Theory of Computing Systems

, Volume 53, Issue 3, pp 503–506

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

Commentary

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 $$w\in (A_{F,r})^{*}\Longleftrightarrow w \not\in\mbox{Test}$$ does not work.

Herewith, 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)

1. (a)

LetAbe any finite nonsolvable monoid. Then there exists a groupFAand a constantr>0 such that the (AF,r)closure problem is$$\mbox {\mathrm {NC}^{1}}$$-complete.

2. (b)

LetAbe any finite monoid, and letFbe a group contained inA, with the same identityeas the monoid identity. Then the (AF,r)closure problem is reducible via AC0-Turing reductions to the word problem over the finite monoidA.

3. (c)

IfAis a finite solvable monoid andFis a group in it with the same identity asA, then the (AF,r)closure problem is in$$\mbox {\mathrm {ACC}^{0}}$$. Furthermore, ifAis an aperiodic monoid then the (AF,r)closure problem is in$$\mbox {\mathrm {AC}^{0}}$$.

We now proceed to give a modification to Definition 7.1, with the property 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:AA 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 AF,rA is defined as AF,r={wA∣|w|≤r,v(ew)∈F}.

The original definition required that v(w) be a group element; instead, we now require v(ew) to be a group element.

The (AF,r)closure problem is the decision problem (AF,r). Since AF,r is finite, (AF,r) is a regular language, and thus the (AF,r) closure problem is always in $$\mbox {\mathrm {NC}^{1}}$$.

With the revised definition, Corollary 7.2 still holds (with the same proof), because the monoid A=S5 is itself a group, and F is a subgroup.

We now state and prove Theorem 7.3 (using the revised definition of AF,r).

### Theorem 7.3

1. (a)

LetAbe any nonsolvable monoid. Then there exists a groupFAand a constantr>0 such that the (AF,r)closure problem is$$\mbox {\mathrm {NC}^{1}}$$-complete.

2. (b)

The (AF,r)closure problem is reducible via AC0-Turing reductions to the word problem over the finite monoidA.

3. (c)

IfAis a solvable monoid then the (AF,r)closure problem is in$$\mbox {\mathrm {ACC}^{0}}$$. Furthermore, ifAis an aperiodic monoid then the (AF,r)closure problem is in$$\mbox {\mathrm {AC}^{0}}$$.

### Proof

(a) Since A is a nonsolvable monoid, A contains a nontrivial nonsolvable group G with identity e.1 Since the word problem over G is $$\mbox {\mathrm {NC}^{1}}$$-complete [2], it suffices to show an $$\mbox {\mathrm {AC}^{0}}$$ reduction from the word problem over G to an appropriate $$A_{F,r}^{*}$$ closure problem. To be precise, the word problem we consider is
$$W:=\{w\in G^*\mid v(w)=e\}$$
Let G={g1,g2,…,gm}. Consider the word $$u=\prod_{1\leq i\leq m}g_{i}^{-1}g_{i}$$ in A. Let w=w1w2wn be an instance of W. We map the instance w to the word z=(∏1≤in−1wiu)wn. Notice that v(z)=v(w). Furthermore, it is not hard to see that by virtue of inserting the word u between wi and wi+1 for 1≤in−1 we have ensured that the word z can be decomposed into z=α1α2αn, where for 1≤in−1 we have |αi|<4m, wi is included in αi, and v(αi)=e. Since v(z)=v(w), it follows that wW iff z can be decomposed as α1α2αn, where each αi is of length at most 4m−1 and v(αi)=e for all i. Clearly, v(i)=v(e)v(αi)=e as well.

### Note

The last sentence above is the only new thing in the proof of part (a).

Letting F={e} and r=4m−1 the above argument shows that wz is an $$\mbox {\mathrm {AC}^{0}}$$ reduction from the $$\mbox {\mathrm {NC}^{1}}$$-complete word problem W to the (AF,r) closure problem.

(b) We devise a test that characterizes membership in (AF,r), using the following claim.

### Claim 7.4

Letx,ybe words inA, and supposev(ex)∈F. Then
$$v(ey) \in F \quad \Longleftrightarrow\quad v(exy)\in F$$

### Proof

For any w=w1w2wn with each wiA, and for 0≤i<jn, let w[i,j] denote the subword wi+1wj. We construct a circuit for (AF,r) that uses oracle gates for the following word problem W over the monoid A:
$$W:=\{w\in A^*\mid v(ew)\in F\}$$
The circuit will have an oracle gate for w[0,j] for each 1≤jn. Let the output of the oracle gate be the bit bj; thus
$$\mbox{~For~} 1 \le j \le n,\quad b_j = \begin{cases} 1 & \mbox{if v(ew[0,j]) \in F}\\ 0 & \mbox{otherwise} \end{cases}$$
We set b0=1. Now we place circuitry to check that
1. (a)

bn=1, and

2. (b)

the string b=b0b1bn does not have r consecutive zeroes.

It is clear that these checks can be performed in AC0. To see why these checks characterize membership in (AF,r), note that:

If w∈(AF,r), then we can decompose w into short strings w=x1x2xm such that each xi has length at most r and each v(exi) is in F. By the claim above, v(ey)∈F for each prefix y of the form x1x2xj. Thus at each such position, the string b will have a 1, and these positions are at most r positions apart.

If the 1s in b are never separated by r or more zeroes, then there is a sequence 0=l0<l1<l2<⋯<lm=n such that for each j, ljlj−1r, and v(ew[0,lj])∈F. By the above claim, each v(ew[lj−1,lj]) is also in F. This gives the required decomposition witnessing w∈(AF,r).

This completes the proof of part (b).

(c) This is an immediate consequence of part (b) and the results of [2, 3]. □

## Footnotes

1. 1.

Notice that e could be different from the monoid identity.

## Notes

### Acknowledgements

The discussion about the flaw and the possible work-arounds took place while Eric Allender, Pierre McKenzie and Meena Mahajan were at Dagstuhl seminar 11121 (March 2011).

### References

1. 1.
Allender, E., Arvind, V., Mahajan, M.: Arithmetic complexity, Kleene closure, and formal power series. Theory Comput. Syst. 36, 303–328 (2003)
2. 2.
Barrington, D.A.: Bounded-width polynomial size branching programs recognize exactly those languages in NC1. J. Comput. Syst. Sci. 38, 150–164 (1989)
3. 3.
Barrington, D.A., Thérien, D.: Finite monoids and the fine structure of NC1. J. ACM 35, 941–952 (1988)