Minimização de Funções Booleanas: K-Maps e Quine-McCluskey, Resumos de Direito Digital

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Chapter 3

Simplification of Switching Functions

Simplification Goals

• (^) Goal -- minimize the cost of realizing a switching function
• (^) Cost measures and other considerations
  • (^) Number of gates
  • (^) Number of levels
  • (^) Gate fan in and/or fan out
  • (^) Interconnection complexity
  • (^) Preventing hazards
• (^) Two-level realizations
  • (^) Minimize the number of gates (terms in switching function)
  • (^) Minimize the fan in (literals in switching function)

Minimization Methods

• (^) Commonly used techniques
  • (^) Boolean algebra postulates and theorems
  • (^) Karnaugh maps
  • (^) Quine-McCluskey method
  • (^) Petrick’s method
  • (^) Generalized concensus algorithm
• (^) Characteristics
  • (^) Heuristics (suboptimal)
  • (^) Algorithms (optimal)

Minimum SOP and POS Representations

• (^) The minimum sum of products (MSOP) of a function, f , is a SOP

representation of f that contains the fewest number of product terms

and fewest number of literals of any SOP representation of f.

• (^) Example -- f(a,b,c,d) =  m (3,7,11,12,13,14,15) = ab + a  cd + acd

= ab + cd

• (^) The minimum product of sums (MPOS) of a function, f , is a POS

representation of f that contains the fewest number of sum terms and

the fewest number of literals of any POS representation of f.

• (^) Example -- f(a,b,c,d) =  M (0,1,2,4,5,6,8,9,10)

= (a + c)(a + d)(a  + b + d)(b + c 

+ d) = (a +c)(a + d)(b + c)(b + d)

Figure 3.1 Venn diagram and equivalent K-map for two variables

( d )

A B f ( A B )

m 0 m 2 m 1 m 3

A

B

0 2 1 3

A

B

B

A

0 2 1 3

( a ) ( b ) ( c )

( e ) ( f )

B

A

0 2 1 3

( g )

A B B

A

B

A

A B A B A B

m 2 m 1 m 3

A B

m 0

Figure 3.2 Venn diagram and equivalent K-map for three variables m (^0)

B

m 2 m (^6) m 1 m 3 m 7 m (^5) 0 2 6 4 1 3 7 5

B

A

A B

C

A B

( a )

( d )

A B C

A B C A B C

A B C

A B C m^0 m^1

m (^2) m (^3) m (^4) m (^5) m (^6) m (^7) m (^4)

C

A

C

C

A B

0 2 6 4 1 3 7 5

C

A B C

A B C

A B C

C

A B

( b ) ( c )

( e ) ( f )

Figure 3.3 (e) -- (f) K-maps for six variables ( e ) ( f ) A E F B C D 0 0 0 0 0 1 0 1 1 0 1 0 0 4 1 2 8 1 5 1 3 9 3 7 1 5 1 1 2 6 1 4 1 0 0 0 0 0 0 1 0 1 1 0 1 0 1 0 0 1 0 1 1 1 1 1 1 0 1 6 2 0 2 8 2 4 1 7 2 1 2 9 2 5 1 9 2 3 3 1 2 7 1 8 2 2 3 0 2 6 3 2 3 6 4 4 4 0 3 3 3 7 4 5 4 1 3 5 3 9 4 7 4 3 3 4 3 8 4 6 4 2 1 0 0 1 0 1 1 1 1 1 1 0 4 8 5 2 6 0 5 6 4 9 5 3 6 1 5 7 5 1 5 5 6 3 5 9 5 0 5 4 6 2 5 8 0 4 1 2 8 1 5 1 3 9 3 7 1 5 1 1 E 2 6 1 4 1 0 C C B 3 2 3 6 4 4 4 0 3 3 3 7 4 5 4 1 3 5 3 9 4 7 4 3 3 4 3 8 4 6 4 2 D E D 4 8 5 2 6 0 5 6 4 9 5 3 6 1 5 7 5 1 5 5 6 3 5 9 5 0 5 4 6 2 5 8 1 6 2 0 2 8 2 4 1 7 2 1 2 9 2 5 1 9 2 3 3 1 2 7 1 8 2 2 3 0 2 6 F A F

Plotting (Mapping) Functions in Canonical Form

on a K-map

• (^) Let f be a switching function of n variables where n  6.
• (^) Assume that the cells of the K-map are numbered from 0 to 2 n^ where the numbers correspond to the rows of the truth table of f.
• (^) If mi is a minterm of f, then place a 1 in cell i of the K-map_._
• (^) Example -- f(A,B,C) =  m (0,3,5)
• (^) If Mi is a maxterm of f , then place a 0 in cell i.
• (^) Example -- f(A,B,C) =  M (1,2,4,6,7)
• (^) If di is a don’t care of f, then place a d in cell i.

