Toward Better Depth Lower Bounds: A KRW-like theorem for Strong Composition
Or Meir
TL;DR
The paper advances depth lower-bound research by isolating strong composition—KW_f ∘* KW_g—as a knob to study KRW-like behavior independent of the direct-sum obstacle. It develops a comprehensive framework combining a structure theorem for multiplexor-based compositions, a chromatic-number approach to graph-based lower bounds, and prefix-thick set techniques to control information leakage across inputs. The main achievement is a quantitative KRW-like lower bound: CC(KW_f ∘* KW_g) ≥ CC(KW_f) + n − (1 − γ)·m − O(log(mn)) for some γ > 0.04, with formal statements involving log-formula complexities; this constitutes a significant step toward understanding how depth or formula complexity scales under composition and provides methods that could inform future breakthroughs on the weak KRW conjecture. The work also delineates barriers to improving the γ constant, suggesting that new ideas are needed to reach stronger bounds that could impact P vs NC^1.
Abstract
One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., $\mathbf{P}\not\subseteq\mathbf{NC}^{1}$). Karchmer, Raz, and Wigderson (Computational Complexity 5(3/4), 1995) suggested to approach this problem by proving that depth complexity of a composition of functions $f\diamond g$ is roughly the sum of the depth complexities of $f$ and $g$. They showed that the validity of this conjecture would imply that $\mathbf{P}\not\subseteq\mathbf{NC}^{1}$. The intuition that underlies the KRW conjecture is that the composition $f\diamond g$ should behave like a "direct-sum problem", in a certain sense, and therefore the depth complexity of $f\diamond g$ should be the sum of the individual depth complexities. Nevertheless, there are two obstacles toward turning this intuition into a proof: first, we do not know how to prove that $f\diamond g$ must behave like a direct-sum problem; second, we do not know how to prove that the complexity of the latter direct-sum problem is indeed the sum of the individual complexities. In this work, we focus on the second obstacle. To this end, we study a notion called "strong composition", which is the same as $f\diamond g$ except that it is forced to behave like a direct-sum problem. We prove a variant of the KRW conjecture for strong composition, thus overcoming the above second obstacle. This result demonstrates that the first obstacle above is the crucial barrier toward resolving the KRW conjecture. Along the way, we develop some general techniques that might be of independent interest.