Canonical Form-Based Boolean Matching and Symmetry Detection in Logic Synthesis and Verification

Transcription

1 aocal orm-based Boolea Matchg ad Symmetry Detecto Logc Sythess ad Vercato Ash Abdollah Uversty o Souther alora ash@uscedu Massoud edram Uversty o Souther alora edram@ceguscedu Abstract A ecet ad comact caocal orm s roosed or the Boolea matchg roblem uder ermutato ad comlemetato o varables I addto a ecet algorthm or comutg the roosed caocal orm s rovded The ececy o the algorthm allows t to be alcable to large comle Boolea uctos wth o lmtato o the umber o ut varables as aosed to revous aroaches whch are ot caable o hadlg uctos wth more tha seve uts Geeralzed sgatures are used to dee ad comute the caocal orm whle smle symmetres o varables s used to mmze the comutatoal comlety o the algorthm All other symmetry relatos are resulted as a b-roduct o the caocal orm comutato Eermetal results demostrate the ececy ad alcablty o the roosed caocal orm I Itroducto Boolea matchg s the roblem o determg whether a Boolea ucto ca be uctoally euvalet to aother oe uder a ermutato o ts uts ad comlemetato o some o ts uts Boolea matchg algorthms have may alcatos logc sythess cludg celllbrary bdg where t s ecessary to reeatedly determe whether some art cluster) o a Boolea etwor ca be realzed by ay o the cells a lbrary [] Boolea matchg s a crtcal ad U-tesve tas ad thereore there have bee may eorts to eectvely solve the roblem [] Boolea uctos that are euvalet uder egato o uts are N-euvalet uder ermutato o uts are -euvalet ad uder both stated codtos are Neuvalet [3] I all we cosder ermutato o uts ad comlemetato o uts ad outut the uctos are NN-euvalet A ehaustve method or Boolea matchg s comutatoally eesve sce the comlety o such a algorthm or -varable uctos s O! + )

2 Boolea matchg algorthms ca be classed to two categores: ar-wse matchg algorthms ad algorthms based o caocal orms o uctos ar-wse Boolea matchg algorthms are based o a sem-ehaustve search where the search sace s rued by the use o some sgatures whch are comuted rom some roertes o Boolea uctos [] A sgature geeral s a descrto o oe or more) ut varables o a Boolea ucto that s deedet o the ermutato or comlemetato o the varables o the ucto To match a ucto agast a cell lbrary ar-wse matchg algorthms ote eed to erorm ar-wse matchg o the ucto wth all the lbrary cells Thereore these algorthms ca oly coe wth lbrares o modest sze Boolea matchg algorthms that belog to the secod category comute some caocal orm or Boolea uctos [5] - [0] These algorthms are based o the act that two uctos match ad oly ther caocal orms are the same Burch ad Log troduced a caocal orm or matchg uder comlemetato ad a sem-caocal orm or matchg uder ermutato o the varables [5] I ther soluto order to hadle comlemetato ad ermutato o uts smultaeously a large umber o orms or each cell are reured Other researchers cludg Wu et al [6] Debath ad Sasao [8] ad rc ad Seche [9] have also roosed caocal orms that are alcable to Boolea matchg uder ermutato o the varables oly but do ot hadle comlemetato o uts Hsberger ad Kolla [7] ad Debath ad Sasao [0] have troduced a caocal orm or solvg the geeral Boolea matchg roblem However ther aroach s maly based o maulatg the truth table o the ucto ad emloyg a table loo-u whch results a eormous sace comlety thus restrctg ther algorthm to lbrary cells wth seve or ewer ut varables I ths aer a ew caocal orm or reresetg Boolea uctos s troduced The roosed caocal orm or a arbtrary Boolea ucto s the uue Boolea ucto that s obtaed ater alyg some caocty-roducg ) trasormato o the ut varables The caocal orms o NN-euvalet Boolea uctos are detcal I artcular a eectve techue s reseted or geeratg ths caocal orm The roosed method s based o usg geeralzed sgatures sgatures o oe or more varables) to d a hase assgmet ad orderg or varables rom here o hase assgmet ad orderg or varables s reerred to as a trasormato o varables or most Boolea uctos sglevarable ad two-varable sgatures are eough to recogze all varables e to obta a