Figure 3.5 K-maps for f(a,b,Q,G) in Example 3.

(a) Minterm form. (b) Maxterm form. Q G a b 1 0 0 0 1 1 1 1 0 0 4 1 2 8 1 5 1 3 9 3 7 1 5 1 1 2 6 1 4 1 0 0 0 0 1 1 1 1 0 b G 1 1 1 1 1 1 1 1 1 a Q Q G a b 0 0 0 1 1 1 1 0 0 4 1 2 8 1 5 1 3 9 3 7 1 5 1 1 2 6 1 4 1 0 0 0 0 1 1 1 1 0 b G 0 0 a Q 0 0 0 0 ( a ) ( b ) f(a,b,Q,G) =  m (0,3,5,7,10,11,12,13,14,15) =  M (1,2,4,6,8,9)

Figure 3.6 K-map of Figure 3.5(a) with variables

reordered: f(Q,G,b,a).

b a Q G 0 0 0 1 1 1 1 0

0 0 0 1 1 1 1 0 G a 1 1 Q b 1 1 1 1 1 1 1 1

f(Q,G,b,a) =  m (0,12,6,14,9,13,3,7,11,15) =  m (0,3,6,7,9,11,12,13,14,15)

Figure 3.7 -- Example 3.6.

(a) Venn diagram form. (b) Sum of minterms. (c) Maxterms. C A B 0 0 0 1 1 1 1 0 0 2 6 4 1 3 7 5 0 1 B 0 C^0 A C A B 0 0 0 1 1 1 1 0 0 2 6 4 1 3 7 5 0 C^1 A B ( b ) ( c ) U n i v e r s a l s e t B C A B (^) A B A B C B C 1 1 1 0 0 ( a ) 0 f(A,B,C) = AB + BC 

Figure 3.8 -- Example 3.7.

(a) Maxterms, (b) Minterms, (c) Minterms of f .

C D A B 0 0 0 0 1 1 1 1 0 0 4 1 2 8 1 5 1 3 9 3 7 1 5 1 1 2 6 1 4 1 0 0 0 0 1 1 1 1 0 B D 0 0 0 0 0 0 0 C ( A + C )^ A ( B + C + D ) C D A B 0 0 0 1 1 1 1 0 0 4 1 2 8 1 5 1 3 9 3 7 1 5 1 1 2 6 1 4 1 0 0 0 0 1 1 1 1 0 B D 1 1 1 1 1 1 1 1 C A ( a ) ( b ) C D A B 1 0 0 0 1 1 1 1 0 0 4 1 2 8 1 5 1 3 9 3 7 1 5 1 1 2 6 1 4 1 0 0 0 0 1 1 1 1 0 B D 1 1 1 1 1 1 1 C A C^ A B C D ( B + C ) B C ( c ) f(A,B,C,D) = (A + C)(B + C)(B  + C  + D)

S.Isrie, MSc.

Simplification of Switching Functions

Using K-maps

• (^) K-map cells that are physically adjacent are also logically adjacent. Also, cells on an edge of a K-map are logically adjacent to cells on the opposite edge of the map.
• (^) If two logically adjacent cells both contain logical 1s, the two cells can be combined to eliminate the variable that has value 1 in one cell’s label and value 0 in the other.
• (^) This is equivalent to the algebraic operation, a P + a P =P where P is a product term not containing a or a .
• (^) Example -- f(A,B,C,D) =  m (1,2,4,6,9)

Figure 3.10 K-map for Example 3. C D A B 0 0 0 1 1 1 1 0 0 4 1 2 8 1 5 1 3 9 3 7 1 5 1 1 2 6 1 4 1 0 0 0 0 1 1 1 1 0 B D 1 1 1 1 1 A C S t e p 2 S t e p 1 S t e p 3 f(A,B,C,D) =  m (1,2,4,6,9)