Digital Circuit Engineering

Transcription

1 Digitl Ciruit Engineering DIGITAL VLSI Consensus, use Krnugh + + = n Distriutive (X + A)(X + ) = X + A DESIGN Simplifition YX + X = X Generl DeMorgn Asorption Y + XY = X + Y F(,,... z,+,.,,) F(,,... z,.,+,,) Crleton University 26 ig3krnghmps_xortena.fm p. Revise; Jnury 22, 27 Slie i Krnugh Mps Another form of the truth tle Lelling Mps Cirling Mps Minimizing Alger y Mps Preutions when irling Don t Cres Where on t res ome from - inry-coe Deiml (CD) igits Don t res on Krnugh Mps Appenix XORs logi mnipultion Using Krnugh Mps with XORs Crleton University Digitl Ciruits II p., ig3krnghmps_xortena.fmrevise; Jnury 22, 27 Comment on Slie i

2 Krnugh Mps; Equtions From Truth Tles A truth tle with F not yet fille in. F This orer is importnt This orer is importnt Rerw tle with hlves sie y sie hlf for = F hlf for = F Compt the tle = = F F Lel on the sies mp of F Chek tht moving one squre only hnges one input it it hnges 2 its hnge Lyout for Krnugh mp Lelling of the xes Lel on the sies Another wy of lelling Arevite lelling Comine lelling ig3krnghmps_xortena.fm p. 2Revise; Jnury 22, 27 Slie 2 Krnugh Mps; Equtions From Truth Tles Krnugh Mps Krnugh Mps The mp is like truth tle Eh squre on the mp represents ifferent input omintion. All possile input omintions re represente on the mp. The inputs re lelle roun the eges of the mp. Not insie the squres s shown on the right. Arrngement of the squres As one steps from one squre to the next, either up, own, left or right, only one it shoul hnge in single step. If one goes to the nerest igonl neighour, two its will hnge. one it hnge one it hnge Crleton University Digitl Ciruits II p. 3, ig3krnghmps_xortena.fmrevise; Jnury 22, 27 Comment on Slie 2

3 Krnugh Mps; Representing AND Terms Different Funtions using AND F= F= () F= F= All the squres where =, =. All the squres where =. F= F=? All the squres where Wrp roun F=? F=? ig3krnghmps_xortena.fm p. 4Revise; Jnury 22, 27 Slie 3 Krnugh Mps; Representing AND Terms Representing AND Terms Representing AND Terms Any single squre (Top row, first two mps) On these mps, ny single squre represents speifi vlues for three vriles, this is the sme s three term AND like Any two jent squres We me the mps so tht only one vrile hnges t time if one moves vertilly or horizontlly. (This is not true for igonl movements). Thus two jent squres lwys hve one ommon vrile. In the top row, thir mp, the squres n re. We n sy + = ( + ) = This shows tht ny two jent squres n e represente y two term AND. Any three jent squres You n only irle if the numer of squres is power of 2. Any four jent squres (Top row, fourth mp) There re two wys to look t this. One is tht ll the squres where = hve in them, hene one n esrie them s. Alterntely one n note tht squres tht re re = ( ) =. (ottom row, first two mps) These represent n respetively. Any eight jent squres If ll the squres on mp re, the funtion is F=. Crleton University Digitl Ciruits II p. 5, ig3krnghmps_xortena.fmrevise; Jnury 22, 27 Comment on Slie 3

4 Krnugh Mps; Joining AND Terms With ORs Joining F = F = F + F 2 + F 3 on the mp + + = F 2 = F 3 = F = + + OR together the AND terms, n ple them on one mp. The terms n overlp. + + = F 4 = F 5 = F 6 = F = + + Using the lrger terms (F 4, F 5, n F 6 ) gives smller expression for F. igger irles give smller gtes. ig3krnghmps_xortena.fm p. 6Revise; Jnury 22, 27 Slie 4 Krnugh Mps; Joining AND Terms With ORs Cirling Mp in Different Wys Cirling Mp in Different Wys Comining Mps The OR of two mps, is new mp in whih the squres re me if there is in either (or oth) of the initil mps Sme Mp, Different Expressions Tke the truth tle for oolen funtion F, written s mp. One n irle it in severl wys. First one n irle iniviul s. This gives long expression for F. Another wy is to rek it up s F, F 2 n F 3 s shown on the top line on the slie. A thir wy is to rek it into F 4, F 5, n F 6, s shown on the ottom line in the slie. This gives smller eqution. F F= F= F 4 + F 5 + F 6 = + + F= F + F 2 + F 3 = + + Crleton University Digitl Ciruits II p. 7, ig3krnghmps_xortena.fmrevise; Jnury 22, 27 Comment on Slie 4

