Constant-depth circuits vs. monotone circuits

Abstract

We establish new separations between the power of monotone and general (non-monotone) Boolean circuits: - For every $k \geq 1$, there is a monotone function in ${\sf AC^0}$ that requires monotone circuits of depth $Ω(\log^k n)$. This significantly extends a classical result of Okol'nishnikova (1982) and Ajtai and Gurevich (1987). In addition, our separation holds for a monotone graph property, which was unknown even in the context of ${\sf AC^0}$ versus ${\sf mAC^0}$. - For every $k \geq 1$, there is a monotone function in ${\sf AC^0}[\oplus]$ that requires monotone circuits of size $\exp(Ω(\log^k n))$. This makes progress towards a question posed by Grigni and Sipser (1992). These results show that constant-depth circuits can be more efficient than monotone circuits when computing monotone functions. In the opposite direction, we observe that non-trivial simulations are possible in the absence of parity gates: every monotone function computed by an ${\sf AC^0}$ circuit of size $s$ and depth $d$ can be computed by a monotone circuit of size $2^{n - n/O(\log s)^{d-1}}$. We show that the existence of significantly faster monotone simulations would lead to breakthrough circuit lower bounds. In particular, if every monotone function in ${\sf AC^0}$ admits a polynomial size monotone circuit, then ${\sf NC^2}$ is not contained in ${\sf NC^1}$ . Finally, we revisit our separation result against monotone circuit size and investigate the limits of our approach, which is based on a monotone lower bound for constraint satisfaction problems established by Göös et al. (2019) via lifting techniques. Adapting results of Schaefer (1978) and Allender et al. (2009), we obtain an unconditional classification of the monotone circuit complexity of Boolean-valued CSPs via their polymorphisms. This result and the consequences we derive from it might be of independent interest.