By Yves Crama, Peter L. Hammer

ISBN-10: 0521847516

ISBN-13: 9780521847513

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Additional info for Boolean Functions: Theory, Algorithms, and Applications (Encyclopedia of Mathematics and its Applications 142)

Example text

11. A minterm (respectively, maxterm) on Bn is an elementary conjunction (respectively, disjunction) involving exactly n literals. Let f be a Boolean function on B n , let T (f ) be the set of true points of f , and let F (f ) be its set of false points. The DNF φf (x1 , x2 , . . 11) j |yj =0 is the minterm expression (or canonical DNF) of f , and the terms of φf are the minterms of f . The CNF ψf (x1 , x2 , . . 12) i|yi =1 is the maxterm expression (or canonical CNF) of f , and the terms of ψf are the maxterms of f .

The rows of the map are indexed by the values of the variable x1 ; its columns are indexed by the values of the pair of variables (x2 , x3 ); and each cell contains the value of the function in the corresponding Boolean point. For instance, the cell in the second row, fourth column, of the map contains a 0, since f (1, 1, 0) = 0. 10 Monotone Boolean functions 33 Because of the special way in which the columns are ordered, two adjacent cells always correspond to neighboring vertices of the unit hypercube U ; that is, the corresponding points differ in exactly one component.

Remark. So that we can get rid of parentheses when writing Boolean expressions, we assume from now on a priority ranking of the elementary operations: Namely, we assume that disjunction has lower priority than conjunction, which has lower priority than complementation. When we compute the value of a parentheses-free expression, we always start with the operations of highest priority: First, all complementations; next, all conjunctions; and finally, all disjunctions. 1). 4. 3) can be rewritten with fewer parentheses as f (x, y, z) = ψ1 (x, y, z) = (x ∨ y)(y ∨ z) ∨ xyz.

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Boolean Functions: Theory, Algorithms, and Applications (Encyclopedia of Mathematics and its Applications 142) by Yves Crama, Peter L. Hammer


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