Logic Optimization 1. Logic Optimization. Optimization vs. Tradeoff. Two-level Logic Optimization. ECE 474A/57A Computer-Aided Logic Design

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1 ECE 474A/57A Computer-Aided Logic Design Logic Optimition Logic Optimition We now know how to build digitl circuits How cn we build better circuits? Let s consider two importnt design criteri Del the time from inputs chnging to new correct stble output Sie the number of trnsistors Assumption Ever gte hs del of gte-del Ever gte input requires 2 trnsistors Ignore inverters w w = w + w sie (trnsistors) trnsistors 2 gte-dels 5 = w(+ ) = w 2 4 trnsistors gte-del w 2 2 = w del (gte-dels) Trnsforming to 2 represents n optimition: Better in ll criteri of interest ECE 474/575 of 49 Digitl Design Copright 26 rnk Vhid ECE 474/575 2 of 49 Two-level Logic Optimition Optimition vs. Trdeoff Two-level logic Circuit with onl two levels (ORed AND gtes) Bsicll sum-of-products form An eqution written s n ORing of product terms Are these two-level logic? = + b c c d e Optimition - Defined s better in ll criteri of interest Del nd sie - we consider sie minimition onl (2-level logic onl) In relit requires blnce of mn criteri metrics Cost, relibilit, time-to-mrket, etc... Trdeoff - Improves some, but worsens other, criteri of interest = bc + c + de j k l m n = ((jk) + l) + mn r s t u = (rs) (tu) technicll es, but not wht we men in terms of logic minimition 32 trnsistors 2 gte-dels = Reduced number of gtes Reduced sie of gte (i.e. number of inputs) 2 trnsistors 2 gte-dels 4 trnsistors w 2 gte-dels G w G = w + w + w Reduced number of gtes Incresed del 2 trnsistors 3 gte-dels Digitl Design Copright 26 rnk Vhid ECE 474/575 3 of 49 Digitl Design Copright 26 rnk Vhid = + ECE 474/575 = w(+) + 4 of 49

2 Preto Points Combintionl Logic Optimition nd Trdeoffs We obviousl prefer optimitions, but often must ccept trdeoffs You cn t build cr tht is the most comfortble, nd hs the best fuel efficienc, nd is the fstest ou hve to give up something to gin other things Mn options in solution spce Preto point Point in solution spce in which no other point better in ll metrics Shown in red Preto points ield the trde-off curve sie sie del del Optimitions All criteri of interest re improved (or t lest kept the sme) Trdeoffs Some criteri of interest re improved, while others re worsened Two-level sie optimition using lgebric methods Gol: circuit with onl two levels (ORed AND gtes), with minimum trnsistors Though trnsistors getting cheper (Moore s Lw), the still cost something Define problem lgebricll Sum-of-products ields two levels = bc + bc is sum-of-products; G = w( + ) is not. Trnsform sum-of-products eqution to hve fewest literls nd terms Ech literl nd term trnsltes to gte input, ech of which trnsltes to bout 2 trnsistors Ignore inverters for simplicit Emple = = ( + ) + ( + ) = * + * = + m n = 6 gte inputs * 2 trnsistor/input = 2 trnsistors sie Digitl Design Copright 26 rnk Vhid del ECE 474/575 5 of 49 Digitl Design Copright 26 rnk Vhid ECE 474/575 6 of 49 Boolen Algebr Algebric Two-Level Sie Minimition Uniting Theorem How do we use Boolen lgebr to obtin fewest literls nd terms?. = b. + = 2. = 2b. + = 3. = = 3b. + = + = 4. If =, then = 4b. If =, then = 5. = 5b. + = 6. = 6b. + = 7. = 7b. + = 8. = 8b. + = 9. =. = (Commuttive) b. + = +. ( ) = ( ) (Associtive) b. + ( + ) = ( + ) + 2. ( + ) = + (Distributive) 2b. + ( ) = ( + ) ( + ) 3. + = (Absorption) 3b. ( + ) = 4. + = (Combining) 4b. ( + ) ( + ) = 5. ( ) = + (DeMorgn s Theorem) 5b. ( + ) = 6. + = + 6b. ( + ) = ECE 474/575 7 of 49 Multipl out to sum-of-products, then ppl Uniting Theorem b + b = (b + b ) = * = Combining terms to eliminte vrible (ormll clled the Uniting theorem ) Sometimes fter combining terms, cn combine resulting terms = = ( + ) + ( + ) = * + * = + G = G = ( +) + (+ ) G = + (now do gin) G = ( +) G = Digitl Design Copright 26 rnk Vhid del = 2 gte del sie = 6 * 2 = 32 trnsistors del = 2 gte del sie = 6 * 2 = 2 trnsistors ECE 474/575 8 of 49 2