5 Mps for, 2, 3, 4, 5 n 6 Inputs; Legl Cirles Mps for ifferent numers of input vriles One Two Three Four Five e e Six Simplifying Logi Cirle jent squres Must irle,2,4,8,6... squres (ones) 6 squres Digonl squres not jent Cirling with wrp roun ig3krnghmps_xortena.fm p. 8Revise; Jnury 22, 27 Slie 5 Mps for, 2, 3, 4, 5 n 6 Inputs; Legl Cirles Krnugh Mp Properties Krnugh Mp Properties Mps my hve ny numer of vriles, ut- Most hve 2, 3, 4, or 5 vrile. One vrile is simple, (2 squres). five veriles hs two 4x4 mps, one for when e=, n one for when e=. Six vriles hve 64 squres n were use in pre-omputer ys. Seven vriles is pst the limit of snity. You will hve 8 loks of 6 squres eh. Rules for irling s Ones on the mps n e irle to simplify logi. Only jent squres n e irle. Digonlly jent squres re not onsiere jent. Cirles must surroun, 2, 4, 8, 6... squres. Not 3, 5,6,7,9,... The mps wrp roun. A squre on n ege re jent to the squre on the opposite ege in the sme olumn (row). Lrger irles give simpler logi, ut- One must oey the ove rules. There re exeptions, prtiulrly with multi-output mps.. PROLEM irle the s on the two mps shown. Crleton University Digitl Ciruits II p. 9, ig3krnghmps_xortena.fmrevise; Jnury 22, 27 Comment on Slie 5

6 Krnugh Mps; Simplifying Equtions Cirling to Minimize the Logi Expression Simplify + + F = + + The term is reunnt F = The Consensus Theorem + + = + OR together the terms, n ple them on one mp. Simplify F = These re the s one must irle Use igger irle in the mile F = + + Using the lrger terms (irles) gives smller expression for F. ig3krnghmps_xortena.fm p. Revise; Jnury 22, 27 Slie 6 Krnugh Mps; Simplifying Equtions Simplifition of oolen Funtion Simplifition of oolen Funtion A oolen funtion n e efine in mny wys A truth tle A Krnugh mp without irles A irle Krnugh mp. A Σ of Π expression. A Π of Σ expression. et. Any funtion hs only one truth tle n only one unirle mp. There re usully severl wys of irling the mp or writing the lgeri expression for the sme funtion. One tries to fin the est efinition for some ojetive. Possile Ojetives Smller iruitry. Lower powere iruitry. Fster iruitry. Mking the iruit smller is usully goo strt towr mking the iruit fster n lower powere. Crleton University Digitl Ciruits II p., ig3krnghmps_xortena.fmrevise; Jnury 22, 27 Comment on Slie 6

7 Krnugh Mps; est Cirling to Reue Logi Simplifition With 4-Input Mps Simplify the funtion efine y this mp John s Solution F= + + Tom s Solution F= + + Usully igger is etter for irles Simplify the funtion efine y this mp John s Solution Tom s Solution F= F= + + ig3krnghmps_xortena.fm p. 2Revise; Jnury 22, 27 Slie 7 Krnugh Mps; est Cirling to Reue Logi Simplifition Simplifition Some Things to Do Use the lrgest Cirle possile John use when he shoul hve use. Try to voi unneessry overlp John hs two overlpping irles. Tom voie oth. 2. PROLEM Fin the simplest S of P expression for the logi funtion F efine y the Krnugh mp on the right. Get F with 9 letters. If it tkes more, o Pro. A. Mp of F Crleton University Digitl Ciruits II p. 3, ig3krnghmps_xortena.fmrevise; Jnury 22, 27 Comment on Slie 7

