Logic Functions and their Representation

Slide 2 Combinational Networks x1x1 x2x2 xnxn f

Logic Functions and their Representation Slide 3 Logic Operations Truth tables xy AND x  y OR x  y NOT  x NAND  x  y NOR  x   y EXOR x  y

Logic Functions and their Representation Slide 4 SOP and POS Definition: A variable x i has two literals x i and  x i. A logical product where each variable is represented by at most one literal is a product or a product term or a term. A term can be a single literal. The number of literals in a product term is the degree. A logical sum of product terms forms a sum-of-products expression (SOP). A logical sum where each variable is represented by at most one literal is a sum term. A sum term can be a single literal. A logical product of sum terms forms a product-of-sums expression (POS).

Logic Functions and their Representation Slide 5 Minterm A minterm is a logical product of n literals where each variable occurs as exactly one literal A canonical SOP is a logical sum of minterms, where all minterms are different. Also called canonical disjunctive form or minterm expansion

Logic Functions and their Representation Slide 6 Maxterm A maxterm is a logical sum of n literals where each variable occurs as exactly one literal A canonical Pos is a logical product of maxterms, where all maxterms are different. Also called canonical conjunctive form or maxterm expansion Show an example

Logic Functions and their Representation Slide 7 Shannon Expansion Theorem: An arbitrary logic function f(x 1,x 2,…,x n ) is expanded as follows: f(x 1,x 2,…,x n ) =  x 1 f(0,x 2,…,x n )  x 1 f(1,x 2,…,x n ) (Proof) When x 1 = 0, = 1  f(0,x 2,…,x n )  0  f(1,x 2,…,x n ) = f(0,x 2,…,x n ) When x 1 = 1, similar

Logic Functions and their Representation Slide 8 Expansions into Minterms Example: Expand f(x 1,x 2,x 3 ) = x 1 (x 2   x 3 ) Example: minterm expansion of an arbitrary function Relation to the truth table Maxterm expansion (duality)

Logic Functions and their Representation Slide 9 Reed-Muller Expansions EXOR properties (x  y)  z = x  (y  z) x(y  z) = xy  xz x  y = y  x x  x = 0 x  1 =  x

Logic Functions and their Representation Slide 10 Reed-Muller Expansions Lemma xy = 0  x  y = x  y (Proof) (  ) Let xy = 0 x  y =  xy  x  y = (  xy  xy)  (x  y  xy) = x  y (  ) Let xy ≠ 0 x = y = 1. Thus x  y = 0, x  y = 1 Therefore, x  y ≠ x  y

Logic Functions and their Representation Slide 11 An arbitrary 2-varibale function is represented by a canonical SOP f(x 1,x 2 ) = f(0,0)  x 1  x 2  f(0,1)  x 1 x 2  f(1,0)x 1  x 2  f(1,1) x 1 x 2 Since the product terms have no common minterms, the  can be replaced with  f(x 1,x 2 ) = f(0,0)  x 1  x 2  f(0,1)  x 1 x 2  f(1,0)x 1  x 2  f(1,1) x 1 x 2 Next, replace  x 1 = x 1  1, and  x 2 = x 2  1 Show results!

Logic Functions and their Representation Slide 12 PPRM An arbitrary n-variable function is uniquely represented as f(x 1,x 2,…,x n ) = a 0  a 1 x 1  a 2 x 2  …  a n x n  a 12 x 1 x 2  a 13 x 1 x 3  …  a n-1,n x n-1 x n  …  a 12…n x 1 x 2 …x n