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Erschienen in:
01.05.2015
New Lower Bounds on Circuit Size of Multi-output Functions
Erschienen in: Theory of Computing Systems | Ausgabe 4/2015
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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,
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 \(C_{U_{2}}(f) \ge 5n-o(n)\)).
$$C_{U_{2}}(g(f)) \ge \min \{C_{U_{2}}(f|_{x_{i} = a, x_{j} = b}) \colon x_{i} \neq x_{j}, a,b, \in \{0,1\}\} +2n-\varTheta (1)$$