New Lower Bounds on Circuit Size of Multi-output Functions
Theory of Computing Systems, 2014
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 ... more 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. 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 , $$C_{U_{2}}(Ax \oplus b)\ge 5(n-m).$$ 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 \(C_{U_{2}}(f) \ge 5n-o(n)\) ).
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