Toward better depth lower bounds: the XOR-KRW conjecture

In this paper, we propose a new conjecture, the XOR-KRW conjecture, which is a relaxation of the Karchmer-Raz-Wigderson conjecture [10]. This relaxation is still strong enough to imply P ⊈ NC1 if proven. We also present a weaker version of this conjecture that might be used for breaking n3 lower bound for De Morgan formulas. Our study of this conjecture allows us to partially answer an open question stated in [5] regarding the composition of the universal relation with a function. To be more precise, we prove that there exists a function g such that the composition of the universal relation with g is significantly harder than just a universal relation. The fact that we can only prove the existence of g is an inherent feature of our approach. The paper's main technical contribution is a new approach to lower bounds for multiplexer-type relations based on the non-deterministic hardness of non-equality and a new method of converting lower bounds for multiplexer-type relations into lower bounds against some function. In order to do this, we develop techniques to lower bound communication complexity in half-duplex and partially half-duplex communication models.