3 Algebric Two-Level Sie Minimition Dupliction Algebric Two-Level Sie Minimition Comple nd Error Prone Duplicting term sometimes helps Note tht doesn t chnge function c + d = c + d + d = c + d + d + d + d... Algebric Mnipultion Which rules to use nd when? Es to miss seeing possible opportunities to combine terms = + + = = (+ ) + ( +) = + (, b, c) = b c + bc + b + b (, b, c) = b c + bc + b (, b, c, d) = b cd + c d + b d + cd + bcd + c d (, b, c, d) = d (, b, c, d, e, f, g) = b c + d e f + f + eg + bcd e f g + bc efg + c (, b, c, d, e, f, g) =? Digitl Design Copright 26 rnk Vhid ECE 474/575 9 of 49 Digitl Design Copright 26 rnk Vhid ECE 474/575 of 49 K-mps (Krnugh Mps) Two-Level Sie Minimition Using K-mps Grphicl method to help us find opportunities to combine terms Grphicl method to help us find opportunities to combine terms Crete mp where djcent minterms differ in one vrible Cn clerl see opportunities to combine terms look for djcent s w G H w ECE 474/575 of 49 Generl K-mp method. Convert the function s eqution into sum-ofproducts form 2. Plce s in the pproprite K-mp cells for ech term 3. Cover ll s b drwing the fewest lrgest circles, with ever included t lest once; write the corresponding term for ech circle 4. OR ll the resulting terms to crete the minimied function. Emple: Minimie G = + b c + b*(c + bc ) Step - Convert to sum-of-products Step 2 - Plce s in the pproprite cells b c Step 3 - Cover s Step 4 - OR terms G = + b c + bc + bc G bc G bc G = + c ECE 474/575 2 of 49 bc c 3

4 Two-Vrible K-Mple Emple Generlied Three-Vrible K-Mp ill in ech cell with corresponding vlue of Drw circles round djcent s Groups of, 2 or 4 Circle indictes optimition opportunit We cn remove vrible To obtin function OR ll product terms contined in circles Mke sure ll s re in t lest one circle 2 2 Three-Vrible Mp b c m m m2 m3 m4 m5 m6 m7 Truth tble bc m m m3 m2 m4 m5 m7 m6 Truth tble REMEMBER: K-mp grphicll plce minterms net to ech other when the differ b one vrible = ( + ) (2 + 2) 2 () () 2 ECE 474/575 3 of 49 m cnnot be plced net to m2 ( b c, bc ) m cn be plced net to m3 ( b c, bc) m2 cn be plced net to m3 ( bc, bc) ECE 474/575 4 of 49 Three-Vrible K-Mp Optimition Guidelines Three-Vrible K-Mp Optimition Guidelines Circles cn cross left/right sides Remember, edges re djcent Minterms differ in one vrible onl Circles must hve, 2, 4, or 8 cells 3, 5, or 7 not llowed 3/5/7 doesn t correspond to lgebric trnsformtions tht combine terms to eliminte vrible Circling ll the cells is OK unction just equls K L E Two djcent s mens one vribles cn be eliminted Sme s in two-vrible K-mps our djcent s mens two vribles cn be eliminted Mkes intuitive sense those two vribles pper in ll combintions, so one must be true Drw one big circle shorthnd for the lgebric trnsformtions bove our djcent cells cn be in shpe of squre G G H G = G = ( ) (must be true) G = ( ( +) + (+ )) G = ( +) G = G = + G = (+ ) G = Drw the biggest circle possible, or ou ll hve more terms thn rell needed H = ( ppers in ll combintions) ECE 474/575 5 of 49 ECE 474/575 6 of 49 4

