Comments on Arithmetic Complexity, Kleene Closure, and Formal Power Series

摘要

In reference to our earlier work [1], Pierre McKenzie and Sambuddha Roy pointed out that the proofs of statements (b) and (c) in Theorem 7.3 are buggy. The main flaw is that the identity e of the group F may not be the identity of the monoid, and so the claim that w ∈ (A_(F,r))~* (↔)w is not an element of $. Test does not work. Herewith, we show: 1. With a slight change to Definition 7.1, the statement of Theorem 7.3 holds unchanged. In our opinion, this is the most interesting way to correct the error in the original paper. We present a complete proof below. For completeness, we also mention another way to correct the error: 2. Leaving Definition 7.1 unchanged, a weaker version of Theorem 7.3 holds (with only minor adjustments to the proof given in the paper).