Better Lower Bounds for Monotone Threshold Formulas

Jaikumar Radhakrishnan
1997 Journal of computer and system sciences (Print)  
We show that every monotone formula that computes the threshold function TH k, n , 2 k nÂ2, has size at least wkÂ2x n log(nÂ(k&1)). The same lower bound is shown to hold in the stronger monotone directed contact networks model. ] The computation of threshold functions by formulas has been widely studied. Over the complete binary basis, Paterson, Pippenger, and Zwick [18] showed that all threshold functions can be computed by formulas of size O(n 3.13 ). For this basis, Pudla k [20] showed a
more » ... r bound of 0(n log log n) for computing TH k, n , 2 k nÂ2; Fischer, Meyer, and Paterson [6] showed a lower bound of 0(n log n) for the majority function TH W nÂ2 X , n . Over the basis [AND, OR, NOT], Paterson, Pippenger, and Zwick [18] showed that TH k, n can be computed by formulas of size O(n 4.57 ). Lower bounds on the size of such formulas were shown by Hansel [8], Krichevskii [13], and Khrapchenko [12] . Hansel and Krichevskii showed a lower bound of 0(n log n) for computing TH 2, n . This implies an 0(n log n) lower bound for all threshold functions TH k, n , 2 k n&1. Khrapchenko showed that any such formula computing TH k, n has size at least k(n&k+1). The existence of polynomial size monotone formulas for computing TH k, n is implied by the O(log n) depth sorting network due to Ajtai, Komolo s, and Szemere di [1]. The existence of more efficient monotone threshold formulas was shown by Valiant [27] and Boppana [2] . Valiant showed that the majority function (TH W nÂ2 X , n ) can be computed using montone formulas of size O(n 5.3 ). Boppana generalized Valiant's result and showed that TH k, n can be computed by monotone formulas of size O(k 4.3 n log n). The lower bounds due to Hansel, Krichevskii, and Khrapchenko, stated above, hold for monotone formulas as well. Before this work, these were the best lower bounds known for monotone formulas. The result of Hansel and Krichevskii was generalized by Snir [26] to obtain an 0(kn log(nÂ(k&1)) lower bound in the context of hypergraph covering. Snir's result implies an 0(kn log(nÂ(k&1)) lower bound on the size of certain restricted depth three formulas computing TH k, n (see [17, 23] ). However, it is not clear how Snir's result may be used to derive our results for monotone threshold formulas. Related to the monotone formulas model is the model of
doi:10.1006/jcss.1997.1287 fatcat:pp4eh4quefdbdihtpiphq23u5q