3 trasormato) However use o sgle-varable ad two-varable sgatures aloe may ot result o a caocal ut trasormato I ths aer t s show that by usg geeralzed sgatures o oe or more varables t s always ossble to create a trasormato o varables o the ucto Eermetal results rovded ths aer demostrate that the roosed aroach or comutg the caocal orm does ot have the lmtatos o revous wors; e t comutes the caocal orm o a Boolea ucto wth ay umber o varables uder both ermutato ad comlemetato o varables A mortat advatage o the roosed techue s the way t hadles ad uses the symmetry o varables to mmze the comlety o the algorthm comared to some o the revous aroaches whch are ot able to cosder symmetres [7][0] Hece the roosed techue s alcable to logc vercato o large crcuts ad to techology mag wth a large ASI lbrary wth cells o ay umber o uts I secto II detos ad termology are troduced I secto III symmetry relatos are dscussed I secto IV sgatures that are utlzed the method are descrbed I secto V the caocal orms s deed ad the detals o comutg the caocal orm s rovded ollowed by eermetal results ad coclusos sectos VI ad VII II relmares We deote vectors ad matrces catal letters e X ) where X deotes a vector o Boolea varables A lteral s a varable or ts comlemet We wll reer to lteral as the ostve hase o varable ad to lteral as ts egatve hase I geeral a lteral ca be deoted as The hase o the lteral s descrbed usg the Boolea varable B {0} where ostve hase) ad 0 egatve hase) or the varable vector X ) ad hase vector ) where X ad B cota the same umber o varables ad B vector X s deed as X ) ) the hase assgmet o to varable As a eamle or X ) ad 00 ) the result o hase assgmet s 3 X ) Also or X ) ad 0 ) the result o hase assgmet s 3 X ) 3 3 3

4 or a set A we use A to deote the cardalty o A Sce deret hase assgmets to X ) s B the umber o ossble The detty hase assgmet s deoted by ) Obvously X X The verse o hase assgmet s tsel e X ) X The cascade o two hase assgmets ) ad Q ) s Q K ) sce Q Q X ) X The s the XNOR oerato e or Boolea varables ad y y y ) A ermutato s a rearragemet o the elemets o a ordered lst or a vector rst we dee ermutato o a set o the orm { } whch wll serve as dces o a ordered lst or a vector A Deto: A ermutato o set A { } s a becto rom A to A e : A A ad ) ) or a subset B A the rage o a ermutato o doma B s deed as: B) { ) B} Based o ths deto A ) A whch s euvalet to the reversblty o ermutato as a ucto e A A st : ) The detty ermutato ι o { } s A deed as ollows: A ) ι We deote the set o all ermutatos o A by Π The cascade o two ermutatos ad o A { } deoted by s deed as: ) )) A The cascade oerato amog ermutatos s ot a commutatve oerato e geeral Π However t s a assocatve oerato Π ) ) ermutatos are reversble The verse o a ermutato deoted by s deed as: ) ) Based o these roertes set Π wth cascade oerato creates a grou The umber o members o ths grou s! Ay ermutato Π ca be aled to a vector o legth eg X ) The result o alcato o ermutato Π to vector s determed A grou G s a te or te set o elemets together wth a bary oerato called the grou oerato) that together satsy the our udametal roertes o closure assocatvty detty ad verse roerty 4

