Logic Functions and their Representation

1. Logic Functions andtheir Representation

2. Combinational Networks x1 x2 f xn Logic Functions and their Representation

3. Logic Operations • Truth tables Logic Functions and their Representation

4. SOP and POS • Definition: A variable xi has two literals xi and xi. 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

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

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

7. Shannon Expansion • Theorem: An arbitrary logic function f(x1,x2,…,xn) is expanded as follows: f(x1,x2,…,xn) = x1f(0,x2,…,xn)  x1f(1,x2,…,xn) (Proof) When x1 = 0, = 1f(0,x2,…,xn)  0f(1,x2,…,xn) = f(0,x2,…,xn) When x1 = 1, similar Logic Functions and their Representation

8. Expansions into Minterms • Example: Expand f(x1,x2,x3) = x1(x2  x3) • Example: minterm expansion of an arbitrary function • Relation to the truth table • Maxterm expansion (duality) Logic Functions and their Representation

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

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

11. An arbitrary 2-varibale function is represented by a canonical SOP f(x1,x2) = f(0,0)x1x2 f(0,1)x1x2 f(1,0)x1x2 f(1,1) x1x2 Since the product terms have no common minterms, the  can be replaced with  f(x1,x2) = f(0,0)x1x2 f(0,1)x1x2 f(1,0)x1x2 f(1,1) x1x2 Next, replace x1= x1 1, and x2= x2 1 Show results! Logic Functions and their Representation

12. PPRM • An arbitrary n-variable function is uniquely represented as f(x1,x2,…,xn) = a0  a1x1  a2x2 …  anxn  a12 x1x2  a13 x1x3 …  an-1,nxn-1xn  …  a12…nx1x2…xn Logic Functions and their Representation