Probabilistic Construction of Monotone Formulae for Positive Linear Threshold Functions
نویسنده
چکیده
We extend Valiant's construction of monotone formulae for the majority function to obtain an eecient probabilistic construction of small monotone formulae for arbitrary positive linear threshold functions. We show that any positive linear threshold function on n boolean variables which has weight complexity q(n) can be computed by a monotone boolean formula of size O(q(n) 3:3 n 2): Our technique also yields a probabilistic construction of small monotone formulae which compute large margin approximators to arbitrary positive linear threshold functions.
منابع مشابه
Monotone circuits for monotone weighted threshold functions
Weighted threshold functions with positive weights are a natural generalization of unweighted threshold functions. These functions are clearly monotone. However, the naive way of computing them is adding the weights of the satisfied variables and checking if the sum is greater than the threshold; this algorithm is inherently non-monotone since addition is a non-monotone function. In this work w...
متن کاملValiant’s Polynomial-Size Monotone Formula for Majority
The existence of polynomial-size (monotone) formulae is known to be equivalent to the existence of logarithmic-depth (monotone) circuits of bounded fan-in. Anyhow, we shall prove the existence of logarithmic-depth monotone formulae (of bounded fan-in) for majority. Actually, two radically different proofs are known: The first proof uses a rather complicated construction of sorting networks of l...
متن کاملSecret Sharing and Secure Computing from Monotone Formulae
We present a construction of log-depth formulae for various threshold functions based on atomic threshold gates of constant size. From this, we build a new family of linear secret sharing schemes that are multiplicative, scale well as the number of players increases and allows to raise a shared value to the characteristic of the underlying field without interaction. Some of these schemes are in...
متن کاملGENERALIZED POSITIVE DEFINITE FUNCTIONS AND COMPLETELY MONOTONE FUNCTIONS ON FOUNDATION SEMIGROUPS
A general notion of completely monotone functionals on an ordered Banach algebra B into a proper H*-algebra A with an integral representation for such functionals is given. As an application of this result we have obtained a characterization for the generalized completely continuous monotone functions on weighted foundation semigroups. A generalized version of Bochner’s theorem on foundation se...
متن کاملFunction Evaluation Via Linear Programming in the Priced Information Model
Wedetermine the complexity of evaluatingmonotone Boolean functions in a variant of the decision tree model introduced in [Charikar et al. 2002]. In thismodel, reading different variables can incur different costs, and competitive analysis is employed to evaluate the performance of the algorithms. It is known that for a monotone Boolean function f, the size of the largest certificate, aka PROOF ...
متن کامل