properties amount to the existence of a certain set of vertices of size k. Usually this can only be expressed by a sentence of quantifier rank at least k. We use the color coding method to demonstrate that some (hyper)graph problems can be defined in FO_q where q is independent of k. This property of a graph problem is equivalent to the question of whether the corresponding parameterized problem is in the class para-AC^0. It is crucial for our results that the FO-sentences have access to built-in addition and multiplication (and constants for an initial segment of natural numbers whose length depends only on k). It is known that then FO corresponds to the circuit complexity class uniform AC^0. We explore the connection between the quantifier rank of FO-sentences and the depth of AC^0-circuits, and prove that FO_q is strictly contained in FO_{q+1} for structures with built-in addition and multiplication.
  • first-order logic
  • quantifier rank
  • parameterized AC^0
  • circuit depth


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  1. N. Alon, R. Yuster, and U. Zwick. Color-coding. Journal of the ACM, 42(4):844-856, 1995. Google Scholar
  2. A. Backurs and P. Indyk. Edit distance cannot be computed in strongly subquadratic time (unless SETH is false). In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 51-58, 2015. Google Scholar
  3. M. Bannach, C. Stockhusen, and T. Tantau. Fast parallel fixed-parameter algorithms via color coding. In Proceedings of the 10th International Symposium on Parameterized and Exact Computation, IPEC 2015, September 16-18, 2015, Patras, Greece, pages 224-235, 2015. Google Scholar
  4. D. A. Mix Barrington, N. Immerman, and H. Straubing. On uniformity within NC¹. Journal of Computer and System Sciences, 41(3):274-306, 1990. Google Scholar
  5. R. B. Boppana and M. Sipser. The complexity of finite functions. In Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity (A), pages 757-804. MIT Press, 1990. Google Scholar
  6. Y. Chen and J. Flum. Some lower bounds in parameterized AC⁰. In Proceedings of the 41st International Symposium on Mathematical Foundations of Computer Science, MFCS 2016, August 22-26, 2016 - Kraków, Poland, pages 27:1-27:14, 2016. Google Scholar
  7. M. Elberfeld, C. Stockhusen, and T. Tantau. On the space and circuit complexity of parameterized problems: Classes and completeness. Algorithmica, 71(3):661-701, 2015. Google Scholar
  8. J. Flum and M. Grohe. Fixed-parameter tractability, definability, and model-checking. SIAM Journal on Computing, 31(1):113-145, 2001. Google Scholar
  9. J. Flum and M. Grohe. Describing parameterized complexity classes. Information and Computation, 187(2):291-319, 2003. Google Scholar
  10. J. Flum and M. Grohe. Parameterized Complexity Theory. Springer, 2006. Google Scholar
  11. J. Håstad. Almost optimal lower bounds for small depth circuits. In Randomness and Computation, pages 6-20. JAI Press, 1989. Google Scholar
  12. N. Immerman. Descriptive Complexity. Graduate Texts in Computer Science. Springer, 1999. Google Scholar
  13. R. Niedermeier and P. Rossmanith. An efficient fixed-parameter algorithm for 3-hitting set. Journal of Discrete Algorithms, 1(1):89-102, 2003. Google Scholar
  14. B. Rossman. On the constant-depth complexity of k-clique. In Proceedings of the 40th Annual ACM Symposium on Theory of Computing, STOC 2008, Victoria, British Columbia, Canada, pages 721-730, 2008. Google Scholar
  15. N. Segerlind, S. R. Buss, and R. Impagliazzo. A switching lemma for small restrictions and lower bounds for k-DNF resolution. SIAM Journal on Computing, 33(5):1171-1200, 2004. Google Scholar
  16. M. Sipser. Borel sets and circuit complexity. In Proceedings of the 15th Annual ACM Symposium on Theory of Computing, 25-27 April, 1983, Boston, Massachusetts, USA, pages 61-69, 1983. Google Scholar
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