8 Krnugh Mps; est Cirling to Reue Logi 4-Input Mps, Cirling Four Corners Simplify the funtion John s Solution efine y this mp F= + + Tom s Solution F= + + Don t forget 4 orners Simplify the funtion John s Solution efine y this mp F= Your Solution essentil F= + ig3krnghmps_xortena.fm p. 4Revise; Jnury 22, 27 Slie 8 Krnugh Mps; est Cirling to Reue Logi Simplifition Theory Essentil Terms A term (irle) is essentil if it ontins t lest one tht nnot e irle y ny other irle, of the sme or lrger size.y Your solution, The term, is essentil in tht no other irle (exept smller ones) will over the squre =., is essentil in tht no other irle (exept smller ones) will over =., is essentil in tht no other irle (exept smller ones) will over = n. One must hve the irles for these essentil terms. Squres Not Covere y Essentil Terms All the squres exept n re overe y the essentil terms. One must look for terms to over these squres. This is the only ple there is hoie. There re three terms tht over one or oth of these squres. Choose the ones to give the simplest expressions. 3. PROLEM Fin the simplest Σ of Π expression for the mp ) on the right. Simplifition ) ) ) ). This is often lle n essentil prime implint. Crleton University Digitl Ciruits II p. 5, ig3krnghmps_xortena.fmrevise; Jnury 22, 27 Comment on Slie 8

9 Exmple of How One Gets Don t Cre Outputs inry Coe Deimls (CD) eiml igit inry representtion eiml igit inry representtion not use x x x x x x inry representtion Representing 3-igit numer in eiml with 2 its. Digit 2 7 Digit 9 Digit 3 Mp showing the vlues of for eh squre Mp showing the eiml equivlent of the input its x x x x 8 9 x x ig3krnghmps_xortena.fm p. 6Revise; Jnury 22, 27 Slie 9 Exmple of How One Gets Don t Cre Outputs inry-coe Deimls inry-coe Deimls These re use minly for sening numers to isplys whih people hve to re. Mny yers go they were use to o ommeril rithmeti. The story ws tht onverting eiml frtions to inry use smll errors whih oul umulte n throw off your nk ount. For exmple $.7 (eiml) = $.,,,... (inry) inry-oe eiml igits use 4 its. The tle shows tht four of the sixteen 4-it omintions re unuse. If one hs iruit whih hs inry-oe eiml inputs there will e four input omintions whih never hppen. If they never hppen, then one oes not hve to worry out the iruits output for these input omintions. These omintions re lle on t re inputs n n e use to simplify the iruit. 4. PROLEM Write the yer in inry oe eiml. In 23, you shoul hve 6 its. Crleton University Digitl Ciruits II p. 7, ig3krnghmps_xortena.fmrevise; Jnury 22, 27 Comment on Slie 9

10 Don t Cre Outputs; Where They Come From Rouning CD numers Roun 2-igit CD numers to -igit. 83 roun to 8 86 roun to 9 85 roun to 9 (ritrry hoie) If lest sig igit 5 - Sen inrement sig to next ig 8 6 Inrm 5 9 Design Ciruit Detet if CD igit x x x x 8 9 x x Mp lotions of CD igits F Digit Dig 5 Ciruit x x x x x x F=igit 5 re s show igits 5 ig3krnghmps_xortena.fm p. 8Revise; Jnury 22, 27 Slie Don t Cre Outputs; Where They Come From Rouning CD numers. Rouning CD numers. There re three prts to this prolem: () Design iruit to hek if CD igit 5. (2) Moify n er iruit to inrement CD igit. (3) Put it ll together. We will only o prt () in the slie ove. Prt 2 is prolem elow. Rouning Using the Don t Cres 5. PROLEM Design the prt (2) of the rouning iruit. A iruit tht inrement CD igit - One only nees hlf-ers, not full ers. - there is rry in from the rouning signl. If the originl igit ws 9, inrementing it will overflow. We will let + = A rel CD inrementer woul inrement 9 to give with rry out. 7 Inrm 8 Crleton University Digitl Ciruits II p. 9, ig3krnghmps_xortena.fmrevise; Jnury 22, 27 Comment on Slie

