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| | 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. |
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| − | ==Algebraic normal form==
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| − | 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).
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| − | {| cellpadding="4"
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| − | |<math>f(x_1, x_2, \ldots , x_n) = \!</math>
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| − | |<math>a_0 + \!</math>
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| − | |<math>a_1x_1 + a_2x_2 + \ldots + a_nx_n + \!</math>
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| − | |<math>a_{1,2}x_1x_2 + a_{n-1,n}x_{n-1}x_n + \!</math>
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| − | |<math>\ldots + \!</math>
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| − | |<math>a_{1,2,\ldots,n}x_1x_2\ldots x_n \!</math>
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| − | |}
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| − | 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).
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| | ==See also== | | ==See also== |
