# Quine-McCluskey Algorithm

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1 Quine-McCluskey Algorithm Useful for minimizing equations with more than 4 inputs. Like K-map, also uses combining theorem Allows for automation Chapter <>

2 Edward McCluskey (99-06) Pioneer in Electrical Engineering First president of IEEE Professor at Princeton, then Stanford 955: Quine-McCluskey Algorithm Chapter <>

3 Quine-McCluskey Algorithm Method:. s Table: List each minterm, sorted by the number of s it contains. Combine minterms.. Prime Implicant Table: When you can t combine anymore, list prime implicants and the minterms they cover. 3. Select Prime to cover all minterms. Chapter <3>

4 Quine-McCluskey Example ABC Y Chapter <4>

5 s Table Quine-McCluskey Example ABC Y of 's 0 Size 000 m0 00 m 00 m 0 m3 0 m5 0 m6 Order the minterms by the number of s they have. Chapter <5>

6 s Table Quine-McCluskey Example ABC Y of 's 0 Size 000 m0 00 m 00 m 0 m3 0 m5 0 m6 Size 00- m(0,) 0-0 m(0,) 0- m(,3) -0 m(,5) 0- m(,3) -0 m(,6) Combine minterms in adjacent groups (starting with the top group). To combine terms, minterm can differ (be or 0) in only place - indicates don t care Chapter <6>

7 s Table Quine-McCluskey Example ABC Y of 's 0 Size 000 m0 00 m 00 m 0 m3 0 m5 0 m6 Size 00- m(0,) 0-0 m(0,) 0- m(,3) -0 m(,5) 0- m(,3) -0 m(,6) Size m(0,,,3) Combine minterms in adjacent groups (starting with the top group). To combine terms, minterm can differ (be or 0) in only place - (don t care) acts like another variable (0,, -) Match up - first. Chapter <7>

8 s Table Quine-McCluskey Example ABC Y of 's 0 Size 000 m0 00 m 00 m 0 m3 0 m5 0 m6 Size 00- m(0,) 0-0 m(0,) 0- m(,3) -0 m(,5)* 0- m(,3) -0 m(,6)* Size m(0,,,3)* List prime implicants: Largest implicants containing a given minterm For example m(0,) is an implicant but not a prime implicant, because m(0,,,3) is a larger implicant containing those minterms. However, m(,5) is a prime implicant no larger implicant exists that contains minterm 5. Chapter <8>

9 Prime Implicant Table s Table Quine-McCluskey Example ABC Y of 's 0 Size 000 m0 00 m 00 m 0 m3 0 m5 0 m6 Size 00- m(0,) 0-0 m(0,) 0- m(,3) -0 m(,5)* 0- m(,3) -0 m(,6)* Size m(0,,,3)* List prime implicants. Prime m(0,,,3) m(,5) m(,6) Minterms ABC Show which of the required minterms each includes. Chapter <9>

10 Prime Implicant Table s Table Quine-McCluskey Example ABC Y of 's 0 Prime m(0,,,3) m(,5) m(,6) Size 000 m0 00 m 00 m 0 m3 0 m5 0 m6 Minterms Size 00- m(0,) 0-0 m(0,) 0- m(,3) -0 m(,5)* 0- m(,3) -0 m(,6)* Size m(0,,,3)* ABC Select columns with only. Corresponding prime implicants must be included in equation. Chapter <0>

11 Prime Implicant Table s Table Quine-McCluskey Example ABC Y of 's 0 Prime m(0,,,3) m(,5) m(,6) Size 000 m0 00 m 00 m 0 m3 0 m5 0 m6 Minterms Size 00- m(0,) 0-0 m(0,) 0- m(,3) -0 m(,5)* 0- m(,3) -0 m(,6)* Size m(0,,,3)* ABC Select columns with only. Corresponding prime implicants must be included in equation. Chapter <>

12 Prime Implicant Table s Table Quine-McCluskey Example ABC Y of 's 0 Prime m(0,,,3) m(,5) m(,6) Size 000 m0 00 m 00 m 0 m3 0 m5 0 m6 Minterms Size 00- m(0,) 0-0 m(0,) 0- m(,3) -0 m(,5)* 0- m(,3) -0 m(,6)* Y = A + BC + BC ABC Chapter <> Size m(0,,,3)* Select columns with only. Corresponding prime implicants must be included in equation.

