Every hereditary permutation property is testable

In 2010 Moreira and three of the current authors introduced a notion of property testing for permutations based on a distance that measures the randomness of a permutation. They proved that every hereditary permutation property is testable in this sense, or weakly testable in their terminology. In this paper, we show that hereditary permutation properties are testable in a stronger sense, namely when testability is defined by means of the usual ℓ1-distance for permutations. This is the permutation analogue of the well-known Alon-Shapira result on the testability of hereditary graph properties.