m2 m3 m1 (a) (b) (c) n2 n3

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1 Outline LOGIC SYNTHESIS AND TWO-LEVEL LOGIC OPTIMIZATION Giovnni De Miheli Stnford University Overview of logi synthesis. Comintionl-logi design: { Bkground. { Two-level forms. Ext minimiztion. Covering lgorithms. Boolen reltions. Logi synthesis nd optimiztion Determine mirosopi struture of the iruit. Explore (re-dely)trde-o: { Comintionl iruits: I/O dely. { Sequentil iruits: yle-time. Explore (power-dely)trde-o: Enhne iruit testility. Ciruit implementtion issues Implementtion styles: { Two-level (e.g. PLA mro ells). { Multi-level (e.g. ell-sed, rry-sed). Opertion: { Comintionl. { Sequentil: Synhronous Asynhronous.

2 Design ow in logi synthesis Ciruit pture: { Tulr speitions of funtions or nite-stte mhines (FSMs). { Shemti pture. { Hrdwre Desription Lnguges (HDLs). Synthesis nd optimiztion: { Mp iruit representtion to strt model. { Trnsformtions on strt model. { Lirry inding. Astrt models Models sed on grphs. Useful for: { Mhine-level proessing. { Resoning out properties. Derived from lnguge models y ompiltion. Struturl views Netlists: { Modules, nets, inidene. { Ports. { Hierrhy. n n p p m n p p m p m p p7 m m m m m m () () () n n n Inidene (sprse) mtrix of grph.

3 Logi funions Logi networks Blk-ox model of omintionl module. Dened on Boolen Alger. Support vriles orrespond to moduleinputs. Logi funtions my hve multiple outputs nd e inompletely speied. Mixed struturl/ehviorl views. Useful for multiple-level logi (omintionl nd sequentil). Interonnetion of modules: { Logi gtes. { Logi funtions. Stte digrms +r/0 r/0 r /0 s 0 r/0 r /0 p = x r /0 s r / r/0 s r /0 q = p + y r / r / s v r / v v p v x Model ehvior of sequentil iruits. v v q v y Grph: { Verties = sttes. { Edges = trnsitions.

4 Mjor logi synthesis prolems Optimiztion of logi funtion representtion. { Minimiztion of two-level forms. { Optimiztion of Binry Deision Digrms (BDDs). Synthesis of omintionl multiple-level logi networks. { Optimiztion or re, dely, power, testility. Optmiztion of FSM models. { Stte minimiztion, enoding. Synthesis of sequentil multiple-level logi networks. { Optimiztion or re, dely, power, testility. Lirry inding. { Optiml seletion of lirry ells. Boolen lger: Comintionl logi design kground { Quintuple (B + 0 ) { Binry Boolen lger B = f0 g Boolen funtion: { Single output: f : B n! B. { Multiple output: f : B n! B m. { Inompletely speied: don't re symol *. f : B n! f0 g m. The don't re onditions We don't re out the vlue of the funtion. Relted to the environment: { Input ptterns tht never our. { Input ptterns suh tht some output is never oserved. Very importnt for synthesis nd optimiztion. Slr funtion: Denitions { ON ; set: suset of the domin suh tht f is true. { OFF ; set: suset of the domin suh tht f is flse. { DC ; set: suset of the domin suh tht f is don't re. Multiple-output funtion: { Dened for eh omponent.

5 Denitions Cuil representtion Boolen vriles. 0 Boolen literl: vrile nd omplement Produt or ue: produt of literls Implint: produt implying vlue of funtion (usully TRUE). { Hyperue in the Boolen spe. Minterm: produt of ll input vriles implying vlue of funtion (usully TRUE). { Vertex in the Boolen spe. Truth tle: Tulr representtions { List of ll minterms of funtion. Implint tle or over: of truth tle x = + 0 y = + + xy { List of implints of funtion suient to dene funtion. Remrk: { Implint tles re smller in size.

6 of implint tle x = + 0 y = + + Cuil representtion of minterms nd implints xy 00 0 * 0 0 * 00 α 0 γ δ f f f = f = Two-level logi optimiztion motivtion Progrmmle logi rrys Redue size of the representtion. Mro-ells with retngulr struture. Diret implementtion: { PLAs { redue size nd dely. Other implementtion styles (e.g. multi-level): { Redue mount of informtion. { Simplify lol funtions nd onnetions. Implement ny multi-output funtion. Lyout esily generted y module genertors. Firly populr in the seventies/eighties (NMOS). Still used for ontrol-unit implementtion.

