1. CS 140 Lecture 4 Professor CK Cheng 4/11/02

2. Part I. Combinational Logic Implementation K-Map Given F R D Obj: Minimize sum of products Proc: Draw K-Map Derive prime implicants Derive the essential prime implicants Derive minimum expression

3. Example Given F =  m (0, 3, 4, 14, 15) D =  m (1, 11, 13) K-map 0 4 12 8 1 5 13 9 3 7 15 11 2 6 14 10 b c a d 1 1 0 0 - 0 - 0 1 0 1 - 1 0 1 0

4. Prime Implicants: Largest rectangles that intersect On Set but not Off Set that correspond to product terms. E.g.  m (0, 4),  m (0, 1),  m (1, 3),  m (3, 11),  m (14, 15),  m (11, 15),  m (13, 15) Essential Primes: Prime implicants covering elements in F that are not covered by any other primes. E.g.  m (0, 4),  m (14, 15) Min exp:  m (0, 4),  m (14, 15), (  m (3, 11) or  m (1,3) ) f(a,b,c,d) = a’b’c’ + abc’ + b’cd (or a’b’d)

5. Corresponding circuit f(a,b,c,d) a’ c’ d’ a b c b’ c d

6. Another example Given F =  m (3, 5), D =  m (0, 4) 0 2 6 4 1 3 7 5 b c a - 0 0 - 0 1 0 1 Primes:  m (3),  m (4, 5) Essential Primes:  m (3),  m (4, 5) Min exp: f(a,b,c) = a’bc + ab’

7. 5 variable K-map 0 4 12 8 c d b e 1 5 13 9 3 7 15 11 2 6 14 10 16 20 28 24 c d b e a 17 21 29 25 19 23 31 27 18 22 30 26 Neighbors of 5 are: 1, 4, 13, 7, and 21 Neighbors of 10 are: 2, 8, 10,14, and 26

8. 6 variable K-map 0 4 12 8 d e c f 1 5 13 9 3 7 15 11 2 6 14 10 16 20 28 24 d e c 17 21 29 25 19 23 31 27 18 22 30 26 d e c f 48 52 60 56 d e c a 49 53 61 57 51 53 63 59 50 54 62 58 b 32 36 44 40 33 27 45 41 35 39 47 43 34 38 46 42 f f

9. Min product of sums Given F =  m (3, 5), D =  m (0, 4) 0 2 6 4 1 3 7 5 b c a - 0 0 - 0 1 0 1 Prime Implicates:  M (0,1),  M (0,2,4,6),  M (6,7) Essential Primes Implicates:  M (0,1),  M (0,2,4,6),  M (6,7) Min exp: f(a,b,c) = (a+b)(c )(a’+b’)

10. Corresponding Circuit a b a’ b’ c f(a,b,c,d)

11. Another min product of sums example Given F =  m (0, 3, 4, 14, 15) D =  m (1, 11, 13) K-map 0 4 12 8 1 5 13 9 3 7 15 11 2 6 14 10 b c a d 1 1 0 0 - 0 - 0 1 0 1 - 0 0 1 0

12. Prime Implicates:  M (2,6),  M (2,10),  M (1,5,9,13),  M (5,7),  M (6,7),  M (8,9,10,11),  M (8,9,12,13) Essential Primes:  M (8,9,12,13) Min exp:  M (8,9,12,13)  M (5,7),  M (2,6),  M (8,9,10,11) or  M (6,7),  M (1,5,9,13),  M (2,10) f(a,b,c,d) = (a+b’+d’)(a’+c’+d)(a’+b)(a’+c)