1ECE2030 ECE2030 Introduction to Computer Introduction to
Computer EngineeringEngineeringLecture 8: Quine-McCluskey
MethodLecture 8: Quine-McCluskey MethodProf.Hsien-Hsin Sean
LeeProf.Hsien-Hsin Sean LeeSchool of ECESchool of ECEGeorgia
Institute of Georgia Institute of TechnologyTechnology H.-H. S. Lee
H.-H. S. Lee2Quine-McCluskey MethodA systematic solution to -Map
!hen more comple" #unction !ith more literals is gi$enIn principle%
can &e applied to an ar&itrary large num&er o# inputs%
i'e' !orks #or ((nn !here nn can &e ar&itrarily large)ne
can translate Quine-McCluskey method into a computer program to
per#orm minimi*ation H.-H. S. Lee H.-H. S. Lee+Quine-McCluskey
Method ,!o &asic steps-inding all prime implicants o# a gi$en
(oolean #unction.elect a minimal set o# prime implicants that co$er
this #unction H.-H. S. Lee H.-H. S. Lee/Q-M Method 0I1= 30) 29, 19,
18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0, E) D, C, B,
F(A,,rans#orm the gi$en (oolean #unction into a canonical .)2
#unctionCon$ert each Minterm into &inary #ormatArrange each
&inary minterm in groupsAll the minterms in one group contain
the same num&er o# 314 H.-H. S. Lee H.-H. S. Lee5Q-M Method:
6rouping minterms= 30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6,
4, 2, m(0, E) D, C, B, F(A,(29) 1 1 1 0 1(30) 1 1 1 1 0(2)0 0 0 1
0(4)0 0 1 0 0(8)0 1 0 0 0(16) 1 0 0 0 0 A B C D E(0)0 0 0 0 0(6)0 0
1 1 0(10) 0 1 0 1 0(12) 0 1 1 0 0(18) 1 0 0 1 0(7)0 0 1 1 1(11) 0 1
0 1 1(13) 0 1 1 0 1(14) 0 1 1 1 0(19) 1 0 0 1 1 H.-H. S. Lee H.-H.
S. Lee7Q-M Method 0II1Com&ine terms !ith 8amming distance91
#rom ad:acent groupsCheck 01 the terms &eing com&ined,he
checked terms are 3co$ered4 &y the com&ined ne! termeep
doing this till no com&ination is possi&le &et!een
ad:acent groups H.-H. S. Lee H.-H. S. Lee;Q-M Method: 6rouping
minterms= 30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2,
m(0, E) D, C, B, F(A,(29) 1 1 1 0 1(30) 1 1 1 1 0(2)0 0 0 1 0(4)0 0
1 0 0(8)0 1 0 0 0(16) 1 0 0 0 0 A B C D E(0)0 0 0 0 0(6)0 0 1 1
0(10) 0 1 0 1 0(12) 0 1 1 0 0(18) 1 0 0 1 0(7)0 0 1 1 1(11) 0 1 0 1
1(13) 0 1 1 0 1(14) 0 1 1 1 0(19) 1 0 0 1 1 A B C D E(0,2)0 0 0
0(0,4)0 0 - 0 0(0,8)0 - 0 0 0(0,16) - 0 0 0 0(2,6)0 0 - 1 0(2,10) 0
- 0 1 0(2,18) - 0 0 1 0(4,6)0 0 1 - 0(4,12) 0 - 1 0 0(8,10) 0 1 0 -
0(8,12) 0 1 - 0 0(16,18)1 0 0 - 0(6,7)0 0 1 1 -(6,14) 0 - 1 1
0(10,11)0 1 0 1 -(10,14)0 1 - 1 0(12,13)0 1 1 0 -(12,14)0 1 1 -
0(18,19)1 0 0 1 - A B C D E(13,29)- 1 1 0 1(14,30)- 1 1 1 0 H.-H.
S. Lee H.-H. S. Lee8Q-M Method: 6rouping mintermsA B C D E(0,2)0 0
0 0(0,4)0 0 - 0 0(0,8)0 - 0 0 0(0,16) - 0 0 0 0(2,6)0 0 - 1 0(2,10)
0 - 0 1 0(2,18) - 0 0 1 0(4,6)0 0 1 - 0(4,12) 0 - 1 0 0(8,10) 0 1 0
- 0(8,12) 0 1 - 0 0(16,18)1 0 0 - 0(6,7)0 0 1 1 -(6,14) 0 - 1 1
0(10,11)0 1 0 1 -(10,14)0 1 - 1 0(12,13)0 1 1 0 -(12,14)0 1 1 -
0(18,19)1 0 0 1 -(13,29)- 1 1 0 1(14,30)- 1 1 1 0 A B C D
E(0,2,4,6)0 0 - 0(0,2,8,10) 0 - 0 0(0,2,16,18)- 0 0 0(0,4,8,12) 0 -
- 0 0(2,6,10,14)0 - - 1 0(4,6,12,14)0 - 1 - 0(8,10,12,14) 0 1 - - 0
A B C D E(0,2,4,6 0 - - - 08,10,12,14) = 30) 29, 19, 18, 16, 14,
13, 12, 11, 10, 8, 7, 6, 4, 2, m(0, E) D, C, B, F(A, H.-H. S. Lee
H.-H. S. Leenchecked terms are prime implicants H.-H. S. Lee H.-H.