11 Krnugh Mps: Using Don t Cres Don t Cres In Krnugh Mps Detet if CD igit is 5 or more. Use of Don t Cres. The CD igits > 9 never hppen We on t re out output for them. Mke these outputs on the mp x x x x 8 9 x x Mp lotions of CD igits x x x x x x F= igit 5 F Digit Dig 5 Ciruit my e irle or not s esire. Cirle to minimize logi Here we irle 4 out of 6 s This mkes F= + + Wht s with this? Inputs to use the six outputs never hppens, ut if they i: - the 4 irle outputs woul now e, n - the 2 unirle ones woul now e. ig3krnghmps_xortena.fm p. 2Revise; Jnury 22, 27 Slie Krnugh Mps: Using Don t Cres Don t Cres In Krnugh Mps Don t Cres In Krnugh Mps If one hs input omintions tht never hppen, one oes not re wht outputs they generte euse those outputs n never hppen. We put on t res on the mp squres for input omintions tht never hppen. One n irle these on t res or not s onvenient. There is ommon error on the slie. The irle oul e extene to inlue ll of. This woul give the finl eqution s F= PROLEM Tke the hex igits,,2,3,4,5,6,7,8,9,a,,c,d,e,f. Plot their lotion on Krnugh mp in the sme wy the CD igits were plotte. Then esign logi iruit whih will use four its w,x,y,z (efining hex igit s input, n give high output if the igit is ivisile y 3, i.e. it is 3, 6, 9, C or F. PROLEM Design iruit whih will use four its,,, efining CD igit s input, n gives high output if the igit is ivisile y 3. Utilize the inputs tht nnot hppen, to give on t re outputs, n hene simplify the logi. Crleton University Digitl Ciruits II p. 2, ig3krnghmps_xortena.fmrevise; Jnury 22, 27 Comment on Slie

12 Krnugh Mps; Exmples With Don t Cres Simplifition With Don t Cres Simplify the funtion efine y this mp John s Solution F= + + Don t hve to irle ll the. Tom s Solution F= + 4 orners not so goo Simplify the funtion efine y this mp John s Solution F= Tom s Solution F= + + ig3krnghmps_xortena.fm p. 22Revise; Jnury 22, 27 Slie 2 Krnugh Mps; Exmples With Don t Cres Don t Cres In Krnugh Mps 8. PROLEM. One wy to esign omprtor for two inry numers is shown. It is me of loks, eh of whih ompres two its. A omprtor for two, 3-it, inry numers X=x 2 x x n Y=y 2 y y is shown. It uses three loks. Eh lok ompres x i with y i n the inputs A i+, i+, from the omprison one for higher orer its, to give outputs A i n i A typil lok is shown. We rop the umersome susripts n write A -, - to tell the A, inputs from the outputs. The - - in the 5th line of the truth tle inputs mens tht if A, =, then, no mtter wht x n y re, the output is A -, - =. Do not onfuse these with the on t res in the outputs,, whih re the result of input omintions tht never hppen. Complete the Krnugh mp for A - inluing the on t res. Then eue the expression for A - whih shoul hve 4 letters. x 2 y 2 x y x y A 3 = A 2 A A 3 = 2 A lok x y A - - A x y A lok lok A - = if x>y or A= - = if x<y or A= A= never hppens y xy A A x Mp of A - y xy A A x Mp of - Crleton University Digitl Ciruits II p. 23, ig3krnghmps_xortena.fmrevise; Jnury 22, 27 Comment on Slie 2

13 Mps With Don t Cres; Common Mistkes Don t Cres Common mistkes. Cirling on t res when there is no nee. Rememer you irle them only if onvenient. (top) 2. Over irling. (mi) 3. Forgetting wrp-roun. (mi, ot) 4. Not enlosing on t res whih woul mke the irle lrger. (mi, ot) ig3krnghmps_xortena.fm p. 24Revise; Jnury 22, 27 Slie 3 Mps With Don t Cres; Common Mistkes Common Mistkes with Don t Cres Common Mistkes with Don t Cres Top: The four orners woul e etter. There is no nee to inlue the upper two s. Mile: The four orners re reunnt. Also the top ovl oul e oule with wrp roun. ottom: The re ovl oul e wrppe roun to over four squres. Crleton University Digitl Ciruits II p. 25, ig3krnghmps_xortena.fmrevise; Jnury 22, 27 Comment on Slie 3

