Boolean functions. Theory, algorithms, and applications by Crama Y., Hammer P.

By Crama Y., Hammer P.

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Extra resources for Boolean functions. Theory, algorithms, and applications

Example text

The DNF ψ(X, Y ) can be regarded as providing an implicit DNF representation of the function φ(X), in the following sense: in order to compute the value of φ at a point X∗ ∈ Bn , one can solve the equation ψ(X∗ , Y ) = 0 and read the value of z in the unique solution of the equation. 2 and to the solution of Boolean equations in Chapter 2. The procedure recursively processes each of the subexpressions of φ and then recombines the resulting DNFs into a single one using additional variables. Intuitively, each additional variable yi (i = 1, 2, .

The time complexity of the procedure is easily established by induction. 22 1 Fundamental concepts and applications Procedure Expand(φ) Input: A Boolean expression φ(x1 , x2 , . . , xn ). Output: A DNF ψ(x1 , x2 , . . , xn , y1 , y2 , . . , ym ), with a distinguished literal z among the literals on {x1 , x2 , . . , xn , y1 , y2 , . . , ym }. begin if φ = xi for some i ∈ {1, 2, . . , n} then return ψ(x1 , x2 , . . , xn ) = 0 n and the distinguished literal xi else if φ = φ1 for some expression φ1 then begin let ψ1 := Expand(φ1 ) and let z be the distinguished literal of ψ1 ; return ψ := ψ1 and the distinguished literal z; end else if φ = (φ1 ∨ φ2 ∨ .

Xn ) on Bn , the procedure associates a DNF ψ(X, Y ) = ψ(x1 , x2 , . . , xn , y1 , y2 , . . , ym ) (where (y1 , y2 , . . , ym ) are additional variables, and possibly m = 0) and a distinguished literal z among the literals on {x1 , x2 , . . , xn , y1 , y2 , . . , ym }. These constructs have the properties that, for all X∗ ∈ B n , there is a (unique) point Y ∗ ∈ B m such that ψ(X ∗ , Y ∗ ) = 0. Moreover, in every solution (X ∗ , Y ∗ ) of the equation ψ(X, Y ) = 0, the distinguished coordinate of the point Y ∗ takes the value z∗ = φ(X∗ ).

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