S. Lee1?2rime Implicants= 30) 29, 19, 18, 16, 14, 13, 12, 11, 10,
8, 7, 6, 4, 2, m(0, E) D, C, B, F(A,CD B A =D C B A =(6,7)0 0 1 1
-(10,11)0 1 0 1 -(12,13)0 1 1 0 -(18,19)1 0 0 1 -(13,29)- 1 1 0
1(14,30)- 1 1 1 0(0,2,16,18)- 0 0 0(0,2,4,6 0 - - - 0 8,10,12,14) A
B C D E=>nchecked terms are prime implicantsD BC A =D C B A =E D
BC =E BCD =E C B =E A = H.-H. S. Lee H.-H. S. Lee11Q-M Method
0III1-orm a 2rime Implicant ,a&le @-a"is: the mintermA-a"is:
prime implicantsAn is placed at the intersection o# a ro! and
column i# the corresponding prime implicant includes the
corresponding product 0term1 H.-H. S. Lee H.-H. S. Lee12Q-M Method:
2rime Implicant ,a&le0 2 4! " #0###2#3#4#
#"#$2$30%&!' @ @ @ @%#0#' @ @ @ @%#2' @ @ @ @%#"$' @ @ @
@%#3&2$' @ @ @ @%#4&30' @ @ @ @%0&2"' @ @ @ @ @ @ @
@%0&2&4&&"'@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @= 30) 29,
19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0, E) D, C, B,
F(A, H.-H. S. Lee H.-H. S. Lee1+Q-M Method 0IB1Locate the essential
ro! #rom the ta&le,hese are essential prime implicants,he ro!
consists o# minterms co$ered &y a single ()Mark all minterms
co$ered &y the essential prime implicants-ind non-essential
prime implicants to co$er the rest o# minterms-orm the .)2 #unction
!ith the prime implicants selected% !hich is the minimal
representation H.-H. S. Lee H.-H. S. Lee1/Q-M Method0 2 4! "
#0###2#3#4#
#"#$2$30%&!' @ @ @ @%#0#' @ @ @ @%#2' @ @ @ @%#"$' @ @ @
@%#3&2$' @ @ @ @%#4&30' @ @ @ @%0&2"' @ @ @ @ @ @ @
@%0&2&4&&"'@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @= 30) 29,
19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0, E) D, C, B,
F(A, H.-H. S. Lee H.-H. S. Lee15Q-M Method0 2 4! " #0###2#3#4#
#"#$2$30%&!' @ @ @ @%#0#' @ @ @ @%#2' @ @ @ @%#"$' @ @ @
@%#3&2$' @ @ @ @%#4&30' @ @ @ @%0&2"' @ @ @ @ @ @ @
@%0&2&4&&"'@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @= 30) 29,
19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0, E) D, C, B,
F(A,= .elect 0?%2%/%7%8%1?%12%1/1 H.-H. S. Lee H.-H. S. Lee17Q-M
Method0 2 4! " #0###2#3#4#
#"#$2$30%&!' @ @ @ @%#0#' @ @ @ @%#2' @ @ @ @%#"$' @ @ @
@%#3&2$' @ @ @ @%#4&30' @ @ @ @%0&2"' @ @ @ @ @ @ @
@%0&2&4&&"'@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @= 30) 29,
19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0, E) D, C, B,
F(A,= .elect 0?%2%/%7%8%1?%12%1/1 H.-H. S. Lee H.-H. S. Lee1;Q-M
Method0 2 4! " #0###2#3#4#
#"#$2$30%&!' @ @ @ @%#0#' @ @ @ @%#2' @ @ @ @%#"$' @ @ @
@%#3&2$' @ @ @ @%#4&30' @ @ @ @%0&2"' @ @ @ @ @ @ @
@%0&2&4&&"'@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @= 30) 29,
19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0, E) D, C, B,
F(A,= .elect 0?%2%/%7%8%1?%12%1/1% 07%;1 H.-H. S. Lee H.-H. S.
Lee18Q-M Method0 2 4! " #0###2#3#4#
#"#$2$30%&!' @ @ @ @%#0#' @ @ @ @%#2' @ @ @ @%#"$' @ @ @
@%#3&2$' @ @ @ @%#4&30' @ @ @ @%0&2"' @ @ @ @ @ @ @
@%0&2&4&&"'@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @= 30) 29,
19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0, E) D, C, B,
F(A,= .elect 0?%2%/%7%8%1?%12%1/1% 07%;1 H.-H. S. Lee H.-H. S.
Lee1
