Corrigendum for Arithmetic Complexity, Kleene Closure, and Formal Power Series

Pierre McKenzie and Sambuddha Roy pointed out that the proof 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 ∈ Test does not work. In this corrigendum, we show: • 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: • Leaving Definition 7.1 unchanged, a weaker version of Theorem 7.3 holds (with only minor adjustments to the proof given in the paper). First, we present the modified version of Theorem 7.3 that holds using the original version of Definition 7.1 Theorem 7.3 (Variant) (a) Let A be any finite nonsolvable monoid. Then there exists a group F ⊆ A and a constant r > 0 such that the (A F,r) * closure problem is NC 1-complete. (b) Let A be any finite monoid, and let F be a group contained in A, with the same identity e as the monoid identity. Then the (A F,r) * closure problem is reducible via AC 0-Turing reductions to the word problem over the finite monoid A. (c) If A is a finite solvable monoid and F is a group in it with the same identity as A, then the (A F,r) * closure problem is in ACC 0. Furthermore, if A is an aperiodic monoid then the (A F,r) * closure problem is in AC 0. We now proceed to give a modification to Definition 7.1, with the property that that both Corollary 7.2 and Theorem 7.3 are true, as stated in the original paper. Definition 7.4 (Modified from Definition 7.1 in the paper) Let A be a finite monoid. There is a natural homomorphism v : A * → A that maps a word w to its valuation v(w) in the monoid A. Let F be a group contained in A, let e denote the identity of F , and let r be a positive integer. The language A F,r ⊆ A * is defined as A F,r = {w ∈ A * | |w| ≤ r, v(ew) ∈ F …