Deterministically Counting Satisfying Assignments for Constant-Depth Circuits with Parity Gates, with Implications for Lower Bounds

We give a deterministic algorithm for counting the number of satisfying assignments of any AC^0[oplus] circuit C of size s and depth d over n variables in time 2^(n-f(n,s,d)), where f(n,s,d) = n/O(log(s))^(d-1), whenever s = 2^o(n^(1/d)). As a consequence, we get that for each d, there is a language in E^{NP} that does not have AC^0[oplus] circuits of size 2^o(n^(1/(d+1))). This is the first lower bound in E^{NP} against AC^0[oplus] circuits that beats the lower bound of 2^Omega(n^(1/2(d-1))) due to Razborov and Smolensky for large d. Both our algorithm and our lower bounds extend to AC^0[p] circuits for any prime p

Smaller ACC0 Circuits for Symmetric Functions

What is the power of constant-depth circuits with $MOD_m$ gates, that can count modulo $m$? Can they efficiently compute MAJORITY and other symmetric functions? When $m$ is a constant prime power, the answer is well understood: Razborov and Smolensky proved in the 1980s that MAJORITY and $MOD_m$ require super-polynomial-size $MOD_q$ circuits, where $q$ is any prime power not dividing $m$. However, relatively little is known about the power of $MOD_m$ circuits for non-prime-power $m$. For example, it is still open whether every problem in $EXP$ can be computed by depth-$3$ circuits of polynomial size and only $MOD_6$ gates. We shed some light on the difficulty of proving lower bounds for $MOD_m$ circuits, by giving new upper bounds. We construct $MOD_m$ circuits computing symmetric functions with non-prime power $m$, with size-depth tradeoffs that beat the longstanding lower bounds for $AC^0[m]$ circuits for prime power $m$. Our size-depth tradeoff circuits have essentially optimal dependence on $m$ and $d$ in the exponent, under a natural circuit complexity hypothesis. For example, we show for every $\varepsilon > 0$ that every symmetric function can be computed with depth-3 $MOD_m$ circuits of $\exp(O(n^{\varepsilon}))$ size, for a constant $m$ depending only on $\varepsilon > 0$. That is, depth-$3$ $CC^0$ circuits can compute any symmetric function in \emph{subexponential} size. This demonstrates a significant difference in the power of depth-$3$ $CC^0$ circuits, compared to other models: for certain symmetric functions, depth-$3$ $AC^0$ circuits require $2^{\Omega(\sqrt{n})}$ size [H{\aa}stad 1986], and depth-$3$ $AC^0[p^k]$ circuits (for fixed prime power $p^k$) require $2^{\Omega(n^{1/6})}$ size [Smolensky 1987]. Even for depth-two $MOD_p \circ MOD_m$ circuits, $2^{\Omega(n)}$ lower bounds were known [Barrington Straubing Th\'erien 1990].Comment: 15 pages; abstract edited to fit arXiv requirement

Satisfiability and Derandomization for Small Polynomial Threshold Circuits

A polynomial threshold function (PTF) is defined as the sign of a polynomial p : {0,1}^n ->R. A PTF circuit is a Boolean circuit whose gates are PTFs. We study the problems of exact and (promise) approximate counting for PTF circuits of constant depth. - Satisfiability (#SAT). We give the first zero-error randomized algorithm faster than exhaustive search that counts the number of satisfying assignments of a given constant-depth circuit with a super-linear number of wires whose gates are s-sparse PTFs, for s almost quadratic in the input size of the circuit; here a PTF is called s-sparse if its underlying polynomial has at most s monomials. More specifically, we show that, for any large enough constant c, given a depth-d circuit with (n^{2-1/c})-sparse PTF gates that has at most n^{1+epsilon_d} wires, where epsilon_d depends only on c and d, the number of satisfying assignments of the circuit can be computed in randomized time 2^{n-n^{epsilon_d}} with zero error. This generalizes the result by Chen, Santhanam and Srinivasan (CCC, 2016) who gave a SAT algorithm for constant-depth circuits of super-linear wire complexity with linear threshold function (LTF) gates only. - Quantified derandomization. The quantified derandomization problem, introduced by Goldreich and Wigderson (STOC, 2014), asks to compute the majority value of a given Boolean circuit, under the promise that the minority-value inputs to the circuit are very few. We give a quantified derandomization algorithm for constant-depth PTF circuits with a super-linear number of wires that runs in quasi-polynomial time. More specifically, we show that for any sufficiently large constant c, there is an algorithm that, given a degree-Delta PTF circuit C of depth d with n^{1+1/c^d} wires such that C has at most 2^{n^{1-1/c}} minority-value inputs, runs in quasi-polynomial time exp ((log n)^{O (Delta^2)}) and determines the majority value of C. (We obtain a similar quantified derandomization result for PTF circuits with n^{Delta}-sparse PTF gates.) This extends the recent result of Tell (STOC, 2018) for constant-depth LTF circuits of super-linear wire complexity. - Pseudorandom generators. We show how the classical Nisan-Wigderson (NW) generator (JCSS, 1994) yields a nontrivial pseudorandom generator for PTF circuits (of unrestricted depth) with sub-linearly many gates. As a corollary, we get a PRG for degree-Delta PTFs with the seed length exp (sqrt{Delta * log n})* log^2(1/epsilon)

Luby-Velickovic-Wigderson Revisited: Improved Correlation Bounds and Pseudorandom Generators for Depth-Two Circuits

We study correlation bounds and pseudorandom generators for depth-two circuits that consist of a $\mathsf{SYM}$-gate (computing an arbitrary symmetric function) or $\mathsf{THR}$-gate (computing an arbitrary linear threshold function) that is fed by $S$ $\mathsf{AND}$ gates. Such circuits were considered in early influential work on unconditional derandomization of Luby, Veli\v{c}kovi\'c, and Wigderson [LVW93], who gave the first non-trivial PRG with seed length $2^{O(\sqrt{\log(S/\varepsilon)})}$ that $\varepsilon$-fools these circuits. In this work we obtain the first strict improvement of [LVW93]'s seed length: we construct a PRG that $\varepsilon$-fools size-$S$ $\{\mathsf{SYM},\mathsf{THR}\} \circ\mathsf{AND}$ circuits over $\{0,1\}^n$ with seed length $$2^{O(\sqrt{\log S })} + \mathrm{polylog}(1/\varepsilon),$$ an exponential (and near-optimal) improvement of the $\varepsilon$-dependence of [LVW93]. The above PRG is actually a special case of a more general PRG which we establish for constant-depth circuits containing multiple $\mathsf{SYM}$ or $\mathsf{THR}$ gates, including as a special case $\{\mathsf{SYM},\mathsf{THR}\} \circ \mathsf{AC^0}$ circuits. These more general results strengthen previous results of Viola [Vio06] and essentially strengthen more recent results of Lovett and Srinivasan [LS11]. Our improved PRGs follow from improved correlation bounds, which are transformed into PRGs via the Nisan--Wigderson "hardness versus randomness" paradigm [NW94]. The key to our improved correlation bounds is the use of a recent powerful \emph{multi-switching} lemma due to H{\aa}stad [H{\aa}s14]