We prove tight lower bounds, of up to n , 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. For every3. More generally: For any integer function D(n), up to n (for some > 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 , 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.

Abstract. Determining the complexity of the reachability problem for vector addition systems with states (VASS) is a long-standing open problem in computer science. Long known to be decidable, the problem to this day lacks any complexity upper bound whatsoever. In this paper, reachability for two-dimensional VASS is shown PSPACE-complete. This improves on a previously known doubly exponential time bound established by Howell, Rosier, Huynh and Yen in 1986. The coverability and boundedness problems are also noted to be PSPACE-complete. In addition, some complexity results are given for the reachability problem in two-dimensional VASS and in integer VASS when numbers are encoded in unary.

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