7 00X 0 Progrmmle logi rry Two-level optimiztion X0 () Assumptions: X 0 () f f { Primry gol is to redue the numer of implints. { All implints hve the sme ost. () { Seondry gol is to redue the numer of literls. f f Rtionle: { Implints orrespond to PLA rows. f = f = 0 { Literls orrespond to trnsistors. Denitions Minimum over: { Cover of the funtion with minimum numer of implints. () α δ { Glol optimum. f f γ Miniml over or irredundnt over: { Cover of the funtion tht is not proper superset of nother over. () α f δ f { No implint n e dropped. γ δ { Lol optimum. () α f f Miniml over w.r.t. -implint ontinment. { No implint is ontined y nother one. { Wek lol optimum. f = f =

8 Denitions Logi minimiztion Prime implint: { Implint not ontined y ny other implint. Prime over: { Cover of prime implints. Essentil prime implint: { There exist some minterm overed only y tht prime implint. Ext methods: { Compute minimum over. { Often impossile for lrge funtions. { Bsed on Quine MCluskey method. Heuristi methods: { Compute miniml overs (possily minimum). { Lrge vriety of methods nd progrms: MINI, PRESTO, ESPRESSO. Quine's theorem: Ext logi minimiztion { There is minimum over tht is prime. Prime implint tle Rows: minterms. Columns: prime implints. Consequene: { Serh for minimum over n e restrited to prime implints. Quine MCluskey method: { Compute prime implints. { Determine minimum over. Exponentil size: { n minterms. { Up to n =n prime implints. Remrk: { Some funtions hve muh fewer primes. { Minterms n e grouped together.

9 Funtion: f = Primes: 00 0 γ δ 00 0 δ 00 0 γ δ 00* *0 * * α 000 () 0 α 0 α () () 0 Implint tle: Minimum over erly methods Petrik's method Redue tle: { Itertively identify essentils, sve them in the over, remove overed minterms. Petrik's method. { Write overing luses in pos form. { Multiply out pos form into sop form. { Selet ue of minimum size. { Remrk: Multiplying out luses is exponentil. pos luses: { ()( + )( + )( + )() = sop form: { + = Solutions: { f g { f g

10 Mtrix representtion View tle s Boolen mtrix: A. Seletion Boolen vetor for primes: x. Determine x suh tht: = 7 { A x. { Selet enough olumns to over ll rows. Minimize rdinlity of x: { : x = [0] T Set overing prolem: Covering prolem edge-over of hypergrph { A set S. (Minterm set). { A olletion C of susets. (Implint set). e { Selet fewest elements of C to over S. d d Intrtle. () () Ext method: { Brnh nd ound lgorithm. Heuristi methods.

11 Brnh nd ound lgorithm Tree serh of the solution spe: { Potentilly exponentil serh. Use ounding funtion: { If the lower ound on the solution ost tht n e derived from set of future hoies exeeds the ost of the est solution seen so fr: { Kill the serh. Good pruning my redue run-time. Brnh nd ound lgorithm BRANCH AND BOUND f Current est = nything Current ost = S = s 0 while (S = ) do f Selet n element in s S Remove s from S Mke rnhing deision sed on s yielding sequenes fs i i= ::: mg for ( i = to m) f Compute the lower ound i of s i if ( i Current ost) Kill s i else f if (s i is omplete solution ) f Current est = s i Current ost = ost of s i g else Add s i to set S g g g g Brnh nd ound lgorithm for overing Redution strtegies r r Prtitioning: x y w z x y w z Bound = Killed sutree { If A is lok digonl: Solve overing prolem for orresponding loks. 9 8 () () Essentils (EPI): { Column inident to one (or more) row with single : Selet olumn. Remove overed row(s) from tle.

