a b v a v b v c v = a d + bd +c d +ae r = p + a 0 s = r + b 0 4 ac + ad + bc + bd + e 5 = a + b = q 0 c + qc 0 + qc (a) s v (b)
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1 Outlin MULTIPLE-LEVEL LOGIC OPTIMIZATION Gionni D Mihli Stnfor Unirsit Rprsnttions. Tonom of optimition mthos: { Gols: r/l. { Algorithms: lgri/booln. { Rul-s mthos. Empls of trnsformtions. Booln n lgri mols. Motition Ciruit moling Multipl-ll ntorks: Logi ntork: { Smi-ustom lirris. { Gts rsus mros (PLAs): Mor iilit. Bttr prformn. Applil to rit of signs. { Intronntion of logi funtions. { Hri struturl/hiorl mol. Boun (mpp) ntorks: { Intronntion of logi gts. { Struturl mol.
2 Empl of oun ntork p p q q p q Empl of ntork = + = + r = p + 0 s = r + 0 t u = = q 0 + q 0 + q = = = s = t = u Empl of ntork = Empl iruit trminl hior p =+ r = p + s = r + t = q = + u = q + q + q f = () p t q r u s ()
3 Ntork optimition Estimtion Minimi r (por) stimt: { sujt to l onstrints. Minimi mimum l: { sujt to r (por) onstrints. Mimi tstilit. Minimi por. Ar: { Numr of litrls. { Numr of funtions/gts. Dl: { Numr of stgs. { Rn gt l mols. { Snsitil pths. Prolm nlsis Multipl-ll optimition is hr. Et mthos: { Eponntil omplit. { Imprtil. Approimt mthos: { Huristi lgorithms. { Rul-s mthos. Strtgis for optimition Impro iruit stp stp. { Ciruit trnsformtions. Prsr ntork hior. Mthos ir in: { Tps of trnsformtions. { Sltion n orr of trnsformtions.
4 Empl limintion Empl limintion = Elimint on funtion from th ntork. p =+ r = p + s = r + Prform ril sustitution. t = q = + u = q + q + q Empl: { s = r + 0 r = p + 0 = { ) s = p p =+ s = p + + t = q = + u = q + q + q Empl omposition Empl omposition Brk on funtion into smllr ons. = p =+ r = p + s = r + Introu n rtis in th ntork. t = q = + u = q + q + q Empl: { = j = ++ p =+ = j + r = p + s = r + { ) j = = j + 0 t = q = + u = q +
5 Empl trtion Empl trtion Fin ommon su-prssion of to (or mor) prssions. = Etrt su-prssion s n funtion. Introu n rt in th ntork. p =+ r = p + s = r + t = q = + u = q + q + q Empl: = { p = + t = { p = ( + ) t = ( + )( + ) + { ) k = + p = k t = k + k + p = k r = p + s = r + k = + t = k + k + q = + u = q + q + q Empl simplition Empl simplition Simplif lol funtion. = p =+ r = p + s = r + Empl: { u = q 0 + q 0 + q t = q = + u = q + q + q { ) u = q + = p =+ r = p + s = r + t = q = + u = q +
6 Empl sustitution Empl sustitution Simplif lol funtion using n itionl input tht s not priousl in its support st. Empl: = p = k r = p + s = r + k = + t = k + k + q = + u = q + q + q { t = k + k +. = { ) t = kq + p = k r = p + s = r + { Bus q = +. k = + q = + t = kq + u = q + q + q Empl squn of trnsformtions j = k = + q = + s = k t = kq + u = q + = j + 0 Optimition pprohs Algorithmi pproh: { Dn n lgorithm for h trnsformtion tp. j = ++ = j + s = k + + { Algorithm is n oprtor on th ntork. k = + q = + t = kq + u = q + Rul-s pproh: { Rul-t s: St of pttrn pirs. { Pttrn rplmnt rin ruls.
7 Algorithmi pproh Eh oprtor hs ll-n proprtis: { Huristi mthos still us. { Wk optimlit proprtis. Squn of oprtors: { Dn sripts. Empl limintion lgorithm St thrshol k (usull 0). Emin ll prssions. Elimint prssions if th inrs in litrls os not th thrshol. { Bs on prin. Empl limintion lgorithm ELIMINAT E( G n (V E), k)f rpt f = slt rt ith lu < k if ( = ) rturn rpl f in th ntork g g sp limint -1 simplif -m noomp limint -1 sp limint 5 simplif -m noomp rsu - Empl MIS/SIS rugg sript f rsu - sp limint -1 sp full-simplif -m noomp
8 Booln n lgri mthos Empl Booln mthos: { Eploit proprtis of logi funtions. { Us on't r onitions. Booln sustitution: { h = + + q = + { ) h = + q + { Compl t tims. Algri mthos: { Vi funtions s polnomils. { Eploit proprtis of polnomil lgr. { Simplr, fstr ut kr. { Bus + q + = + ( + ) + = = + +. Algri sustitution: { t = k + k +. { ) t = kq + { Bus q = +. Summr Multill logi snthsis is prform stp-is trnsformtions. Algorithms r s on oth th Booln n th lgri mols. Rul-s sstms.
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