5 Three-Vrible K-Mp Optimition Guidelines our-vrible K-Mp Optimition Guidelines Ok to cover twice Just like duplicting term Remember, c + d = c + d + d I our-vrible K-mp follows sme principle Left/right djcent Top/bottom lso djcent No need to cover s more thn once Yields etr terms not minimied The two circles re shorthnd for: I = I = I = ( + ) + ( ) I = ( ) + () J Adjcent cells differ in one vrible Two djcent s men two vribles cn be eliminted our djcent s mens two vribles cn be eliminted Eight djcent s mens three vribles cn be eliminted =w + G= ECE 474/575 7 of 49 ECE 474/575 8 of 49 our-vrible K-Mple Emple our-vrible K-Mple Emple - Continued Minimie: H = b (cd + c d ) + b c d + b cd + bd + bcd Minimie: H = b (cd + c d ) + b c d + b cd + bd + bcd. Convert to sum-of-products H = b cd + b c d + b c d + b cd + bd + bcd H c d b. Convert to sum-of-products H = b cd + b c d + b c d + b cd + bd + bcd b c d b cd bd bcd H c d b 2. Plce s in K-mp cells b cd b c d ECE 474/575 9 of 49 ECE 474/575 2 of 49 5

6 our-vrible K-Mple Emple - Continued our-vrible K-Mple Emple - Continued Minimie: H = b (cd + c d ) + b c d + b cd + bd + bcd Minimie: H = b (cd + c d ) + b c d + b cd + bd + bcd. Convert to sum-of-products H = b cd + b c d + b c d + b cd + bd + bcd H c d b. Convert to sum-of-products H = b cd + b c d + b c d + b cd + bd + bcd H c d b 2. Plce s in K-mp cells 2. Plce s in K-mp cells 3. Cover s bc 3. Cover s bc bd 4. OR resulting terms bd b d unn-looking circle, but remember tht left/ right djcent, nd top/bottom djcent ECE 474/575 2 of 49 b d H = b d + bc + bd ECE 474/ of 49 Lrger N-Vrible K-Mps Don t Cre Input Combintions Grphicl minimiing b hnd 5 nd 6 vrible mps eist, but hrd to use M not ield minimum cover depending on order we choose Is error prone Minimition thus tpicll done b utomted tools b cd cd b m m m3 m2 m6 m7 m9 m8 m4 m5 m7 m6 m2 m2 m23 m22 m2 m3 m5 m4 m28 m29 m3 m3 m8 m9 m m m24 m25 m27 m26 ef = ef = Don t Cre Input Input combintion tht the designer doesn t cre wht the output is i.e. input condition cn never occur Thus, mke output be or for those cses in w tht best minimies the eqution Represented s Xs in K-mp bc = nd bc = re unused inputs b c Z b cd m m m3 m2 m4 m5 m7 m6 m2 m3 m5 m4 m8 m9 m m b cd m6 m7 m9 m8 m2 m2 m23 m22 m28 m29 m3 m3 m24 m25 m27 m26 b cd m32 m33 m35 m34 m36 m37 m39 m38 m44 m45 m47 m46 m4 m4 m43 m42 b cd m48 m49 m5 m5 m52 m53 m55 m54 m6 m6 m63 m62 m56 m57 m59 m58 including this term doesn t help us G bc e = e = ive-vrible Mp ef = ef = Si-vrible Mp ECE 474/ of 49 including this term enbles better minimition ECE 474/ of 49 6

7 Simplified Nottion for Sum-of-Products orm Generlied Three-Vrible K-Mp Insted of listing ech product, simpl list the minterm number (, b) = m(, 2) = m + m2 m minterms, M mterms (, b) = m(4, 5, 6, 7) Don t forget column is followed b 2 - b b b c bc b b ECE 474/ of 49 ECE 474/ of 49 Generlied Three-Vrible K-Mp our-vrible K-Mple Emple (, b) = m(, ) + d(4, 5) d don t cres (, b, c, d) = m(4, 5,, 5) Don t forget in 4-vrible K-mp, columns nd rows re out of sequence too (,,, ) b c bc b c d H c d b ECE 474/ of 49 ECE 474/ of 49 7

8 Ect Algorithms vs. Heuristic Quine-McCluske Overview Algorithm inite set of instructions/steps to solve problem Termintes in finite time t known end stte Mn lgorithms cn eist tht solve the sme problem Wht mkes one lgorithm better thn nother? Optimlit best qulit solution found Efficienc good qulit solution found fst Ect Algorithm Developed in the mid-5 s inds the minimied representtion of Boolen function Provides sstemtic w of generting ll prime implicnts then etrcting minimum set of primes covering the on-set Accomplishes this b repetedl ppling the Uniting theorem Uniting theorem: b + b = (b+b ) = * = Ect Algorithm inds optiml solution M not be efficient Heuristic Efficient inds good solution, but not necessril optiml ECE 474/ of 49 ECE 474/575 3 of 49 Review Definitions Review Definitions Minterm product term whose literls include ever vrible of the function ectl once in true or complemented form On-set All minterms tht define when = Off-set All minterms tht define when = (, b, c) = b c + b vribles:, b, c minterms: b c b bc bc on-set: b c, bc, bc off-set: b c, bc, bc, b c, b c Implicnt An product term (minterm or other) tht when cuses = On K-mp, n legl (but not necessril lrgest) circle Prime implicnt Mimll epnded implicnt n further epnsion would cover s not in on-set Essentil prime implicnt The onl prime implicnt tht covers prticulr minterm in function s on-set Importnce: We must include ll essentil PIs in function s cover In contrst, some, but not ll, non-essentil PIs will be included bc b c G bc bc b c bc bc b b 4 implicnts 2 prime Implicnts 4 prime implicnts 2 essentil prime implicnts b b ECE 474/575 3 of 49 Note: We use K-mps re for illustrtion purposes onl ECE 474/ of 49 8

