New Lower Bounds on Circuit Size of Multi-output Functions

Abstract

Let B _{n, m} be the set of all Boolean functions from {0, 1}ⁿ to {0, 1}^m, B _n = B _{n, 1} and U ₂ = B ₂∖{⊕, ≡}. In this paper, we prove the following two new lower bounds on the circuit size over U ₂.

• 1. A lower bound $C_{U_{2}}(f) \ge 5n-o(n)$ for a linear function f ∈ B _{n − 1,logn} . The lower bound follows from the following more general result: for any matrix A ∈ {0, 1}^{m × n} with n pairwise different non-zero columns and b ∈ {0, 1}^m,

  $$C_{U_{2}}(Ax \oplus b)\ge 5(n-m).$$

• 2. A lower bound $C_{U_{2}}(f) \ge 7n-o(n)$ for f ∈ B _{n, n} . Again, this is a consequence of the following result: for any f ∈ B _n satisfying a certain simple property,

  $$C_{U_{2}}(g(f)) \ge \min \{C_{U_{2}}(f|_{x_{i} = a, x_{j} = b}) \colon x_{i} \neq x_{j}, a,b, \in \{0,1\}\} +2n-\varTheta (1)$$

  where g(f) ∈ B _{n, n} is defined as follows: g(f) = (f ⊕ x ₁, … , f ⊕ x _n ) (to get a 7n − o(n) lower bound it remains to plug in a known function f ∈ B _{n, 1} with $C_{U_{2}}(f) \ge 5n-o(n)$).