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In this article, we describe an algorithm to determine whether a permutationclass C given by a finite basis B of excluded patterns contains a finite numberof simple permutations. This is a continuation of the work initiated in[Brignall, Ruskuc, Vatter, Simple permutations: decidability and unavoidablesubstructures, 2008], and shares several aspects with it. Like in this article,the main difficulty is to decide whether C contains a finite number of properpin-permutations, and this decision problem is solved using automata theory.Moreover, we use an encoding of proper pin-permutations by words over a finitealphabet, introduced by Brignall et al. However, unlike in their article, ourconstruction of automata is fully algorithmic and efficient. It is based on thestudy of pin-permutations in [Bassino, Bouvel, Rossin, Enumeration ofpin-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, sis the maximal size of a pin-permutation in B and k is the number ofpin-permutations in B.

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