New Lower Bounds on Circuit Size of Multi-output Functions

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 . A lower bound C U 2 ( f ) ≥ 5 n − o ( n ) for a linear function f ∈ B n − 1,log n . 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 ( A x ⊕ b ) ≥ 5 ( n − m ) . A lower bound C U 2 ( f ) ≥ 7 n − 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 ) ) ≥ min { C U 2 ( f | x i = a , x j = b ) : x i ≠ x j , a , b , ∈ { 0 , 1 } } + 2 n − Θ ( 1 ) where g ( f ) ∈ B n , n is defined as follows: g ( f ) = ( f ⊕ x 1 , … , f ⊕ x n ) (to get a 7 n − o ( n ) lower bound it remains to plug in a known function f ∈ B n , 1 with C U 2 ( f ) ≥ 5 n − o ( n ) ).