Contributions to the Classification for Testability: Four Universal and One Existential Quantifier

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 begun 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 a quantifier pattern ∀∗∃∗. We simplify their untestable example and therefore show that there is an untestable graph property expressible with each of the following quantifier patterns: ∀∃, ∀∃∀, ∀∃∀ and ∀∃∀.