Separation of the Monotone NC Hierarchy

Abstract

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.

Untitled

“…The simulation theorem asserts that the optimal protocol for f • g n is to simulate the decision tree that computes f if g is a hard function. Simulation theorems have been established in various cases, when g is bitwise AND or OR [15], Inner-Product [3], Index Function [13,4]. Our work gives a new simulation theorem when g is an XOR function.…”

“…After almost a decade of efforts, the conjecture has been established for several classes of XOR-function, such as symmetric functions [20], monotone functions and linear threshold functions [12], constant F 2 -degree functions [17]. A different line of work close to ours is the simulation theorem in [13,20,15,3,4]. They study the relationship between the (regular) decision tree complexity of function f and the communication complexity of f • g n where g is a 2-argument function of small size.…”

Incremental Branching Programs

“…However, the monotone circuit depth lower bounds of [RaMc99] for the RELGEN function are sufficient to derive the following statement.…”

“…Let δ > 0 be such that n-RELGEN requires monotone circuit depth n δ (as proved in [RaMc99][Corollary 3.6]). Consider an n-RELGEN instance w ∈ {0, 1} We conclude the proof of the theorem from the claim as follows.…”

“…• We prove that size lower bounds for (deterministic and nondeterministic) syntactic incremental branching programs for GEN can be obtained from the pebbling-based lower bounds on marking machines of [Co74,PTC77], as well as from the monotone circuit depth lower bounds for a version of GEN by Raz and McKenzie [RaMc99].…”