5 based o the relatos ) ) ad X ) K ) ) ) ) ) whch s a rearragemet the etres o vector X Net we descrbe N trasormatos comrsg o hase assgmet ad ermutatos As a eamle or the ermutato Π3 where ) ) 3 ad 3) we have ) ) 3 3 Deto: A N trasormato o vector X ) s deed as a hase assgmet ollowed by a ermutato I artcular or hase assgmet ) ad ermutato Π the N trasormato T o vector X ) s comuted as ollows: T X) X ) Now the ) ) ) ) ) ) ) ) ) T K K K or more smly T X) ) [ X )] The set o all N trasormatos o a vector o sze s deoted by Γ T B Π } { As a eamle or 0 ) ad ) ) we have T ) ) The umber o trasormatos Γ s Γ B Π! The detty trasormato s deoted by T ι where ) s the detty hase assgmet ad ι s the detty ermutato Obvously T ι X) X or ay vector X The cascade o two trasormatos T ad T deoted by T T s deed as T T X) T T )) Oe ca very that T T X T where ad ) ) The verse o trasormato T deoted by T ) satses the relato T T ) T ) T Tι Based o the relato T T T T oe ca er that ) ) ι a grou T ) T ) Set Γ wth cascade oerato creates I the remader o ths aer whe there s o ambguty we deote a trasormato or brevty I addto we may deote a N trasormato o vector as T by T T X or TX ) stead o T X) or T X) ) We usually deote the detty trasormato by regard to ermutatos X wll reer to X) I T ι ally wth 5

6 Let ucto X ) be a sgle-outut comletely-seced Boolea ucto o X ) e : B B The oset o X ) s a subset o ts doma B that results X ) We wll deote the sze o oset o X ) by X ) e X ) The coactor o X ) wth resect to lteral s a ucto X ) o X K + ) deed as X ) X ) X ) A cube s the Boolea coucto o some lterals m L The coactor o X ) m wth resect to a cube s a ucto o varables X that are ot reset ostve or egatve hase) s deed as K where X X } X ) X ) X ) m m { m osder two uctos X ) ad g X ) deed over the same varable set X ) Deto: Two uctos X ) ad g X ) are -euvalet deoted by g ermutato such that X ) g X ) s a tautology there ests a Deto: Two uctos X ) ad g X ) are N-euvalet deoted by ests a N trasormato T such that X ) g TX ) s a tautology N g there The most geeral tye o euvalece s whe we also cosder hase assgmet o the outut We wll deote the hase assgmet B to ucto X ) by X )) or X ) or short Deto: Two uctos X ) ad g X ) are NN-euvalet deoted by g there ests a N trasormato T ad a outut hase assgmet B such that X ) g TX ) s a tautology e T Γ B X B X ) g TX ) Eamle: Let 3 ) + 3 ad g 3 ) It s easy to see that X ) g TX ) where X ) ad T X ) ) Thus X ) ad g X ) are thus NN-euvalet 3 3 NN-euvalece s a euvalece relato Boolea matchg s ote deed terms o N or NN-euvalece I rcle N ad NN-euvalece ca be reduced to!! ad +! tautology checs 6

7 We use the symbol ad to desgate the cosesus ad the smoothg oerators wth resect to varable resectvely Recall that the cosesus oerato corresods to uversal uatcato ad s comuted as whle the smoothg oerato corresods to estetal uatcato ad s comuted as + osesus smoothg) wth resect to a array o varables ca be comuted by reeated alcato o sgle-varable cosesus smoothg) oeratos III Symmetry relatos I ths secto we dscuss varable symmetres Boolea uctos uctoal symmetres rovde sgcat beets or multle tass sythess ad vercato As wll be elaed detal later ths aer cocets o Boolea matchg ad symmetry are closely related I the Boolea matchg algorthm that wll be rovded ths aer ths relatosh maests tsel two levels rst smle tyes o symmetres that are eesve to dscover) are utlzed to reduce the comlety o the Boolea matchg algorthm Secod the roosed Boolea matchg algorthm wll geerate as a b-roduct) the remag more comlcated) symmetres Symmetres rovde sghts to the structure o the Boolea ucto that ca be used to acltate oeratos o t They ca also serve as a gude or reservg that structure whe the ucto s trasormed some way I the cotet o Boolea matchg roblem symmetres that we elore are varable ermutatos wth ossble comlemetato that leave the ucto uchaged I the resece o uctoal symmetres several desg roblems eg crcut restructurg checg satsablty ad comutg seuetal reachablty) are cosderably smled Hece terest uctoal symmetres has bee ee sce the early days o logc desg [] I the cotet o logc sythess whch we vew as a rocess that trasorms a tal reresetato o the ucto eg sum o roducts reresetato or bary decso dagram reresetato) to a al mlemetato as a mult-level Boolea etwor o rmtve cells selected rom a gve ASI cell lbrary whe guded by owledge o uctoal symmetres such a rocess yelds hgher ualty crcut realzatos o the ucto [3] I [4] uctoal symmetry s eloted to otmze a crcut mlemetato or low ower cosumto ad delay uder a area crease costrat Aother beet o owledge about uctoal symmetres s that t ca hel roduce better varable orders or Bary Decso 7