14 Appenix I: Krnugh Mps With XORs Using XORs XOR Sum of Prouts (XSofP) XORs re use inste of the ORs use in SofP Exmple of XSofP: C CD AD A D C D A In CMOS (n most other) logi fmilies XOR gtes re lrger n slower thn OR/NOR. Thus iruits re not oneptully esigne using XORs. However: For some very-fst logi XORs re uner twie the ost of NORs. Some iruits re esier to oneptully esign thinking out XORs. like: ers error-heking iruits Gry-oe to inry onverters. Quntum n DNA logi re esigne with XORs These logi types re not yet prtil ut soon? D ig3krnghmps_xortena.fm p. 26Revise; Jnury 22, 27 Slie 4 Appenix I: Krnugh Mps With XORs Using XORS Using XORS For very fst logi XORs re reltively more vntgeous. Very fst logi, is often ifferentil. Differentil gtes with output X n supply X lso t no ost. With oth X n X ville n XOR n e me for uner twie the ost of NAND. Two suh fst ifferentil logi type re: MOS Complimentry-Moe Logi (MCML) n Emitter-Couple Logi (ECL). Quntum n DNA logi 2 These logi tehnologies nturlly ten to use XOR rther thn NAND or NOR. Smith 3 gives more omplete ut simple review of XOR XNOR iruits. Sum-of-Prouts n Extene (or Exlusive)Sum-of-Prouts The sum-of-prouts form of funtion is strit OR of ANDs wy of writing funtion, there must e no rkets or long overrs; however single inverting rs re llowe. See the notes on ftoring. Exmples: Not sum-of-prout e e + ( + e) + + e The extene sum-of-prout (X-SofP) or (X-ΣofΠ) is sum-of-prout in whih OR is reple y XOR Exmples: e e. Prof. Mrek Perkowski t see lso 2. Mihel A. Nielsen n Is L. Chung, Quntum Computtion n Quntum Informtion, Cmrige Univ Press, Prof. D.W. Smith t Crleton University Digitl Ciruits II p. 27, ig3krnghmps_xortena.fmrevise; Jnury 22, 27 Comment on Slie 4

15 Commonly Use XOR Formuls A A = A A = Inversion of XOR A = A = A (I x) Commuttive X Y = Y X A A( C) = A AC No Dulity with n (Dx) Corllry, There is no (Dx2) Rules using XOR = = No rkets neee Assoitive Thus one n write A C A ( C) = (A ) C A A A C C Distriutive with AND C (A) C) = (A ) (A C) Disjuntive Theorem (Dj) When PQ =, where: PQ = squre-y-squre AND of P n Q Then: P + Q = P Q (if PQ=) P= Q= F = P + Q F = + Cirles on t overlp Hene PQ = F = P Q = ig3krnghmps_xortena.fm p. 28Revise; Jnury 22, 27 Slie 5 Commonly Use XOR Formuls XOR Formul Detils XOR Formul Detils A A = A A = = = An o numer of inversion rs inverts n XOR sequene. A C D = A C D = A C D = A ( C) D A ( C) = (A ) C You n put the rkets nywhere, hene you on t nee them. A( C) = (A) (AC) istriutive with AND (Dx). The wy to rememer 3 A + = A (A) Proof: A + = A (A) using Exmple = A (A) using Exmple 2 = = = = There is no istriutive lw with OR. There is no D2 (ul type) istriutive lw. (see Prolems. n. ) Exmples: A + = A A Proof: if A== then f= =, using =, = otherwise A= n f= A = A using X =X 2 (A) = A With XORs, rkets Proof: shoul not e ssume (A) = () (A) = ( A) using the Dx unless they on t = A using X =X mke ifferene. Writing A mkes it esy to onfuse ( A) with (A) Writing A C is OK sine ( A) C = (A C) Crleton University Digitl Ciruits II p. 29, ig3krnghmps_xortena.fmrevise; Jnury 22, 27 Comment on Slie 5