13 Quine-McCluskey Example ABCD Y Chapter <3>

14 s Table Quine-McCluskey Example ABCD Y of 's 3 Size 000 m 000 m 000 m4 000 m8 00 m3 00 m6 00 m9 00 m0 0 m4 Chapter <4>

15 s Table Quine-McCluskey Example ABCD Y of 's 3 Size 000 m 000 m 000 m4 000 m8 00 m3 00 m6 00 m9 00 m0 0 m4 Size 00- m(,3) -00 m(,9) 00- m(,3) 0-0 m(,6) -00 m(,0) 0-0 m(4,6) 00- m(8,9) 0-0 m(8,0) -0 m(6,4) -0 m(0,4) Chapter <5>

16 s Table Quine-McCluskey Example ABCD Y of 's 3 Size 000 m 000 m 000 m4 000 m8 00 m3 00 m6 00 m9 00 m0 0 m4 Size 00- m(,3) -00 m(,9) 00- m(,3) 0-0 m(,6) -00 m(,0) 0-0 m(4,6) 00- m(8,9) 0-0 m(8,0) -0 m(6,4) -0 m(0,4) Size m(,6,0,4) Chapter <6>

17 s Table Quine-McCluskey Example ABCD Y of 's 3 Size 000 m 000 m 000 m4 000 m8 00 m3 00 m6 00 m9 00 m0 0 m4 Size 00- m(,3)* -00 m(,9)* 00- m(,3)* 0-0 m(,6) -00 m(,0) 0-0 m(4,6)* 00- m(8,9)* 0-0 m(8,0)* -0 m(6,4) -0 m(0,4) Size m(,6,0,4)* Chapter <7>

18 Prime Implicant Table s Table Quine-McCluskey Example ABCD Y of 's 3 Size 000 m 000 m 000 m4 000 m8 00 m3 00 m6 00 m9 00 m0 0 m4 Prime m(,6,0,4) m(,3) m(,9) m(,3) m(4,6) m(8,9) m(8,0) Size 00- m(,3)* -00 m(,9)* 00- m(,3)* 0-0 m(,6) -00 m(,0) 0-0 m(4,6)* 00- m(8,9)* 0-0 m(8,0)* -0 m(6,4) -0 m(0,4) Minterms Chapter <8> Size m(,6,0,4)* ABCD

19 Prime Implicant Table Quine-McCluskey Example ABCD Y Prime m(,6,0,4) m(,3) m(,9) m(,3) m(4,6) m(8,9) m(8,0) Minterms ABCD Chapter <9>

20 Prime Implicant Table Quine-McCluskey Example ABCD Y Prime m(,6,0,4) m(,3) m(,9) m(,3) m(4,6) m(8,9) m(8,0) Minterms ABCD Chapter <0>

21 Prime Implicant Table Quine-McCluskey Example ABCD Y Prime m(,6,0,4) m(,3) m(,9) m(,3) m(4,6) m(8,9) m(8,0) Minterms ABCD Chapter <>

22 Prime Implicant Table Quine-McCluskey Example ABCD Y Prime m(,6,0,4) m(,3) m(,9) m(,3) m(4,6) m(8,9) m(8,0) Minterms ABCD Chapter <>

23 Prime Implicant Table Quine-McCluskey Example ABCD Y Prime m(,6,0,4) m(,3) m(,9) m(,3) m(4,6) m(8,9) m(8,0) Minterms ABCD Y = CD + ABD + ABC Chapter <3>

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