An algorithm for deciding the finiteness of the number of simple permutations in permutation classes

In this article, we describe an algorithm to determine whether a permutation class C given by a finite basis B of excluded patterns contains a finite number of simple permutations. This is a continuation of the work initiated in [Brignall, Ruskuc, Vatter, Simple permutations: decidability and unavoidable substructures, 2008], and shares several aspects with it. Like in this article, the main difficulty is to decide whether C contains a finite number of proper pin-permutations, and this decision problem is solved using automata theory. Moreover, we use an encoding of proper pin-permutations by words over a finite alphabet, introduced by Brignall et al. However, unlike in their article, our construction of automata is fully algorithmic and efficient. It is based on the study of pin-permutations in [Bassino, Bouvel, Rossin, Enumeration of pin-permutations, 2011]. The complexity of the overall algorithm is O(n log n + s^{2k}) where n denotes the sum of the sizes of permutations in the basis B, s is the maximal size of a pin-permutation in B and k is the number of pin-permutations in B.