Untestable Properties Expressible with Four First-Order Quantifiers

  • Charles Jordan
  • Thomas Zeugmann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6031)


In property testing, the goal is to distinguish between structures that have some desired property and those that are far from having the property, after examining only a small, random sample of the structure. We focus on the classification of first-order sentences based on their quantifier prefixes and vocabulary into testable and untestable classes. This classification was initiated by Alon et al. [1], who showed that graph properties expressible with quantifier patterns ∃ * ∀ * are testable but that there is an untestable graph property expressible with quantifier pattern ∀ * ∃ *. In the present paper, their untestable example is simplified. In particular, it is shown that there is an untestable graph property expressible with each of the following quantifier patterns: ∀ ∃ ∀ ∃, ∀ ∃ ∀ 2, ∀ 2 ∃ ∀ and ∀ 3 ∃.


property testing logic 


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  1. 1.
    Alon, N., Fischer, E., Krivelevich, M., Szegedy, M.: Efficient testing of large graphs. Combinatorica 20(4), 451–476 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alon, N., Shapira, A.: A characterization of the (natural) graph properties testable with one-sided error. SIAM J. Comput. 37(6), 1703–1727 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alon, N., Shapira, A.: A separation theorem in property testing. Combinatorica 28(3), 261–281 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Blum, M., Luby, M., Rubinfeld, R.: Self-testing/correcting with applications to numerical problems. J. of Comput. Syst. Sci. 47(3), 549–595 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Börger, E., Grädel, E., Gurevich, Y.: The Classical Decision Problem. Springer, Heidelberg (1997)zbMATHGoogle Scholar
  6. 6.
    Diestel, R.: Graph Theory, 3rd edn. Springer, Heidelberg (2006)Google Scholar
  7. 7.
    Enderton, H.B.: A Mathematical Introduction to Logic, 2nd edn. Academic Press, London (2000)Google Scholar
  8. 8.
    Goldreich, O., Goldwasser, S., Ron, D.: Property testing and its connection to learning and approximation. J. ACM 45(4), 653–750 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Jordan, C., Zeugmann, T.: Contributions to the classification for testability: Four universal and one existential quantifier. Technical Report TCS-TR-A-09-39, Hokkaido University, Division of Computer Science (November 2009)Google Scholar
  10. 10.
    Jordan, C., Zeugmann, T.: Relational properties expressible with one universal quantifier are testable. In: Watanabe, O., Zeugmann, T. (eds.) SAGA 2009. LNCS, vol. 5792, pp. 141–155. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  11. 11.
    Rubinfeld, R., Sudan, M.: Robust characterizations of polynomials with applications to program testing. SIAM J. Comput. 25(2), 252–271 (1996)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Charles Jordan
    • 1
  • Thomas Zeugmann
    • 1
  1. 1.Division of Computer ScienceHokkaido University, N-14, W-9SapporoJapan

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