Testable and untestable classes of first-order formulae

In property testing, the goal is to distinguish structures that have some desired property from those that are far from having the property, based on only a small, random sample of the structure. We focus on the classification of first-order sentences according to their testability. This classification was initiated by Alon et al. [2], who showed that graph properties expressible with prefix ∃∗∀∗ are testable but that there is an untestable graph property expressible with quantifier prefix ∀∗∃∗. The main results of the present paper are as follows. We prove that all (relational) properties expressible with quantifier prefix ∃∗∀∃∗ (Ackermann’s class with equality) are testable and also extend the positive result of Alon et al. [2] to relational structures using a recent result by Austin and Tao [8]. Finally, we simplify the untestable property of Alon et al. [2] and show that prefixes ∀∃, ∀∃∀, ∀∃∀ and ∀∃∀∃ can express untestable graph properties when equality is allowed.