2021-05-13T03:59:32Zhttps://eprints.lib.hokudai.ac.jp/dspace-oai/requestoai:eprints.lib.hokudai.ac.jp:2115/499382017-10-15T15:00:00Zhdl_2115_20053hdl_2115_145Testable and untestable classes of first-order formulaeJordan, CharlesZeugmann, ThomasProperty testingLogicRandomized algorithmsAckermann's classRamsey's class007In 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 there exists*for all* are testable but that there is an untestable graph property expressible with quantifier prefix for all*there exists*. The main results of the present paper are as follows. We prove that all (relational) properties expressible with quantifier prefix there exists*for all there exists* (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 for all(3)there exists, for all(2)there exists for all, for all there exists for all(2) and for all there exists V there exists can express untestable graph properties when equality is allowed.ElsevierJournal Articleapplication/pdfhttp://hdl.handle.net/2115/49938https://eprints.lib.hokudai.ac.jp/dspace/bitstream/2115/49938/1/JCSS78-5_1557-1578.pdf0022-0000Journal of Computer and System Sciences785155715782012-09enginfo:doi/10.1016/j.jcss.2012.01.007author