1. Multi Function Quine-McCluskey 2-Level Minimization
Shantanu Dutt
University of Illinois at Chicago
Acknowledgement: Transcribed to Powerpoint by Huan Ren from Prof. Shantanu Dutt’s handwritten notes

2. Multiple Output Q-M General digital system: Multiple i/p’s
Multiple o/p’s
Share gates as much as possible to reduce cost
Sometimes sharing reduces cost; however indiscriminate sharing (e.g., having canonical SOPs for all functions) can actually increase cost.
Q: How to do this optimally?
A: Multi-function Q-M .

3. Multiple Output Q-M AB CD 00 01 11 10 AD α 00 AD 01 11 10 AB BD α ABD4 12 81 β AD 01 11 51
131 91 β α α α β β
Separate consideration 11 31 71
111 α α
151 α β β 10 2 6 14
101 AB β β A fα D BD α
ABD α B β D
No common terms →
Total cost =12 [six 2-i/p gates] →
12*2=24 transistors B fβ A A D

4. Multiple Output Q-M
Combined consideration (include common implicants—PIs of the function fa.fb—first):
Total cost
Total literals in distinct/different implicants + total # of implicants (non-distinct, i.e., a common impl. occurring multiple times, is counted as many times as it occurs) over all functions
(2+2+3)+4=11
4 trans. A
4 trans. D fα
6 trans. A B
22 transistors
4 trans. D
4 trans. fβ A B

5. Multiple Output Q-M AB CD 00 01 11 10 ACD ABC α 00 01 BD α 11 ACD α 10β
ABC α 00 41 α
121 81 β β 01 1 51
131 9 α α β BD α 11 11 31 71 α α
151 α β
ACD α 10 2 6x
141 10 AB
Taken separately X β X β β β β
ABD α β CD β
Total cost=18

6. Multiple Output Q-M (contd.)
Use heuristic: “Always include common implicants”
Total cost= 14+6=20
So how do we determine optimally when and when not to include a common term in an expression ?
This can be achieved by the multiple-function QM method.

7. Multiple-function QM Difference with single function QM
(1) Each minterm is tagged by “flag(s)” of the function it is in.
(2) For PI formation, the impl./min-term list is formed for all min-terms/DCs over all functions.
(3) Two impl’s (incl. minterms) can be combined only if they contain same common flags. The new impl. formed by this combining is flagged by the set of common (intersecting) flags.
(4) An impl. i that was combined to form a larger impl. can be “checked” only if the combined impl. contain all the flags that i contains.
(5) In the PIT, the cols are separated for each function, and each minterm for a function is listed under its function’s group—thus a minterm will have separate cols for each function it belongs to.

8. Multiple-function QM (contd.)
Difference with single function QM (contd.):
(6) In the PIT, each PI covers minterms only for functions whose flags it contains.
All other rules of PIT reduction applies, including the heuristic rules for cyclic PITs.
Example 1

9. Min term List 1 List 2 List 3 Flags 1 0001 α 1,3 00-1 1,3,5,7 0--1 PI28
1000 β
1,5
0-01
8,9,10,11
10--
PI3 3
0011
8,9
100-
5,7,13,15
-1-1
PI4 5
0101
8,10
10-0
9,11,13,15
1--1
PI5 9
1001
3,7
0-11 10
1010
5,7
01-1 7
0111
5,13
-101 11
1011
9,11
10-1 13
1101
α β
9,13
1-01 15
1111
10,11
101-
7,15
-111
11,15
1-11
13,15
11-1
PI1
Example 1:

10. * * * PI table fα fβ PIs 1 3 5 7 13 15 8 9 10 11 3+1 PI1αβ 2+1 PI2αC 1 * 2 * 3 4
More Rules:
(5) The covering rules apply across the entire PIT, not just in the sub-PIT corresponding to each function.
(6) The EPI defn, however, applies to only single functions, and if a multi-function PI g is an EPI for some funcs and not for all for which it is an impl, it is only chosen for the former function(s) and not right away for the latter. After this partial inclusion (for some funcs), g’s cost is reduced to 1, and it maybe chosen later based on this reduced cost. The reduction in cost corresponds to the AND gate cost of g which has already been incurred. The remaining 1 cost corresponds to the OR gate cost in each remaining function for which it may be included at a later time.
Note: We will provide an alternative for rule (6) in which an EPI/pseudo-EPI for one func is also automatically included in all other funcs for which it is an implicant. This alternative uses a sweep-up phase.

