### Abstract

We prove tight lower bounds, of up to n^{ε}, 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 all i≥1, monotone-NC^{i}≠monotone-NC^{i+1}. 3. More generally: For any integer function D(n), up to n^{ε} (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. NC^{1} from monotone-NC^{2} 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 Ω(log^{2}n). 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 k≤n^{ε}, we get a tight lower bound of Ω(k log n). Only a bound of Ω(k) was previously known.

Original language | English (US) |
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Pages (from-to) | 234-243 |

Number of pages | 10 |

Journal | Annual Symposium on Foundations of Computer Science - Proceedings |

State | Published - Dec 1 1997 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Hardware and Architecture

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## Cite this

*Annual Symposium on Foundations of Computer Science - Proceedings*, 234-243.