The Kahr-Moore-Wang Class Contains Untestable Properties.

Property testing is a kind of randomized approximation in which one takes a small, random sample of a structure and wishes to determine whether the structure satisfies some property or is far from satisfying the property. We focus on the testability of classes of first-order expressible properties, and in particular, on the classification of prefix-vocabulary classes for testability. The main result is the untestability of [for all there exists for all, (0, 1)](=). We also show that this class remains untestable without equality in at least one model of testing. These classes are well-known and (at least one is) minimal for untestability. We discuss what is currently known about the classification for testability and briefly compare it to other classifications.