LATA 2010: Language and Automata Theory and Applications pp 333-343 | Cite as
Untestable Properties Expressible with Four First-Order Quantifiers
Abstract
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 ∃.
Keywords
property testing logicPreview
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References
- 1.Alon, N., Fischer, E., Krivelevich, M., Szegedy, M.: Efficient testing of large graphs. Combinatorica 20(4), 451–476 (2000)MATHCrossRefMathSciNetGoogle Scholar
- 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)MATHCrossRefMathSciNetGoogle Scholar
- 3.Alon, N., Shapira, A.: A separation theorem in property testing. Combinatorica 28(3), 261–281 (2008)MATHCrossRefMathSciNetGoogle Scholar
- 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)MATHCrossRefMathSciNetGoogle Scholar
- 5.Börger, E., Grädel, E., Gurevich, Y.: The Classical Decision Problem. Springer, Heidelberg (1997)MATHGoogle Scholar
- 6.Diestel, R.: Graph Theory, 3rd edn. Springer, Heidelberg (2006)Google Scholar
- 7.Enderton, H.B.: A Mathematical Introduction to Logic, 2nd edn. Academic Press, London (2000)Google Scholar
- 8.Goldreich, O., Goldwasser, S., Ron, D.: Property testing and its connection to learning and approximation. J. ACM 45(4), 653–750 (1998)MATHCrossRefMathSciNetGoogle Scholar
- 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.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.Rubinfeld, R., Sudan, M.: Robust characterizations of polynomials with applications to program testing. SIAM J. Comput. 25(2), 252–271 (1996)MATHCrossRefMathSciNetGoogle Scholar