# Separation of the Monotone NC Hierarchy

DOI: 10.1007/s004930050062

- Cite this article as:
- Raz, R. & McKenzie, P. Combinatorica (1999) 19: 403. doi:10.1007/s004930050062

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, 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.