8 Dagrams BDDs) ad related data structures eg Algebrac Decso Dagrams) The sze o the BDD o a Boolea ucto ca be sgcatly reduced symmetrc varables are laced adacet ostos Based o ths observato a secalzed stg rocedure or dyamc varable orderg was roosed [5] Ths lays a crucal role symbolc model checg I ths aer we study symmetres the most geeral rom e cosderg ut ermutato ut hase assgmet ad outut hase assgmet whch to the best o our owledge has ot bee studed thoroughly eough the ast Deto: A ucto X ) where X ) s symmetrc wth resect to a N trasormato T Γ o ts uts there ests a outut hase assgmet B such that X ) TX ) We wll reer to such a trasormato a symmetry-roducg S) trasormato ad deote the set o all S trasormatos by S { T Γ X ) TX )} S creates a sub-grou o Γ As metoed beore some tyes o symmetry are easly detectable ad are dscovered beore the Boolea matchg algorthm We start by dscussg these tyes o symmetres Deto Smle Symmetry): or a ucto X ) where X ) two varables ad are sad to be symmetrc deoted as X ) s varat uder a echage o ad e K ) ) I the case o smle symmetry the hase o the outut always remas uchaged The N trasormato T S assocated wth ths smle symmetry has the ollowg eect: T K ) ) A smlar tye o smle symmetry betwee varables ad s whe the ollowg codto holds: K ) ) I ths case we use otato or euvaletly We shall reer to ad as beg symmetrc ths case as well Eamle: or the ucto X ) + ) + ) we have ad

9 To accout or both tyes o symmetry wth a ued otato we use Whe the eresso dcates that whereas 0 mles that e K ) ) Varables ad are called symmetrc We wll use W or W ) to deote the N trasormato W K ) ) Wth ths otato W S It s well ow ad ca be readly show by usg Boole s easo theorem [5] that codto s euvalet to Ths euato serves as the comutatoal chec or rst-order symmetry betwee varables ad ucto X ) The symmetry relato s a euvalece relato Hece t s ossble to artto varables K to euvalece classes whch we wll reer to as symmetry classes A overvew o such a rocedure whch s comosed o two ested loos that terate o the varables s as ollows We deote symmetry classes by K m where m s the umber o classes The rst ste s to create { } where s cosdered the seed varable or class Net every varable that s symmetrc to wll be added to The rst remag varable say s used to talze { } Net symmetrc varables to are added to Ths rocedure cotues utl all varables are arttoed to symmetry classes K m Symmetry classes wll clude all ormato about smle symmetres or eamle gve symmetry classes K m oe ca er that the there ests a hase assgmet B such that However the symmetry classes do ot clude ormato as to whether or 0 Oe way to clude hase ormato symmetry classes s to choose arorate hases or varables whle ormg classes oe at a tme or eamle cosder a class wth seed I there ests a varable that s symmetrc to e the lteral wll be added to Ths s because we chose ostve hase or the seed o I we were to 9

10 choose 0 as the seed o the the case o lteral wll be added to sce Suose { } s a symmetry class geerated ths maer Based o the revous dscusso egatg the hases o the lterals { K } wll create a alterate symmetry class o the same varables We shall deote ths alterate class by whch troduces the oto o hase assgmet or symmetry classes e } { The algorthm or geeratg rst-level symmetry classes s gve below I ths algorthm we choose ostve hases or the seeds o all classes Algorthm Ge_ st _Order_Symm ) ; whle X {} do { {} ; select X ; or y X do { y ) the { y }; } X X ; +; } or the symmetry classes geerated ths maer lterals o a class do ot reure ay hase assgmet to become symmetrc the curret hases o lterals wll be e) e Eamle: or ucto X ) + ) + ) there est two symmetry classes: } ad } { { I the remader o ths aer we shall deote lterals by smle letters such as or y whch does ot ecessary mea that the hase o lteral s ostve Wth ths coveto the revous relato may be wrtte as: y y 0