12 Brnh nd ound lgorithm for overing Redution strtegies Column (implint) dominne: e { If ki kj 8k: d d remove olumn j. () () Row (minterm) dominne: { If ik jk 8k : Remove row i. A = redution Fourth olumn is essentil. Fifth olumn is dominted. Fifth row is dominnt. A = g Brnh nd ound overing lgorithm EXACT COV ER(A x ) f Redue mtrix A nd updte orresponding x if (Current estimte jj) return() if ( A hs no rows ) return (x) Selet rnhing olumn x = fa = A fter deleting nd rows inident to it ex = EXACT COV ER( A f x ) if ( jexj < jj) = ex x = 0 fa = A fter deleting ex = EXACT COV ER( A f x ) if ( jexj < jj) = ex return ()

13 Bounding funtion Estimte lower ound on the overs derived from the urrent x. A = The sum of the ones in x, plus ound on over for lol A: { Independent set of rows: No in sme olumn. Row independent from,,. Clique numer is. { Build grph denoting pirwise independene. { Find lique numer. { Approximtion (lower) is eptle. Bound is. A = A = There re no independent rows. Clique numer is (one vertex). Bound is + (lredy seleted essentil). Choose rst olumn: { Reur with A f = []. Delete one dominted olumn. Tke other olumn (essentil). { New ost is. Exlude rst olumn: { Find nother solution with ost (disrded).

14 ESPRESSO-EXACT α γ δ ε 000 ζ Ext minimizer [Rudell]. d Ext rnh nd ound overing. 0 0 () Compt implint tle: { Group together minterms overed y the sme implints () () 0 Very eient. Solves most prolems. 0**0 *0*0 0** 0** *0 *0 Prime implint tle (fter removing essentils) 0000, Reent developments Mny minimiztion prolems n e solved extly tody. Usully ottlenek is tle size. Impliit representtion of prime implints: { Methods sed on BDDs [COUDERT]: To represent sets. To do dominne simplition. { Methods sed on signture ues [MCGEER]: Represent set of primes.

15 Summry Ext two-level minimiztion of logi funtions Boolen reltions Generliztion of Boolen funtions. Bsed on derivtives of Quine-MCluskey method. Mny minimiztion prolems n e now solved extly. Usul prolems re memory size nd time. More thn one output pttern my orrespond to n input pttern. Some degrees of freedom in nding n implementtion: { More generl thn don't re onditions. Prolem: { Given Boolen reltion, nd minimumover of omptile funtion. z z 0 Compre: { + >? { + <? COMPARATOR N x x x 0 ADDER N x f 000, 00, 00 g f 000, 00, 00 g f 000, 00, 00 g f 000, 00, 00 g f 000, 00, 00 g 0 0 f 000, 00, 00 g 0 0 f 0, 00 g 0 0 f 0, 00 g 0 0 f 0, 00 g 0 0 f 0, 00 g 0 0 f 0, 00 g 0 f 0, 00 g 0 f 0, 00 g 0 f 0, 0, g 0 f 0, 0, g f 0, 0, g

16 Remrk: () Minimum implementtion 0 0 x 0 * * 00 * 0 * 00 * * 00 * * 00 * * 00 { Ciruit is no longer n dder. Minimiztion of Boolen reltions Sine there re mny possile output vlues there re mny logi funtions implementing the reltion. { Comptile funtions. Find funtion with minimum rdinlity. Do not enumerte ll possile funtions: { My e too mny. Represent the primes of ll possile funtions: { Comptile primes ( ; primes). Ext: Minimiztion of Boolen reltion { Find set of omptile primes. { Solve inte overing prolem. Consisteny reltions. Heuristi: { Itertive improvement [GYOCRO]. Boolen reltion: Comptile primes: f 00 g 0 0 f 00 g 0 0 f 00 g 0 f 0 g 0 0 f 00 g 0 f 0 g 0 f 00, g f 00, g * 0 * 0 *

17 Binte overing Input 0 { output 0. { Covering luse ( + ). Input { output 00 or. { No implint { 00 { orret. { Either or [ { output { orret. { Only or is seleted { output 0 or 0 { WRONG. { Covering luse { inte. Overll overing luse: ( + ) ( + ) ( ) ( ) Covering prolem with inte luse. Implitions: { The seletion of prime my exlude other primes. No gurntee of nding fesile solution: { Inonsistent luses. Minimum-ost stisility prolem. { Muh hrder to solve thn unte over. { Brnh nd ound lgorithm. { BDD-sed methods. Summry Boolen reltions Generliztion of Boolen funtions. { Mny possile output ptterns. Useful for modeling: { Csded loks. { Portions of multiple-level networks. More degree of freedom in implementtion. Hrder prolem to solve.

Engr354: Digital Logic Circuits

Engr354: Digital Logic Circuits Engr354: Digitl Logi Ciruits Chpter 4: Logi Optimiztion Curtis Nelson Logi Optimiztion In hpter 4 you will lern out: Synthesis of logi funtions; Anlysis of logi iruits; Tehniques for deriving minimum-ost