Changes

919 bytes removed ,  13:10, 22 October 2008
trim
Line 5: Line 5:  
A '''boolean mask operation''' on boolean-valued functions combines values point-wise, for example, by [[exclusive disjunction|XOR]], or other [[boolean operator]]s.
 
A '''boolean mask operation''' on boolean-valued functions combines values point-wise, for example, by [[exclusive disjunction|XOR]], or other [[boolean operator]]s.
   −
==Algebraic normal form==
     −
A boolean function can be written uniquely as a sum ([[exclusive disjunction|XOR]]) of products ([[logical conjunction|AND]]). This is known as the [[algebraic normal form]] (ANF).
  −
  −
{| cellpadding="4"
  −
|-
  −
|<math>f(x_1, x_2, \ldots , x_n) = \!</math>
  −
|<math>a_0 + \!</math>
  −
|-
  −
|
  −
|<math>a_1x_1 + a_2x_2 + \ldots + a_nx_n + \!</math>
  −
|-
  −
|
  −
|<math>a_{1,2}x_1x_2 + a_{n-1,n}x_{n-1}x_n + \!</math>
  −
|-
  −
|
  −
|<math>\ldots + \!</math>
  −
|-
  −
|
  −
|<math>a_{1,2,\ldots,n}x_1x_2\ldots x_n \!</math>
  −
|}
  −
  −
The values of the sequence <math>a_0, a_1, \ldots, a_{1, 2, \ldots, n}</math> can therefore also uniquely represent a boolean function. The algebraic degree of a boolean function is defined as the highest number of <math>x_i</math> that appear in a product term. Thus <math>f(x_1, x_2, x_3) = x_1 + x_3</math> has degree 1 (linear), whereas <math>f(x_1, x_2, x_3) = x_1 + x_1 x_2 x_3</math> has degree 3 (cubic).
      
==See also==
 
==See also==
12,080

edits