11 The classes geerated by Ge_ st _Order_Symm are mamal the sese that or every class o other lteral y s symmetrc to the lterals o the class e y y So ar we have dscussed smle symmetres whch corresod to N trasormatos that volve oly two varables I the seuel we reset a ey theorem whch rovdes a valuable sght or hadlg ad eumeratg symmetres rst we reset a lemma that wll be useul rovg the ma theorem I the ollowg we wll deote the th elemet o vector TX by [ TX] urthermore [ TX] wll deote [ TX ] ) Lemma : or ay N trasormato ) ) T Γ ad W Γ T ) W T W ) ) where roo: Based o revous dscussos ) ) )) [ T X ] ad [ ) X ] T ) ) We comute [ T ) W T X ] or deret values o { K} rst we cosder values o such that { ) )} ad the deal wth { ) )} )) or { ) )} T W T X W T X )) [ T X ) ] [ ] [ ] ) The secod eualty ollows rom ) { } or ) ) ) ) ) ) ) ) ) W T X] ) [ W T X] [ T X] ) ) [ T ) ) ) ) ) W T X ] ) [ W T X] [ T X ] ) ) [ T I addto or Hece we showed that T ) W T ) ) ) ) ) ) whch roves that T ) W T W ) ) Let ad deote two symmetrc varables o ucto X) osder a geeral symmetry relato that volves more tha two varables e cosder X) T X) I the ollowg we wll elore the eect o T S o symmetrc varables Every N trasormato T Γ o X ca be regarded as a mag ucto o lterals wth ) ) the seccato T )

12 We dee the eect o N trasormato T o lteral as T )) ) T Lemma : Let ucto X) be symmetrc wth resect to N trasormato T e T S ; Mags o two symmetrc varables ad uder T are symmetrc e )) T ) T roo: Sce S s a subgrou T S T ) S urthermore sce varables ad are symmetrc W S Based o the revous Lemma ad the act that S s a subgrou W ) ) T ) W T S where ) ) whch roves that ) ) By alyg hase assgmet ) to both sdes o oe obtas ) ) ) ) ) ) ) ) or T T )) ) whch roves the lemma Now we wll vestgate the eect o N trasormato T S o smle symmetry classes The rage o a N trasormato T o a symmetry class or ay other subset o lterals) s deed as T ) { T ) } where geeral reresets a lteral wth ostve or egatve hase) rather tha a varable e there s varable wth hase such that Theorem : Let ucto X) be symmetrc wth resect to N trasormato T e T S ad let be a rst order mamal symmetrc class o varables o X) The rage o T o e T ) ) wll be a mamal symmetry class roo: Based o the revous lemma ay ar o lterals ad y o T ) are symmetrc Recall that or two lterals a symmetry class the symmetry does ot reure addtoal hase assgmet sce arorate hases have already bee assged to the lterals whle geeratg the symmetry classes Now we wll rove that T ) s mamal by showg there s a lteral y symmetrc to lteral T ) e y ) the y s a lteral T ) Sce S s subgrou T S T S rom the revous lemma y T ) T y) rom T ) t ca be see that T ) ad sce s mamal : T ) T y) T y) y T ) Ths roves the theorem

