No abstract
We study depth lower bounds against non-monotone circuits, parametrized by a new measure of non-monotonicity: the orientation 1 of a function f is the characteristic vector of the minimum sized set of negated variables needed in any DeMorgan 2 circuit computing f . We prove trade-off results between the depth and the weight/structure of the orientation vectors in any circuit C computing the CLIQUE function on an n vertex graph. We prove that if C is of depth d and each gate computes a Boolean function with orientation of weight at most w (in terms of the inputs to C), then d×w must be Ω(n). In particular, if the weights are o( n log k n ), then C must be of depth ω(log k n). We prove a barrier for our general technique. However, using specific properties of the CLIQUE function (used in [5]) and the Karchmer-Wigderson framework [12], we go beyond the limitations and obtain lower bounds when the weight restrictions are less stringent. We then study the depth lower bounds when the structure of the orientation vector is restricted. Asymptotic improvements to our results (in the restricted setting), separates NP from NC. As our main tool, we generalize Karchmer-Wigderson game [12] for monotone functions to work for non-monotone circuits parametrized by the weight/structure of the orientation. We also prove structural results about orientation and prove connections between number of negations and weight of orientations required to compute a function.
The class FORMULA[s]∘G consists of Boolean functions computable by size- s De Morgan formulas whose leaves are any Boolean functions from a class G. We give lower bounds and (SAT, Learning, and pseudorandom generators ( PRG s )) algorithms for FORMULA[n 1.99 ]∘G, for classes G of functions with low communication complexity . Let R (k) G be the maximum k -party number-on-forehead randomized communication complexity of a function in G. Among other results, we show the following: • The Generalized Inner Product function GIP k n cannot be computed in FORMULA[s]° G on more than 1/2+ε fraction of inputs for s=o(n 2 /k⋅4 k ⋅R (k) (G)⋅log(n/ε)⋅log(1/ε)) 2 ). This significantly extends the lower bounds against bipartite formulas obtained by [62]. As a corollary, we get an average-case lower bound for GIP k n against FORMULA[n 1.99 ]∘PTF k −1 , i.e., sub-quadratic-size De Morgan formulas with degree-k-1) PTF ( polynomial threshold function ) gates at the bottom. Previously, it was open whether a super-linear lower bound holds for AND of PTFs. • There is a PRG of seed length n/2+O(s⋅R (2) (G)⋅log(s/ε)⋅log(1/ε)) that ε-fools FORMULA[s]∘G. For the special case of FORMULA[s]∘LTF, i.e., size- s formulas with LTF ( linear threshold function ) gates at the bottom, we get the better seed length O(n 1/2 ⋅s 1/4 ⋅log(n)⋅log(n/ε)). In particular, this provides the first non-trivial PRG (with seed length o(n)) for intersections of n halfspaces in the regime where ε≤1/n, complementing a recent result of [45]. • There exists a randomized 2 n-t #SAT algorithm for FORMULA[s]∘G, where t=Ω(n\√s⋅log 2 (s)⋅R (2) (G))/1/2. In particular, this implies a nontrivial #SAT algorithm for FORMULA[n 1.99 ]∘LTF. • The Minimum Circuit Size Problem is not in FORMULA[n 1.99 ]∘XOR; thereby making progress on hardness magnification, in connection with results from [14, 46]. On the algorithmic side, we show that the concept class FORMULA[n 1.99 ]∘XOR can be PAC-learned in time 2 O(n/log n) .
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