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1 CS 20 Lecture 14 Karnaugh Maps Professor CK Cheng CSE Dept. UC San Diego
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2 1.Introduction 2.The Maps 3.Boolean Optimization Karnaugh Maps
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3 Definitions: Literal: x i or x i ’ Product Term:x 2 x 1 ’x 0 Sum Term:x 2 + x 1 ’ + x 0 Minterm of n variables: A product of n literals in which every variable appears exactly once. f(a,b,c,d): ab’cd’, a’bc’d’ Maxterm of n variables: A sum of n literals in which every variable appears exactly once. f(a,b,c,d): (a’+b+c+d), (a’+b’+c+d) Introduction
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4 Input: Boolean expression of n binary variables Goal: Simplification of the expression. E.g. we want to minimize # terms and # literals. Applications: Logic: rule reduction Hardware Design: cost and performance optimization. Cost (wires, gates): # literals, product terms, sum terms Performance: speed, reliability
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5 Introduction IDABf(A,B)minterm 0000 1011A’B 2101AB’ 3111AB An example of 2-variable function f(A,B)=A+B
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6 Function can be represented by sum of minterms: f(A,B) = A’B+AB’+AB This is not minimal however! We want to minimize the number of literals and terms. We factor out common terms – A’B+AB’+AB= A’B+AB’+AB+AB =(A’+A)B+A(B’+B)=B+A Hence, we have f(A,B) = A+B
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7 K-Map: Truth Table in 2 Dimensions A = 0 A = 1 B = 0 B = 1 0 2 1 3 0 1 1 A’B AB’ AB f(A,B) = A + B
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8 IDABf(A,B)minterm 0000 1011A’B 2100 3111AB Another Example f(A,B)=B f(A,B)=A’B+AB=(A’+A)B=B
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9 On the K-map: A = 0 A= 1 B= 0 B = 1 0 2 1 3 0 1 A’B AB f(A,B)=B
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10 IDABf(A,B)Maxterm 0000A+B 1011 2100A’+B 3111 Using Maxterms f(A,B)=(A+B)(A’+B)=(AA’)+B=0+B=B
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The Maps: Representation of k-Variable Functions Boolean Expression Truth Table Cube K Map Binary Decision Diagram 11 (1,1,1)(1,1,0) (0,0,0)(0,0,1) (0,1,0) (0,1,1) (1,0,1) A cube of 3 variables: (A,B,C) C B A
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Representation of k-Variable Func. Boolean Expression Truth Table Cube K Map Binary Decision Diagram 12 (0,1,1,1)(0,1,1,0) (0,0,0,0)(0,0,0,1)(1,0,0,1) (1,1,1,1) (1,1,0,1) (1,0,0,0) (0,0,1,0) (1,1,1,0) (0,0,1,1) (1,0,1,1) (0,1,0,1) (1,0,1,0) A cube of 4 variables: (A,B,C,D) D C B A
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13 Three-Variable K-Map Id a b c f (a,b,c) 0 0 0 0 f(0,0,0) 1 0 0 1 f(0,0,1) 2 0 1 0 f(0,1,0) 3 0 1 1 f(0,1,1) 4 1 0 0 f(1,0,0) 5 1 0 1 f(1,0,1) 6 1 1 0 f(1,1,0) 7 1 1 1 f(1,1,1)
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14 Three-Variable K-Map Id a b c f (a,b,c) 0 0 0 0 1 1 0 0 1 0 2 0 1 0 1 3 0 1 1 0 4 1 0 0 1 5 1 0 1 0 6 1 1 0 1 7 1 1 1 0
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15 Corresponding K-map 0 2 6 4 1 3 7 5 b = 1 c = 1 a = 1 1 1 1 1 0 0 0 0 (0,0) (0,1) (1,1) (1,0) c = 0 Gray code f(a,b,c) = c’
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16 Karnaugh Maps (K-Maps) Boolean expressions can be minimized by combining terms K-maps minimize equations graphically
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17 Find rectangles to cover 1’s in adjacent entries. Rectangles can overlap but should not include 0’s. Use the rectangle that corresponds to a product term. K-map y(A,B)=A’B’C’+A’B’C= A’B’(C’+C)=A’B’
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18 Another 3-Input example Id a b c f (a,b,c) 0 0 0 0 0 1 0 0 1 0 2 0 1 0 1 3 0 1 1 0 4 1 0 0 1 5 1 0 1 1 6 1 1 0 - 7 1 1 1 1 Don’t Care Entry: “-” means the entry is not relevant either at input or output. In other words, we are free to assign either 0 or 1 to reduce the Boolean expression.
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19 Corresponding K-map 0 2 6 4 1 3 7 5 b = 1 c = 1 a = 1 0 1 - 1 0 0 1 1 (0,0) (0,1) (1,1) (1,0) c = 0 f(a,b,c) = a + bc’
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20 Yet another example Id a b c f (a,b,c,d) 0 0 0 0 1 1 0 0 1 1 2 0 1 0 - 3 0 1 1 0 4 1 0 0 1 5 1 0 1 1 6 1 1 0 0 7 1 1 1 0
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21 Corresponding K-map 0 2 6 4 1 3 7 5 b = 1 c = 1 a = 1 1 - 0 1 1 0 0 1 (0,0) (0,1) (1,1) (1,0) c = 0 f(a,b,c) = b’
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22 Quiz Given Boolean function, f(a,b,c)= a’b’+a’c’+bc’+ab+b’c 1.Write the truth table 2.Use Karnaugh map to derive the function in a minimal expression of sum of product form.
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23 4-input K-map
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24 4-input K-map
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25 4-input K-map
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26 K-maps with Don’t Cares
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27 K-maps with Don’t Cares
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28 K-maps with Don’t Cares
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