11. Min term List 1 List 2 List 3 Flags 4 0100 α 4,5 010- PI1 5,7,13,15
Example 2:
Min
term
List 1
List 2
List 3
Flags 4
0100 α
4,5
010-
PI1
5,7,13,15
-1-1
PI5 8
1000 β
8,12
1-00
PI2
12,13,14,15
11--
PI6 3
0011
3,7
0-11
PI3 5
0101
5,7
01-1 6
0110
5,13
-101 12
1100
6,14
-110
PI7 7
0111
12,13
110- 13
1101
α β
12,14
11-0 14
1110
7,15
-111 15
1111
13,15
11-1
PI4
14,15
111-

12. * * * * * fα fβ PIs C 3 4 5 7 13 15 8 12 14 PI1α3+1 PI2β3+1 PI3α3+16 C 7 * 5 * C 4
Total cost =13+5=18

13. Example 3
A=0
A=1 BC BC DE 00 01 11 10 DE 00 01 11 10 00 41 α
121 8 00 16 20 28 24 β 01 1 51
131 91 01 17
211
291 25 α β α β α β β 11 11 3 71 11
191
231 α β
151 α α
311
271 α β α α 10 2 61
141 10 10 18
221
301 26 α α β β
ABC
ACE
ABC
ABDE α α α
CDE
ADE β β α β
PI1
PI2
PI3
PI6
PI7
PI4
CDE α
PI5

14. Taking # of MTs covered and cost into consideration
PIs 4 5 6 7 9 13 15 19 23 27 31 12 14 21 29
3+1
PI1α
PI2αβ
PI3β
4+1
PI4α
PI5α
PI6β
PI7α 1 2 8 3 4 5 6 1 * * β 4 1 8 α 5 * C 2 * 7 6 * 3 *
Note: For every choice of a multi-func PI g as an EPI/pseudo-EPI for a function, mark each inclusion choice of g by the flag of the function(s) for which it is an EPI/pseudo-EPI. At the end, the flags will determine which functions g should be included in
Total cost=26

15. Three Output Function Example

16. Min
term
List 1
List 2
List 3
Flag
0000 αγ
0,2
00-0
PI2
4,5,6,7
01-- β
PI1 2
0010
PI10
0,8
-000 γ
PI3 4
0100
2,6
0-10
PI4 8
1000 βγ
PI11
2,10
-010
α β
PI5 5
0101
4,5
010- 6
0110
4,6
01-0 10
1010 α
8,10
10-0
PI6 12
1100 Α
PI12
5,7
01-1
β γ
PI7 7
0111
PI13
5,13
-101
PI8 13
1101
6,7
011- 15
1111
7,15
-111
PI9 β β β

17. γ γ γ fα fβ fγ PIs 2 7 10 4 5 8 2+1 PI1 β 3+1 PI2αγ PI3γ PI4β PI5αβ2 7 10 4 5 8
2+1
PI1 β
3+1
PI2αγ
PI3γ
PI4β
PI5αβ
PI6β
PI7βγ
PI8γ
PI9α
4+1
PI10α
PI11
PI12α
PI13α 5 1 α 2 5 γ C γ
New cost = 1 3 γ β 4 γ β γ β 2 2 3 3 1 1

18. Reduced PIT γ γ fα f γ PIs 7 8 3+1 PI3γ PI7βγ PI9α 4+1 PI11β PI13α A B5 1 C γ 2 γ β α 3 6 γ γ
New cost = 1

19. Multi-function QM: Rule 7: Alternative to Rule 6
If a PI is found to be essential for one or more functions that it covers, delete it completely. Also, include it in other functions fj (for which it is currently non-essential) whose minterms it covers only if not all such MTs of fj have been deleted so far.
At the end of the PIT phase when all MTs have been covered do the following sweep-up phase
In each function fj’s expression, mark each implicant of fj that was included in it because it was either an essential-PI or pseudo-essential PI for fj.
For each unmarked implicant PIk in a function fj , delete PIk from fj’s expression only if each MT of fj that it covers is also covered by some marked implicant of fj.

20. Utility of the Sweep-up Phase
Example: 1 2 4
PIs fα fβ
PIjαβ
PIkβ
PIlβ
PIrβ
*αβ 1 …. *β 4 C 3 *β 2
fα =PIj+……
fβ=PIj+PIk+PIr
We thus get fβ =PIk+PIr, reducing the cost by 1.
Deleted in the sweep-up phase.
Exercise: Use this alternative to rule 6 to find a minimum-cost covering of the problem in Example 3 (PIT given on slide 14)