16 sis for XK-mps; K-Mp Prity Disjuntive Theorem (ont) When PQ...R = then P + Q +... R = P Q... R Extension of (Dj) to Prity When the o prity of P,Q,R = on ll squres of the mp then P+Q+R = P Q R The loops my overlp with prity, 2 loops, 4 loops... overlpping, provie G= on those squres. R= F = P + Q + R P= Cirles on t overlp Q= Hene PQR = squre-y-squre AND of P,Q n R = F = P Q R Prity= ( loop overs squre) Prity= (3 loops over squre) Prity= (no loops over squre) F = P Q R mp of F F = overlp on squre Prity= (2 loops over squre) ut G= in this squre G = mp of G Using norml Sum of prouts G = + + mp of G ig3krnghmps_xortena.fm p. 3Revise; Jnury 22, 27 Slie 6 sis for XK-mps; K-Mp Prity XOR Formul Detils 9. Alger Prolems PROLEM Prove tht: A C = A C = (A ) C. PROLEM Prove or isprove (i) (A+) C = (A ) C lterntely (A ) C = (A+) C (ii) (A+) C = (A C) + ( C). PROLEM Prove or isprove (i) A+( C) = A C (ii) A+( C) = (A ) + (A C) 2. PROLEM Prove the Xonsensus theorem (y x)(x z) = (y z) Crleton University Digitl Ciruits II p. 3, ig3krnghmps_xortena.fmrevise; Jnury 22, 27 Comment on Slie 6

17 XK-Mp Exmples Three Exmples funtion H funtion G H = + + P Q R= P+Q+R 2 numer of overs for eh squre of G funtion K numer of overs H = when PQR= let P= Q= R= PQR= on ll squres + equls exept on squre, there: Prity (,) = ( ) () = Thus = G = on K= on squres overe y terms the prity is K= on squres overe y 2 terms the prity is K= on squres overe y 3 terms the prity is K= ig3krnghmps_xortena.fm p. 32Revise; Jnury 22, 27 Slie 7 XK-Mp Exmples XOR Formul Detils 2. Trnsposition theorem; very powerful tool. Given three funtions of the sme vriles, n f = g h, then: g = f h n h = g f Proof: Given f = g h f h = g h h XOR oth sies with h = g h h = ; g = g qe. Exmple: A Gry oe to inry oe onverter The Reflete Gry oe, g N,...g 2,g,g, is erive from the inry oe, N,... 2,,, it-y-it using: g k = k k+, exept for the most signifint its where N = g N. To onvert Gry oe to inry oe, use the trnsposition theorem to give: 3-it Gry Coe 2 g 2 g g k = g k k+. (k<n) Thus to onvert the four-it Gry oe, g 3,g 3,g,g, to 3 = g 3 2 = g 2 3 = g 2 g 3 = g 2 = g g 2 g 3 = g = g g g 2 g 3 3, 2,, use: De When ounting in Gry Coe, only one it hnges t time. Crleton University Digitl Ciruits II p. 33, ig3krnghmps_xortena.fmrevise; Jnury 22, 27 Comment on Slie

18 XK-Mp Exmples Three More Exmples of X-SofP Mps funtion L 2 L= funtion M Showing numer of overs of squres M= funtion P P= ig3krnghmps_xortena.fm p. 34Revise; Jnury 22, 27 XK-Mp Exmples XOR Formul Detils Dulity Equivlents While there is no interhnge ulity, there re two powerful wys to trnsform ientities. () f g = f g = f g (2) f = g h g = f h Using these lrge numer of XOR formuls n e trnsforme into eh other. Exmple: A( C) = (A) (AC) gives (AC) = [A( C)] (A) The nonil forms of the ExOR lger The isjuntion theorem (Dj) n e use to express funtion given in terms AND n opertors. First the funtion is expne in unreue minterm form using the + opertor. Sine ll the loops over only one squre, they re isjoint. Then the + opertors n then e reple with opertors using (Dj). Thus n unsimplifie nnonil ExOR form omes iretly from its truth tle. For exmple, with 3 vriles, get: f() = (α ) (α ) (α 2 ) (α 3 ) (α 4 ) (α 5 ) (α 6 ) (α 7 ) where: α i =, epening on whether the ith term is present or not. An lterntive nonil form, foun y expning ny inverte vriles in the equlity ove with x=x, n expning using (Dx2) = ( )( ) = ( )( ) = [( )] [ ) ] = [( ()] [ ()] = () () When ll suh terms re XORe together mny will rop out leving: f() = β (β ) (β 2 ) (β 3 ) (β 4 ) (β 5 ) (β 6 ) (β 7 ) where: β i =, epening on whether the ith term is present or not.. Minterm form is the funtion otine y looping ll s in the K-mp with one-squre loop. No minimiztion. Crleton University Digitl Ciruits II p. 35, ig3krnghmps_xortena.fmrevise; Jnury 22, 27 Comment on Slie