9 Quine-McCluske Algorithm Quine-McCluske Emple. ind ll the prime implicnts 2. ind ll the essentil prime implicnts 3. Select miniml set of remining prime implicnts tht covers the on-set of the function Minimie = b c + b c + b c + bc + bc Step : ind ll the prime implicnts List ll elements of on-set nd don t cre set, represented s binr number Group minterms ccording to the number of s in the minterm b c () G () group G contins ll minterms contining ero s b c () G () group G contins ll minterms contining one b c bc (5) (6) (5) (6) group contins ll minterms contining two s bc (7) G3 (7) group G3 contins ll minterms contining three s this grouping strteg will help us compre the minterms sstemticll ECE 474/ of 49 ECE 474/ of 49 Quine-McCluske Emple Quine-McCluske Emple Step : ind ll the prime implicnts(cont ) Compre ech entr in Gi to ech entr in Gi+ If the differ b bit, we cn ppl the uniting theorem nd eliminte literl Add check to minterm/implicnt to remind us tht it is not prime implicnt (combined with nother element to form lrger implicnt) Step 2: ind ll essentil prime implicnts Crete prime implicnt chrt Columns re minterm indicies, rows re the prime implicnts we determined = b c + b c + b c + bc + bc () () () () () G G () () G G (,) - (,5) - no new implicnts re generted end of step (,) - (,5) - (5) (6) (5,7) - (6,7) - we hve found ll prime implicnts (ones without check mrks) (5,7) - (6,7) - G3 (7) ECE 474/ of 49 derived in Step ECE 474/ of 49 9

10 Quine-McCluske Emple Quine-McCluske Emple Step 2: ind ll essentil prime implicnts (cont ) Plce X in row if the prime implicnt covers the minterm Essentil prime implicnts re found b looking for rows with single X If minterm is covered b one nd onl one prime implicnt it s n essentil prime implicnt Add essentil prime implicnts to the cover Step 3: Select miniml set of remining prime implicnts tht covers the on set of the function Step 2 determined essentil prime implicnts, nd dded to cover Essentil prime implicnts m cover other minterms, cross out ll minterms covered b the prime implicnts Minterm onl needs to be covered once essentil prime implicnts (,) - = b + b (,) - = b + b (,5) - (,5) - (5,7) - (5,7) - (6,7) - (6,7) - ECE 474/ of 49 ECE 474/ of 49 Quine-McCluske Emple Quine-McCluske Emple Step 3: Select miniml set of remining prime implicnts tht covers the on set of the function (cont ) Bsed on which minterms re left, dd miniml set of prime implicnts to cover Summr Is this n optiml solution? YES. We generte ll the minterms nd mke sure the re ll covered b the prime implicnts Onl minterm 5 remins either prime implicnt (,5) or (5,7) will work Is the solution unique? NOT NECESSARILY. There could be different sets of minimum covers. (,) - (,5) - = b + b + b c (5,7) - (6,7) - ECE 474/ of 49 ECE 474/575 4 of 49

