Testable and Untestable Classes of First-Order Logic

Property testing is essentially a kind of constant-time randomized approximation. Alon et al. [3] were the first to consider the idea of testing properties expressible in syntactic subclasses of first-order logic. They proved the testability of all properties of undirected, loop-free graphs expressible with quantifier prefix ∃∗∀∗, and also that there exist untestable properties of undirected, loop-free graphs expressible with quantifier prefix ∀∗∃∗. In this dissertation, we continue the study of testing subclasses of first-order logic. In particular, we focus on the classification of prefix-vocabulary classes, or classes defined by quantifier prefix and vocabulary, according to their testability. The main results are as follows. First, we develop a framework for relational property testing including variations corresponding to the different models considered in the literature for non-uniform hypergraph testing. We then use this framework to prove the following. 1. All (relational) properties expressible by formulae in Ackermann’s class with equality ([∃∗∀∃∗, all ]=) are testable in all of our models. 2. All (relational) properties expressible in Ramsey’s class ([∃∗∀∗, all ]=) are testable in all of our models. This extends the result by Alon et al. [3] to the full class. i