13 The theorem has a strog mlcato that s ay N trasormato T S mas mamal symmetry classes to other mamal symmetry classes Ths result ca be cosdered as a costrat or ay trasormatos o T S It s esecally mortat the rocess o detyg N S sce t wll lmt the sace o trasormatos to be elored I other words to elore ossble N trasormatos T S t s sucet to oly elore N trasormatos that are seced terms o hgher order symmetry classes stead o dvdual varables Sce the umber o classes s usually cosderably ewer tha the umber o varables ths theorem teds to greatly reduce the search sace Let K m rereset the mamal symmetry classes or varables o ucto X) The corresodg N trasormato T S T ) must satsy IV Sgatures ovetoally a sgature a lter or a ecessary codto) s deed as some characterstcs o a Boolea ucto wth resect to oe o ts ut varables We shall reer to such a sgature as a rst order sgature or st -sgature) sce t oly deeds o oe ut varable rst order sgatures have bee used to dety varables that ca be echaged ermuted) wthout aectg the ucto tsel e ay ossble corresodece betwee the ut varables o two uctos s restrcted to a corresodece betwee varables wth the same st -sgature So each varable o a ucto has a uue st -sgature the there ca be at most oe ossble corresodece to ay o the varables o some other ucto That s why the ualty o ay st -sgature s characterzed by ts ablty to be a uue detcato o a varable o a ucto ad o course by ts ablty to be comuted ast The st -sgatures that have bee troduced lterature der terms o ther ualty gure Although the st order sgatures have bee successul a large umber o ractcal cases they do ot utlze the ull otetal o sgatures the Boolea matchg roblem There s o set o st -sgatures that ca uuely dety all the varables However ths goal ca be acheved by usg hgher order sgatures as descrbed below The st -sgatures have bee tradtoally deed or varables However sce we ted to cosder hase assgmet addto to ermutato o ut varables we dee the st - sgatures or lterals as oosed to varables) 3

14 A well-ow st -sgature or a lteral o a Boolea ucto X) s the mterm cout o the ONSET o the coactor o ths ucto wrt e I ar-wse matchg methods or checg -euvalece) a st -sgature must be able to mae out a ut varable deedet o ut varable ermutato so that t ca establsh a corresodece betwee varable o X) wth a varable o some other Boolea ucto g X ) It oly maes sese to try to establsh a corresodece betwee these two varables oly varable o X ) has the same st -sgature as varable o g X ) The ma dea o ths ar-wse matchg aroach s clear: I we are able to comute a uue sgature or each ut varable o X) the the varable mag roblem wll have bee solved there s oly oe or o ossble varable corresodece or -euvalece o ucto X) wth ay other ucto g X) I we d or each varable o X) a varable o g X) that has the same uue sgature the we wll have establshed a corresodece Otherwse we wll ow mmedately that these two uctos are ot -euvalet The ma roblem that arses ths aradgm s whe more tha oe varable o a ucto X) has the same st -sgature t s ot ossble to dstgush betwee these varables e there s o uue corresodece that ca be establshed wth the uts o some other ucto I ths aer we wll geeralze the cocet o rst order sgatures to hgher order sgatures that have comlete eressve ower to hadle the Boolea matchg roblem Recall that a cube roduct term) s the coucto o some lterals Let a a K a be lterals created rom varables K e a a wth the restrcto that both hases o the same lteral are ot reset a a a K e or each varable at most oe o lterals ad belog to { a a K a} Deto: The th order sgature o ucto X ) wth resect to lterals a a K a s the mterm cout o coactor o X) wth resect to cube a a Ka e X ) where X deotes those varables o X that are ot a a K a } ay hase The 0 th order sgature s X) { 4

15 We wll reer to a cube a a Ka as a ostve cube lterals a a a K are ther ostve hases e a a or euvaletly a a a } { } { We also reer to sgatures assocated wth ostve cubes as ostve sgatures Wth resect to varables K the umber o -lteral cubes or the umber o th -order sgatures) s whereas the umber o ostve th -order sgatures s!! )! I the ollowg we troduce a method to eumerate ostve sgatures o ucto X) A -lteral cube ca be reseted as L where } { K } { We mose the costrat that ossble to d K that satsy ths costrat < < K< Obvously or each -lteral cube t s always Beore we roceed urther we must dee the lecograhcal comarso o two vectors Deto: osder two vectors A ) ad B β β β ) where } { K } ad β β β } { K } Let be the smallest de such that { β { The the order relato betwee A ad B s deed as < β A B Wth ths deto a order relato ca be deed betwee -lteral ostve cubes Deto: osder two cubes A L ad B L where β β β < < K< ad β < β < K< β The order relato betwee A ad B s deed as ) β β β ) A We deote the set o all -lteral ostve cubes by Q L < < K< } Sce we have deed a orderg or members o B { Q these members whch corresod to - lteral ostve cubes) ca be rereseted as K where the suerscrt dcates the umber o lterals each cube ad the ollowg orderg s satsed by ther dces: K Notce that! Q! )! Every ossble ostve cube ca be deoted as The set o all ostve cubes s deoted by 0 Q Q Q Q L Q { } 5

