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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 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 . Multiple Output Q-M f m(1,3,5,7,13,15) f m(8,9,10,11,13,15) Separate consideration f AD BD cost 6 A D B D fα f AB AD cost 6 B A A D fβ CD AB 00 01 11 10 ADα 00 0 8 01 1 11 3 10 2 4 12 1α 5 1α 13 1α 7 1α 15 6 1α β 1α β 14 BDα 1β 9 1 β ADβ 11 1β 10 1β ABβ ABDαβ No common terms → Total cost =12 [six 2-i/p gates] → 12*2=24 transistors Multiple Output Q-M Combined consideration (include common implicants— PIs of the function f.f—first): f AD ABD f AB ABD 22 transistors Total cost A D A B D A B 4 trans. 4 trans. fα 6 trans. 4 trans. 4 trans. fβ 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 Multiple Output Q-M f m(3,4,5,7,13,15) CD ABCα 00 4 1α 12 01 1 5 1α 13 11 3 7 1α 15 6 xβ 14 ACDα 10 f ACD ABC BD cost 8 3 11 f AB ACD cost 5 2 7 Total cost=18 00 01 11 0 f m(8,12,13,14,15) d(6) Taken separately AB 2 1α Xβ ABDαβ 10 ACDβ 1β 8 1α β 9 1α β 1β 1β 11 10 CDβ Xβ BDα ABβ Multiple Output Q-M (contd.) Use heuristic: “Always include common implicants” f ACD ABC ABD f ACD ABD BCD 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. 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. 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 f m(1,3,5, 7,13,15) f m(8,9,10,11,13,15) Min term 1 List 1 Flags 0001 α Min term 1,3 List 2 Flags 00-1 α Min term 1,3,5,7 List 3 Flags 0--1 α PI 2 8 1000 β 1,5 0-01 α 8,9,10,11 10-- β PI 3 3 0011 α 8,9 100- β 5,7,13,15 -1-1 α PI 4 5 0101 α 8,10 10-0 β 9,11,13,15 α Example 1: 9 1001 β 3,7 0-11 10 1010 β 5,7 01-1 α 7 0111 α 5,13 -101 α 11 1011 β 9,11 10-1 β 13 1101 α β 9,13 1-01 β 15 1111 α β 10,11 101- β -111 α 11,15 1-11 β 7,15 1--1 β PI 5 f m(1,3,5, 7,13,15) f m(8,9,10,11,13,15) PI table 1 1 1 1 5 5 2 2 2 fα PIs *5 * 1 * 2 3+1 PI1αβ 2+1 PI2α 2+1 PI3β 3 2+1 PI4α 4 2+1 PI5β C 1 3 5 7 2 5 5 fβ 13 15 8 9 10 11 13 15 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. f PI1 PI2 f PI1 PI3 Example 2: Min term 4 f m(3,4,5,7,13,15) f m(8,12,13,14,15) d(6) List 1 Flags 0100 α Min term 4,5 List 2 Min Flags 010- α List 3 term PI1 5,7,13,15 -1-1 α PI 5 8 1000 β 8,12 1-00 β PI2 12,13,14,15 11-- β PI 6 3 0011 α 3,7 0-11 5 0101 α 5,7 01-1 α 6 0110 β 5,13 -101 α 12 1100 β 6,14 -110 β 7 0111 α 12,13 110- β 13 1101 α β 12,14 11-0 β 14 1110 β 7,15 α 15 1111 α β 13,15 11-1 αβ 14,15 111- β -111 α PI3 PI7 PI4 3 1 1 3 7 7 2 2 fα PIs * * * C 1 PI1α3+1 2 PI2β3+1 3 PI3α3+1 PI4αβ3+1 * 7 * 5 PI5α2+1 C C 6 4 3 4 5 7 5 5 fβ 13 15 8 12 13 14 15 PI6β2+1 PI7β3+1 f PI1 PI3 PI5 f PI 2 PI 6 5 Total cost =13+5=18 Example 3 DE BC A=0 00 01 11 10 00 0 4 1α 12 1β 8 01 1 5 1α β 13 1α β 9 11 3 7 1α β 15 10 2 6 1α 14 1α β 1β DE 1α 11 10 ABCα ACEαβ ABCβ ABDEα PI1 PI3 PI2 PI4 BC A=1 00 01 11 10 00 16 20 28 24 01 17 21 1β 29 25 11 19 23 1α 31 10 18 22 1 α 30 1α CDEβ CDEα PI5 PI6 1β 27 1α 1β 1α 26 ADEα PI7 Taking # of MTs covered and cost into consideration fα C PIs *β1 *α 8 * 5 * 2 3+1 PI1α 3+1 1 PI2αβ 3+1 PI3β 4+1 PI4α 7 3+1 PI5α 3+1 PI6β 3+1 PI7α * * 6 3 4 5 6 7 9 13 15 fβ 19 27 31 5 7 12 13 14 15 21 29 4 5 4 6 6 4 8 C 23 4 5 1 2 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 f PI1 PI2 PI4 +PI7 f PI2 PI3 +PI6 Total cost=26 Three Output Function Example f (A, B, C, D) m(0, 2, 7,10) d(12,15) f (A, B, C, D) m(2,4,5) d(6, 7,8,10) f (A, B, C, D) m(2,7,8) d(0,5,13) Min term List 1 Min term Flag 0 0000 αγ 2 0010 αγβ PI1 List 2 Min Flag term 0,2 00-0 αγ PI2 4,5,6,7 01-- β 0,8 -000 γ PI3 2,6 0-10 β PI4 -010 α β PI5 0 4 0100 β 8 1000 βγ 5 0101 βγ 4,5 010- β 6 0110 β 4,6 01-0 β 10 1010 α 8,10 10-0 β PI6 12 1100 Α 5,7 01-1 β γ PI7 5,13 -101 γ PI8 6,7 011- β 7,15 -111 α 7 PI11 2,10 β β 0111 αγ PI1 2 PI1 3 13 15 1101 γ 1111 α List 3 PI9 PI1 fα PIs αγ 0 2 7 10 1 2+1 PI1 β 2 3+1 PI2αγ New cost = 1 fβ 2 4 fγ 5 2 3+1 PI3γ 3+1 PI5αβ 3+1 PI6β 3+1 PI7βγ C 3+1 PI8γ 3+1 PI9α 4 4+1 PI10αβ γ 4+1 PI11β 4+1 PI12α γ 4+1 PI13αβ 8 5 3+1 PI4β 3 7 γ 2 2 3 3 1 1 5 γ f =PI 2 PI5 PI13 f =PI1 PI5 Reduced PIT PIs γ 4 3+1 PI3γ C 3+1 PI7βγ 1 3+1 PI9 2 4+1 C α 3 4+1 γ fα fγ 7 7 8 f =PI 2 PI3 PI13 A 5 PI13 γ D PI2 αβ γ fα 6 γ New cost = 1 C PI1 α PI11β B 3 4 PI3 PI4 PI5 fβ fγ 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. Utility of the Sweep-up Phase Example: 1 PIs 1 fα 1 2 4 2 fβ αβ PIjαβ β PIkβ 4 PIlβ 3 PIrβ 2 * 1 …. *C β * 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)