Relational Properties Expressible with One Universal Quantifier Are Testable

Abstract

Abstract. In property testing a small, random sample of an object is taken and one wishes to distinguish with high probability between the case where it has a desired property and the case where it is far from having the property. Much of the recent work has focused on graphs. In the present paper three generalized models for testing relational structures are introduced and relationships between these variations are shown. Furthermore, the logical classification problem for testability is considered and, as the main result, it is shown that Ackermann's class with equality is testable.

Untestable Properties Expressible with Four First-Order Quantifiers

“…Informally, we seek the untestable properties that are easiest to express. Jordan and Zeugmann [10] showed that classes with (at most) one universal quantifier are testable, and so there are at least two minimal untestable classes which have either two or three universal quantifiers.…”

“…In a similar way, the number of loops in a graph is asymptotically insignificant compared to the number of possible non-loops. Modifying the definition of distance to account for this makes testing strictly more difficult (see Jordan and Zeugmann [10]) and so we use the more general definition above.…”

“…Definition 6 is a special case of a definition from Jordan and Zeugmann [10], and mrdist abbreviates "maximum relational distance." We use the following simple variation of indistinguishability for graphs that may contain loops.…”

“…The results in Jordan and Zeugmann [10] show that one universal quantifier is not sufficient to express an untestable property (regardless of the vocabulary), and so it would be interesting to determine the status of the remaining prefixes with two universal quantifiers.…”

A Note on the Testability of Ramsey’s Class

“…In property testing, we are interested in distinguishing between structures that have some property and those that are far from having the property, and so we require a distance measure. Jordan and Zeugmann [9] introduced several possible distances and considered the relationship between the resulting notions of testability. We are proving a positive result, and so it suffices to use only the most restricted variant considered there.…”