Multi Function Quine-McCluskey 2-Level Minimization

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. α α β β BD α Multiple Output Q-M AB CD 00 01 11 10 AD 00 0 4 12 81 α β AD 01 11 51 131 91 β α α β • Separate consideration 11 31 71 111 α α 151 β AB 10 2 6 14 101 β β A fα D 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 4 trans. • 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 A 4 trans. D fα 6 trans. A B 4 trans. 22 transistors D 4 trans. fβ A B

5. Multiple Output Q-M α α β β ABD BD α α β CD β X X β β AB CD 00 01 11 10 ACD β ABC 00 0 41 121 81 α α β β 01 1 51 131 9 α 11 11 31 71 α α 151 ACD α 10 2 6x 141 10 AB • Taken separately β β 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. Example 1:

10. * 5 5 5 5 5 * * 3 4 C 2 1 1 1 1 2 2 2 2 1 • PI table 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. Example 2:

12. * * * C C * * 1 1 2 2 2 3 5 7 6 1 4 3 5 5 5 7 7 3 Total cost =13+5=18

13. α α α α β β β β Example 3 A=0 A=1 BC BC DE 00 01 11 10 DE 00 01 11 10 00 0 41 121 8 00 16 20 28 24 α β 01 01 1 51 131 91 17 211 291 25 α β β 11 11 3 71 11 191 231 151 α α 311 271 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. * β 1 α * * * C * * 4 5 1 3 1 2 2 3 4 7 5 8 6 8 4 5 6 6 4 Total cost=26 • Taking # of MTs covered and cost into consideration 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

15. Three Output Function Example

16. β β β

17. γ α C 2 1 1 5 2 2 3 3 3 1 4 5 γ New cost = 1 γ β γ β γ β

18. γ A B C D γ C PI1 fα PI2 C PI3 fβ α New cost = 1 1 2 PI4 fγ 4 4 3 3 5 6 PI5 Reduced PIT γ γ β γ

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