# Proof of $DLOGTIME-CC^0 = MOD[<,bit]$

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)