9
$\begingroup$

Let $CC^0[m]$ be the class of constant-depth, polynomial-sized circuits consisting entirely of $MOD_m$ gates, which put out $1$ iff the sum of their inputs $\equiv 0~(\textrm{mod}~m)$. In the same way that $DLOGTIME$-uniform $AC^0$ is equivalent to $FO[<,bit] = FO[+,*]$, in multiple sources 1, 2 it's asserted that $DLOGTIME$-uniform $CC^0$ is equivalent to $MOD[+,*]$, the set of logical sentences using only the modular quantifier $\exists^{(m,k)}$ that evaluates to true iff the number of satisfying solutions is congruent to $k$ mod $m$.

However, all of the sources that are cited for these claims 3, [4], [5] only provide proofs for the classes $AC^0, ACC^0, TC^0, NC^1$ and rely on the usage of standard quantifiers $\forall, \exists$, which are not proven to be expressible in $CC^0$. Does anyone know of how to modify the original proofs of $DLOGTIME-AC^0 = FO[<,bit] = FO[+,*]$ to achieve $DLOGTIME-CC^0 = MOD[<,bit] = MOD[+,*]$ without using these quantifiers, or a source for such proofs?

[4]: Introduction to Circuit Complexity: A Uniform Approach by Heribert Vollmer

[5]: Descriptive Complexity by Neil Immerman

(These proofs mostly just follow the one laid out in 3)

$\endgroup$
5
$\begingroup$

I have reached out to the authors of papers [1] and [2], and it seems that the claims were both in error, leaving this an open question.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.