11 Quine-McCluske Emple 2 Quine-McCluske Emple 2 Minimie = w + w + w + w + w + w + w + w + w + w Step : ind ll the prime implicnts List ll elements of on-set nd don t cre set, represented s binr number Group minterms ccording to the number of s in the minterm w () w (3) w (2) w (4) w (7) w (6) w (3) w (5) w (9) w () G G G3 () (2) (4) (3) (6) (9) (7) () (3) G4 (5) ECE 474/575 4 of 49 Step : ind ll the prime implicnts (cont ) Compre ech entr in Gi to ech entr in Gi+ G G G3 G4 If the differ b bit, we cn ppl the uniting theorem nd eliminte literl Add check to minterm/implicnt to remind us tht it is not prime implicnt () (2) (4) (3) (6) (9) (7) () (3) (5) (,2)? (,4)? (2,3)? (2,6)? (2,9)? N (4,3)? N (4,6)? (4,9)? N (3,7)? (3,)? (3,3)? N (6,7)? (6,)? N (6,3)? N (9,7)? N (9,)? (9,3)? (7,5)? (,5)? (3,5)? G (,2) - (,4) - G (2,3) - G3 (2,6) - (4,6) - (3,7) - (3,) - (6,7) - (9,) - (9,3) - (7,5) - (,5) - (3,5) - no new implicnts re generted end of step ECE 474/ of 49 G G (,2,4,6) -- (2,3,6,7) -- (3,7,,5) -- (9,,3,5) -- Quine-McCluske Emple 2 Quine-McCluske Emple 2 Step 2: ind ll essentil prime implicnts Crete prime implicnt chrt Columns re minterm indicies, rows re the prime implicnts we determined Plce X in row if the prime implicnt covers the minterm Essentil prime implicnts re found b looking for rows with single X Add essentil prime implicnt to the cover Step 3: Select miniml set of remining prime implicnts tht covers the on set of the function Cross out ll minterms covered b the prime implicnts Bsed on which minterms re left, dd miniml set of prime implicnts to cover Minterm 3 nd 7 remin either prime implicnt (2,3,6,7) or (3,7,,5) will work essentil prime implicnts (,2,4, 6) -- (2,3,6,7) = w + w (,2,4, 6) -- (2,3,6,7) = w + w + (3,7,,5) -- (3,7,,5) -- (9,,3,5) -- (9,,3,5) -- ECE 474/ of 49 ECE 474/ of 49

12 Quine-McCluske Emple 3 Petrick s Method Quine-McCluske Emple 3 Petrick s Method Wht if determining minimum prime implicnt cover is not so es? Assume we hve the implicnt tble below Determine prime implicnts, dd to cover Emple 3 (cont ) Remove minterms covered b prime implicnts Leves 3 minters m7, m3, nd m5 Which remining prime implicnts should we use to obtin the minimum cover? essentil prime implicnts (2,6) - (2,6) - (8,9) - = w + (8,9) - = w + (6,7) - (6,7) - (9, 3) - (9, 3) - (7, 5) - (7, 5) - (3, 5) - (3, 5) - ECE 474/ of 49 ECE 474/ of 49 Quine-McCluske Emple 3 Petrick s Method Quine-McCluske Emple 3 Petrick s Method Petrick s Method used to determine minimum cover. Reduce prime implicnt chrt b eliminting prime implicnt rows nd corresponding columns 2. Lbel rows of reduced prime implicnt chrt P, P2 3. orm logicl eqution which is true when ll columns re covered 4. Reduce to minimum sum of products b multipling out nd ppling X + XY = X 5. Ech term in solution represents covering solution Count number of terms in ech, choose one corresponding to the minimum number P P2 P3 P4 (2,6) - (8,9) - (6,7) - (9, 3) - (7, 5) - (3, 5) - P = (P + P3)(P2 + P4)(P3 + P4) P = (P + P3)(P2P3 +P2P4 +P4P3 + P4P4) P = (P + P3)(P2P3 +P2P4 +P4P3 + P4) P = (P + P3)(P2P3 +P2P4 +P4) P = (P + P3)(P2P3 + P4) P = PP2P3 + PP4 + P3P2P3 + P3P4 P = PP2P3 + PP4 + P2P3 + P3P P4P4 = P4 P4 + P4P3 = P4 P4 + P2P4 = P4 P3P2P3 = P2P3 inl cover = essentil prime implicnts + minimum prime implicnt cover Essentil Prime Implicnts w, Minimum prime implicnt cover list: (option - PP4) w,w (option 2 - P2P3) w, (option 3 - P3P4), w (2,6) - (8,9) - P (6,7) - P2 (9, 3) - P3 (7, 5) - P4 (3, 5) - P = PP4 + P2P3 + P3P An of these provide minimum cover (equl number of circles ) Minimied Eqution = w w Actull - PP2P3 + P2P3 = P2P3, so we cn eliminte term ltogether more terms thn other solutions An of these provide minimum cover ECE 474/ of 49 ECE 474/ of 49 2

13 Quine-McCluske Wht bout don t cres? Alterntive methods to determine Minimum Cover Row vs. Column Dominnce ECE 474/ of 49 3