Paul Valiant’s Canonical Tester is a one-stop-shop solution for testing any symmetric property of a distribution. If the Canonical Tester can’t do it, neither can you. The additional upshot is that the Canonical Tester is quite simple an algorithm, so proving lower bounds (impossibility results) usually takes short 5-line arguments explaining why the Canonical Tester can’t do it.

Testing asymmetric properties is now the next interesting question. Interesting it is indeed because most practical questions actually tend to involve asymmetric properties. A modified example of Paul’s talk goes like this. Imagine Amazon makes a change on their website, and they like to know whether more books compared to electronics were sold after the change. This is an asymmetric test.