Testable and untestable classes of first-order formulae

Abstract

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 ∀ 3 ∃, ∀ 2 ∃∀, ∀∃∀ 2 and ∀∃∀∃ can express untestable graph properties when equality is allowed.

The Kahr-Moore-Wang Class Contains Untestable Properties

“…We suspect that this class remains untestable using dist. These results sharpen some of the results of Jordan and Zeugmann (2012).…”

“…The current paper sharpens some (prefixes ∀ 2 ∃∀, ∀∃∀∃, ∀∃∀ 2 ) of the results of Jordan and Zeugmann (2012) and so we briefly outline the improvement that allows us to minimize the prefix considered. The untestable property considered there is closely related to the untestable property of Alon et al (2000), but modified to minimize the number of quantifiers used.…”

“…These are (restrictions of) well-known classes. Jordan and Zeugmann (2012) extended the positive result to the full Ramsey's class ([∃ * ∀ * , all] = ), proved the testability of Ackermann's class with equality ([∃ * ∀∃ * , all] = ), and sharpened the negative result to prefixes ∀ 3 ∃, ∀ 2 ∃∀, ∀∃∀∃ and ∀∃∀ 2 (on directed graphs with equality). This paper sharpens these last three prefixes and proves that ∀∃∀ is a minimal prefix class for untestability.…”

“…The resulting model makes testing strictly more difficult than using dist, see Jordan and Zeugmann (2012). We refer to properties that are testable using mrdist as mr-testable.…”

“…Jordan and Zeugmann (2012)). We omit the result for [∀∃∀, (0, 1)] because it may depend on the choice of model.…”