16 Now we eted the deto o orderg to members o Q or two cubes ad l l the < l Now l the l < l I ths orderg sgle lteral cubes aear rst ollowed by two lteral ad hgher lteral cubes e 0 Q K Q K K Q K Q } 0 whch ca also be rereseted as ecet or {}): Q Q Q Q K K K L K L L 3 3 We are ow ready to troduce the sgature vector or a ucto X) Deto: or the ucto X) where X ) wth ostve cubes the sgature vector deoted by V s deed as ollows: V st d d sgatures sgatures ) sgatures ) whch ca euvaletly be reseted as: V st d th th sgatures sgatures ) sgatures sgature ) 3 I the ollowg we reset a mortat theorem whch roves that the sgature vector o a ucto s uue e two deret uctos have deret sgature vectors e V g V X ) g X ) To rove ths clam we rst rove that values o all sgatures o all orders ca be obtaed rom the sgature vector whch oly clude ostve sgatures) Ths vector evetually seces the ucto X) uuely or all X B Lemma 3: Values o all th -sgatures ca be uuely obtaed rom the -) th -sgatures ad ostve th -sgatures roo: Gve all -) th -sgatures ad ostve th -sgatures we wat to rove that all th - sgatures ca be comuted A th -sgature geeral ca be deoted by where L { } ad B The roo s by ducto o the umber o egatve 6

17 lterals o L deoted by The value o rages rom 0 to or 0 the cube s ostve ad sce ostve th -sgatures are gve the clam s correct or 0 Assume that t s correct or m e all th -sgatures assocated wth cubes o m egatve lterals are comuted Now we wll comute the th -sgatures assocated wth cubes o m egatve lterals Let s deote such a cube by s + L L + Assume that 0 + oe o the egatve lterals e 0 Let s deote ths cube by L L 0 + Accordgly we create the cube + L L + whch the hase o s + ostve Oe ca easly very that + where L L Hece 0 ca be comuted as Sce s a -)-lteral cube ad cotas m 0 0 egatve lterals the clam s rove by ducto Theorem : or a ucto X ) sgature vector V uuely ad comletely seces ucto X ) roo: Sce the sgature vector cludes all ostve sgatures t ca be see that the values o ca be comuted or all ossble cubes by usg ducto o the umber o lterals ad the revous Lemma However or ths roo we are oly terested -lteral cubes sce + they delver sucet ormato to secy ucto or all ots K ) B e K ) where L Based o ths theorem the ecessary ad sucet codto or two uctos X ) ad g X ) to be eual s that g or all ostve cubes whch are reseted the sgature vectors o X ) ad g X ) ) I ths art we vestgate the mlcato o ths theorem o the geeral symmetry relato osder N trasormato T whch corresods to the symmetry relato X ) TX) Let s deote g X ) TX) Sce uctos X ) ad g X ) are eual ther sgature vectors are eual e g or all L The coactors ad g ca be eressed as X L ad g g X ) TX) X ) L T ) T ) L T ) ) L 7

18 Let s deote t T ) ad T ) t t Lt or L The ecessary ad sucet codto or N trasormato T to be symmetrc s X ) TX) T ) Q Notce that X ) TX) the where deotes ay cube ot oly a ostve T ) cube) A result o the revous theorem s that t roves that a sucet codto or X ) TX) s that or ostve cubes oly We wll revst these results uture sectos V T ) aocal orm I ths secto we reset a caocal orm or Boolea matchg uder NN-euvalece As was roved revous chaters NN-euvalece s a euvalece relato that arttos the set o all sgle outut Boolea uctos to euvalece classes Let s cosder a NNeuvalece class by E X ) X ) )} Every two uctos E are symmetrc to { m X each other e X ) E X ) E X ) X ) ad ay ucto symmetrc to some ucto E s also E The Boolea matchg roblem uder NN-euvalece s reduced to that o veryg whether or ot two target Boolea uctos X ) ad g X ) belog to the same NN-euvalece class I the caocal orm based Boolea matchg a uue reresetatve s selected or every class called the NN-reresetatve o the class Let s deote the NN-reresetatve o a class E by X ) Deto: The NN-reresetatve X ) o a class E s deed as the NN-reresetatve or the caocal orm) o all uctos X ) X ) K ) E Let s deote the caocal m X orm o a ucto X ) by X ) e we use catal letters or caocal orms) We have: X ) X ) X ) K ) Notce that X ) s oe o X ) X ) K ) e m X X ) E m X The NN-reresetatve X ) s selected amog X ) X ) K ) based o some crtera m X that maes X ) uue or eamle oe way s to dee a total orderg or uctos X ) X ) K m X ) ad select the mamum or mmum wth resect to the deed order) as the NN-reresetatve caocal orm) 8

