MUTIPE-EVE OGIC OPTIMIZATION II Booln mthos Eploit Booln proprtis Giovnni D Mihli Don t r onitions Stnfor Univrsit Minimition of th lol funtions Slowr lgorithms, ttr qulit rsults Etrnl on t r onitions Empl & ' : Controllilit on t r st Input pttrns nvr prou th nvironmnt t th ntwork s input s1 s2 DEMUX 1 2 *3 *4 NETWORK N1 1 ))2 1 (2 N2 N3 o1 o2 Osrvilit on t r st : Input pttrns rprsnting onitions whn n output is not osrv th nvironmnt Rltiv to h output Vtor nottion us:! " # $ + Inputs rivn -multiplr, - - /0 1 2 314 65 7 9 : = > 2 83 4 1? 6 @ A 2 1B 3 O P Q Outputs osrv whn 1 9 C D 1E G 6 H 9 I J 6 K M 9 N 4 2 3 2 4 3 4 R 4 S T 1 U V W X YZZ[ \ ] ^ 1 ` _1 44 f
œ g i œ ž Empl ovrll trnl on t r st h Intrnl on t r onitions k l m n DC CDCo p q r s t u wwwv 1 { 2 } ~ 3 š 4 š š 1 ƒ 2 3 ˆ 4 Š4 Œ 2 Ž 3 1 4 2 3 1 CDC in Ÿ out SUBNETWORK NETWORK Intrnl on t r onitions Inu th ntwork strutur Empl Controllilit on t r onitions: Pttrns nvr prou t th inputs of suntwork = + () = + = + () = + Osrvilit on t r onitions: Pttrns suh tht th outputs of suntwork r not osrv ª CDC of inlus ««Minimi ± ² ³ µ ¹ º» to otin:
5 - ¾ å Ý Ý ¼Stisfiilit ½ on t r onitions Invrint of th ntwork: À Á Â Ã Ä À ÁÅ Â Ã Æ Ç È É Ê Ë Ì Í Î Ï Ð Ñ Ò Ó Ô Õ Ö ÙCDC omputtion Ü Ntwork trvrsl lgorithm: Consir iffrnt uts moving from input to output Initil CDC is ß à ß á â Ú Û Ø Usful to omput ontrollilit on t rs ã Mov ut forwr Consir SDC ontriutions of prssor! Rmov unn vrils onsnsus ÙCDC omputtion ä Ý æ ç è é ê ë ì CONTROABIITY(, í î í ï ð ñ ò ) ó ô õ ö ø ù ú û ü ø ý þ forh vrt ÿ in topologil orr st ll ir su of r in forh vrt!! " # $ &!! " # $ ( ' ) * ) +, / 0 1 2 3 / 4 1 2 1 1 8 2 3 () 92 4 1 1 Empl 2 () 3 2 84 1 1 2 6 () 3 7 92 4 :{,} {,} > {,,4} {1,,4} = {1,2,3,4}
ª Â Ä Ó Ã Å À Á A Assum B C B D E G H1I J4 Empl? @ Prturtion mtho K Slt vrt M : Contriution N to O P O Q R S : T U V W 6 X Y 9 Z 2 3 Drop vrils [ \ ] ^ 6 _ ` 9 2 3 onsnsus: f g h i1 k4 l Slt vrt m n : Contriution N to o p o q r s : t u v w 1 { } ~ ƒ 1 ˆ Š 4 1 Œ Drop vril Ž 1 onsnsus: š œ 4 Moif ntwork ing n tr input «Etr input n flip ± ² Rpl lol funtion polrit of signl ³ µ Prtur trminl hvior: f ¹ º» ž Ÿ Ÿ 2 ¾ δ Empl ¾ ¼ ½ ¾ Osrvilit on t r onitions Conitions unr whih hng in polrit of signl is not priv t th outputs Complmnt of th Booln iffrn: () () () Æ Ç È Æ É Ê Ç Ë Ì Í 1 Î Ï Ð Ñ Ò 0 Diffrn of prtur funtion: fô Õ 0Ö fø Ù 1Ú
Ý à Z ß Û Ü æ è ð _ ` ã g Osrvilit on t r omputtion Prolm: Outputs r not prss s funtion of ll vrils If ntwork is flttn to otin f, it m plo in si ç Trvrs At root: Singl-output ntwork with tr strutur ntwork tr ï é ê ë ì í î is givn At intrnl vrtis: ä å Rquirmnt: á ol â Ntwork ruls for trvrsl omputtion ñ ò ó ô õ ö ø ù ú û ü ý þ ñ ò ó ù Empl ÿ 1 1 4 2 Assum 0! " # $ # & ' $ ( ) 5 6 7 8 9 : = >? @ A B : C 1 1 2 4 1* +, - / 0 1 2 0 3 4 D E G H 1 I J K M N O P Q M R P S 1T U V W X Y 1 [ Gnrl nout ronvrgn ntworks \ ] or h vrt with two (or mor) fnout stms: Th ontriution of th long th stms nnot tout ourt Å Intrpl of iffrnt pths Mor lort nlsis ^ 1 4
