Approximating Boolean Functions with Disjunctive Normal Form
share
The theorem states that: Every Boolean function can be ϵ -approximated by a Disjunctive Normal Form (DNF) of size O_ϵ(2^n/logn). This paper will demonstrate this theorem in detail by showing how this theorem is generated and proving its correctness. We will also dive into some specific Boolean functions and explore how these Boolean functions can be approximated by a DNF whose size is within the universal bound O_ϵ(2^n/logn). The Boolean functions we interested in are: Parity Function: the parity function can be ϵ-approximated by a DNF of width (1 - 2ϵ)n and size 2^(1 - 2ϵ)n. Furthermore, we will explore the lower bounds on the DNF's size and width. Majority Function: for every constant 1/2 < ϵ < 1, there is a DNF of size 2^O(√(n)) that can ϵ-approximated the Majority Function on n bits. Monotone Functions: every monotone function f can be ϵ-approximated by a DNF g of size 2^n - Ωϵ(n) satisfying g(x) ≤ f(x) for all x.
READ FULL TEXT