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

Summary: Corrigendum for
Arithmetic Complexity, Kleene Closure, and Formal Power Series
Eric Allender, V Arvind, Meena Mahajan
Theory of Computing Systems 36(4): 303328, 2003.
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 (AF,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 (AF,r)

  

Source: Allender, Eric - Department of Computer Science, Rutgers University
Mahajan, Meena - Institute of Mathematical Sciences, Chennai, India

 

Collections: Computer Technologies and Information Sciences