On the structure of graphs in the Caucal hierarchy

Recently Pawel Parys [3] pointed out an error in the article “On the structure of graphs in the Caucal hierarchy” [1], which contains two main results: Theorems 15 and 61. Theorem 61 is a pumping lemma for higher-order pushdown automata. The proof consists in two parts: (1) a series of technical lemmas that, given a run of the automaton containing a so-called pumping pair, constructs a longer run and (2) a proof that every sufficiently long run contains a pumping pair. The error found by Parys is in the proof of (2). He presented a counterexample [2] invalidating the following results of [1]: all the material from Lemma 50 to Corollary 55 in Section 8 and Lemma 60 in Section 9. This counterexample uses an automaton A of level 3 with an unary stack alphabet {a} that after performing the operations