🤖 AI Summary
Establishing super-logarithmic depth lower bounds for Boolean circuits remains a central open problem in complexity theory; progress hinges on resolving the Karchmer–Raz–Wigderson (KRW) conjecture.
Method: We advance its proof by (i) proving the monotone KRW conjecture for all monotone inner functions satisfying query-to-communication lifting theorems, and (ii) introducing the “semi-monotone composition” model, which decouples the complexity analysis of monotone inner and arbitrary outer functions. Our approach integrates query-to-communication lifting, characterizations of monotone circuit complexity, and the communication-to-circuit-depth correspondence.
Contribution/Results: We verify the KRW conjecture for fundamental monotone functions—including st-connectivity, clique detection, and generation functions. Moreover, within the semi-monotone framework, we extend the non-monotone KRW proof to significantly broader classes of inner functions, substantially widening the applicability of the KRW paradigm.
📝 Abstract
One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., $mathrm{P}
subseteq ext{NC}^{1}$). Karchmer, Raz, and Wigderson [13] suggested to approach this problem by proving that depth complexity behaves “as expected” with respect to the composition of functions $fdiamond g$. They showed that the validity of this conjecture would imply that $mathrm{P}
subseteq ext{NC}^{1}$. Several works have made progress toward resolving this conjecture by proving special cases. In particular, these works proved the KRW conjecture for every outer function, but only for few inner functions. Thus, it is an important challenge to prove the KRW conjecture for a wider range of inner functions. In this work, we extend significantly the range of inner functions that can be handled. First, we consider the monotone version of the KRW conjecture. We prove it for every monotone inner function whose depth complexity can be lower bounded via a query-to-communication lifting theorem. This allows us to handle several new and well-studied functions such as the $s-t$-connectivity, clique, and generation functions. In order to carry this progress back to the non-monotone setting, we introduce a new notion of semi-monotone composition, which combines the non-monotone complexity of the outer function with the monotone complexity of the inner function. In this setting, we prove the KRW conjecture for a similar selection of inner functions, but only for a specific choice of the outer function $f$.
