Separation of the Monotone NC Hierarchy

, for the monotone depth of functions in monotone-P. As a result we achieve the separation of the following classes.

1. monotone-NC ≠ monotone-P.

2. For every i≥1, monotone-≠ monotone-.

3. More generally: For any integer function D(n), up to (for some ε>0), we give an explicit example of a monotone Boolean function, that can be computed by polynomial size monotone Boolean circuits of depth D(n), but that cannot be computed by any (fan-in 2) monotone Boolean circuits of depth less than Const·D(n) (for some constant Const).

Only a separation of monotone- from monotone- was previously known.

Our argument is more general: we define a new class of communication complexity search problems, referred to below as DART games, and we prove a tight lower bound for the communication complexity of every member of this class. As a result we get lower bounds for the monotone depth of many functions. In particular, we get the following bounds:

1.  For st-connectivity, we get a tight lower bound of . That is, we get a new proof for Karchmer–Wigderson's theorem, as an immediate corollary of our general result.

2.  For the k-clique function, with , we get a tight lower bound of Ω(k log n). This lower bound was previously known for k≤ log n [1]. For larger k, however, only a bound of Ω(k) was previously known.