g h i ù þ f ñ C o C D É æ N œ N 7 Two-w fnout stm Comput Comin s f, 1 2, sts ssoit with gs t vrt { formul rivtion } ~ f 1 6 ƒ 2 1 1 f 1ˆ 6 Š Œ 2 Ž 0 0 f 1 6 2 1 1 f 1 6 š 2 0 0 Ž ž f 1Ÿ 2 0 1 f 1 2 0 1ª ª «f 1 2 1± 1² ³ f 1µ 2 0 1¹ ¹ º» Ž ¼ f 1½ ¾ 6 2 0À 1Á  fã 1Ä Å 6 Æ Ç 2 0 0È È Ê Ë Ì Í Î Ñ Ò Ó Ô Õ 2Ï 1 Ð 1Ö 0 Ø Ù Ú Û Ø 6 Ü Ý ß à á â ã à 2 1ä å ç è é ê ç ë ç ì í î ï ð ormul rivtion: Assum two qul prturtions on th gs k l fm 1n o 6 p 2 1q 1r s ft 1u v 6 w 2 0 0 Bus ò ó ò 1 ô õ 2 ö Multi-w Thorm gstms ø t ú û ü ý n intrnl or input vrt th g vrils t ÿ 1 2 1 2 orrsponing to Osrvilit on t r lgorithm 6 OBSERVABIITY(8 9 : = >,? @ A ) E G H B forhj K vrt M N in rvrs topologil for 1 to O PQ R S T U V Q R W U X Y Z 1 [ \ ] ^ _ ` f g h 1 g i 1 k k k l m l n o orr I t 1 2! " # th g s $ & ' () * +,- /+ 0 1 11 2 2 2 1 3 4 1 3 5
* E u r v w s i k l NM ÿ t 1 1 2 3 () 2 4 Empl 1 1 2 () 3 p q { } ~ 0 1 1 ƒ ~ ˆ Š Œ Ž 0 š œ ž Ÿ 1 1 ª «± ² 1 4 4 ³ 1µ ¹ º» ¼ ½ ¾  ÃÀ ÄÁ 4Å Æ Ç Í È ÎÉ Ê ÏÉ1 Ë Ì 4 4 1 Ð Ñ Ò 4 Ó Ô Õ Ö Ø Ù Ú Û ÜÝ Û ß Û à á â ã ä å æç è é êî ì í ë4 1 4 4 ï ð 1ñ ò ó ô 1 õ ö ú ø 1ù 4 1 û ü ý 4 2 4 Trnsformtions with on t rs Booln simplifition: Us Minimi þ stnr minimir (Esprsso) th numr of litrls Booln sustitution: Simplif funtion Equivlnt to simplifition ing n tr input with glol on t r onitions Empl Booln sustitution Sustitut into to gt! " # $ & ' ( ) # $ ) & $ # ) " & ' ) SDC st: Simplif +, - / 0 1 2 / 3 with 4 5 6 7 4 5 8 9 7 4 6 5 : 8 9 5 s on t r Singl-vrt optimition P Q R S T Q U V W X Y Z [ Y X \ ] ^ ^ _ m O rpt ` = slt vrt H f Comput th lol on t r st Optimi th funtion g h until (no mor rution is possil) Simplifition ils = >? @ A B C A D On litrl lss G hnging H th support I of J K
p u º ¼ G G Ý H nm o À Á Â Ä I ¾M Æ ˆ Empl M Š Optimition n prturtions Rpl q r s t δ Œ () () () Prturtion v w w { w Ž No trnl on t r st Conition for fsil rplmnt: Prturtion oun H lol on t r st Rpl AND wir: } ~ DC ƒ Anlsis: l If not primr input onsir lso CDC st š œ š œ œ ž Ÿ µ ª «± ª ² ³ fsil! Dgrs of from M Multipl-vrt optimition k ull rprsnt on t r onitions: Simplif mor thn I on lol funtion t tim Etrnl on t rs ¹ Intrnl osrvilit n G ontrollilit Potntill ttr (mor ¹ gnrl) pproh» Don t rs rprsnt n uppr oun prturtion I on th Anlsis: à Multipl l prturtions Approimtions: ½ Us smllr on t r sts to omputtion sp-up th Conition for fsil rplmnt: Å Uppr n» lowr ouns Booln rltion mol on th prturtion
Ë Ì ò õ Ô Õ # Ô! Õ " É ˆ Empl δ 1 δ2 ÇM È Multipl-vrt optimition Booln rltion mol ÖM () Ê () Ê Ù () () Ø Æ Th two prturtions k r rlt Í Cnnot hng simultnousl: Î Ï Ð Î Ñ Ò Ó Ñ Ú Û Ü Ý ß 0 0 0 à 00, 01, 10 á 0 0 1 â 00, 01, 10 ã 0 1 0 ä 00, 01, 10 å 0 1 1 æ 00, 01, 10 ç 1 0 0 è 00, 01, 10 é 1 0 1 ê 00, 01, 10 ë 1 1 0 ì 00, 01, 10 í 1 1 1 î 11 ï Multipl-vrt optimition Booln rltion mol Comput Booln rltion: ðm ñ ó lttn th ntwork Anl pttrns ô Driv quivln ö Us rltion minimir rltion from s $ Multipl-vrt optimition Booln rltion mol M rpt = slt vrt sust forh vrt Comput Dtrmin th quivln lsss of th Booln rltion of th suntwork inu in n optiml funtion omptil with th rltion using rltion minimir until (no mor rution is possil) ø Empl of minimum funtion: ù ú û ü ý þ 1 ÿ ÿ 10 1 1 01