Untestable Properties Expressible with Four First-Order Quantifiers
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. , 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 ∃.
Keywordsproperty testing logic
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