A very recent paper by Caussinus, McKenzie, Therien, and Vollmer
[CMTV] shows that ACC^0 is properly contained in ModPH, and TC^0
is properly contained in the counting hierarchy. Thus, [CMTV] shows
that there are problems in ModPH that require superpolynomial-size
uniform ACC^0 ... more >>>

We prove that if every problem in $NP$ has $n^k$-size circuits for a fixed constant $k$, then for every $NP$-verifier and every yes-instance $x$ of length $n$ for that verifier, the verifier's search space has an $n^{O(k^3)}$-size witness circuit: a witness for $x$ that can be encoded with a circuit ... more >>>

We prove that all functions that have low degree torus polynomials approximating them with small error also have $MidBit^+$ circuits computing them. This serves as a partial converse to the result that all $ACC$ functions have low degree torus polynomials approximating them with small error, by Bhrushundi, Hosseini, Lovett and ... more >>>

We give a conversion from non-classical polynomials to $\mathit{MidBit}^+$ circuits and vice-versa. This conversion, along with previously known results, shows that torus polynomials, non-classical polynomials and $\mathit{MidBit}^+$ circuits can all be converted to each other. Therefore lower bounds against any of these models lead to lower bounds against all three ... more >>>

The class $ACC$ consists of Boolean functions that can be computed by constant-depth circuits of polynomial size with $AND, NOT$ and $MOD_m$ gates, where $m$ is a natural number. At the frontier of our understanding lies a widely believed conjecture asserting that $MAJORITY$ does not belong to $ACC$. The Boolean ... more >>>