Abstract
We study how big the blow-up in size can be when one switches between the CNF and DNF representations of Boolean functions. For a function f : { 0 , 1 } n → { 0 , 1 } , cnfsize ( f ) denotes the minimum number of clauses in a CNF for f; similarly, dnfsize ( f ) denotes the minimum number of terms in a DNF for f. For 0 ⩽ m ⩽ 2 n - 1 , let dnfsize ( m , n ) be the maximum dnfsize ( f ) for a function f : { 0 , 1 } n → { 0 , 1 } with cnfsize ( f ) ⩽ m . We show that there are constants c 1 , c 2 ⩾ 1 and ε > 0 , such that for all large n and all m ∈ [ 1 ε n , 2 ε n ] , we have 2 n - c 1 ( n / log ( m / n ) ) ⩽ dnfsize ( m , n ) ⩽ 2 n - c 2 ( n / log ( m / n ) ) . In particular, when m is the polynomial n c , we get dnfsize ( n c , n ) = 2 n - θ ( c - 1 ( n / log n ) ) .
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