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. [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: ∀∃∀∃, ∀∃∀, ∀∃∀ and ∀∃.