19 Observato: Two uctos X ) ad g X ) are NN-euvalet ad oly they have the same caocal orm e X ) g X ) X ) G X ) The NN-euvalece class that cludes a ucto X ) deoted by E s the set o all uctos that are NN-euvalet to X ) Hece E ca be created by alyg all N trasormatos ad outut hase assgmets to X ) e E TX) B T Γ } { The umber o deret N trasormatos ad outut hase assgmets to X ) s + B Γ! ; However E s geeral less tha B Γ because o the symmetry relatos dscussed revous chaters Sce X ) ad X ) are NN-euvalet there s a N trasormato T such that TX) X ) ; however T s ot the oly such N trasormato Deto: We call the set o N trasormatos T such that TX) X ) the caocty- roducg ) trasormatos : { T Γ TX) X )} We reset a algorthm to comute the caocal orm o a gve NN-euvalece class as well as the set o all trasormatos S) trasormatos trasormatos S ca be easly obtaed rom S was elaed the revous secto We wll show that the set o symmetry-roducg The mortace o detyg all N or ay set S Γ o N trasormatos ad trasormato Γ T we dee TS ad ST as ollows: TS { TT T S} ad ST { T T T S} I S Γ s a subgrou the TS s called the let coset o S determed by T ad ST s called the rght coset o S determed by T Lemma 4: or ucto X ) let T ad T be two trasormatos T T s a S trasormato e T T T T S roo: learly T TX) X ) ad T T X ) X ) whch result TX) T X ) X ) T T X ) T T S Theorem 3: or ucto X ) ad ay trasormato T s the rght coset o determed by T e T S T S 9

20 roo: rst we rove that T S T T : T S T T S T T T T S T X ) X ) T X ) T TX ) TX ) T T X TX ) X ) ) TX ) X ) T Now we rove that T T S T Based o the revous Lemma T T TT S TT T S T T S T ; hece S T It ca be easly vered that S T T S TT S ; hece or ay T S T whch shows that S ca be easly obtaed rom The set o S trasormatos S cludes trasormatos corresodg to smle symmetres I the algorthm that we wll reset et to dety trasormatos rst smle symmetres are deted sce the comutatoal comlety s relatvely lower tha that o geeral symmetres Ths ormato s used ecetly to comute the remag S trasormatos o V The roosed aocal orm S are comuted Net based o I ths art the caocal orm used ths aer s ormally deed As metoed earler amog uctos o a NN-euvalece class the NN-reresetatve s selected based o a crtero that maes the reresetatve uue amog all uctos the class We revously deed the sgature vector or a ucto ad roved that t s uue or every ucto We wll dee a total orderg or uctos based o the lecograhcal comarso o ther sgature vector Deto: osder two uctos X ) ad g X ) wth sgature vectors V ad V g resectvely The order relato betwee X ) ad g X ) s deed as: V g V X ) g X ) The relato X ) g X ) meas that X ) g X ) or X ) g X ) g Ths orderg s well deed sce we roved that V V X ) g X ) A mortat asect o the sgature vector s that t eables us to comare uctos eve they are ot 0

21 uctos o the same varable vector The oly reuremet s the sze o ther varable vectors should be the same osder uctos X ) ad g Y ) where X ) ad X y y y ) the ad g are eual or euvalet) ther sgature vectors are eual e g g V V Also or the order relato g g V V Usg ths order relato the NN-reresetatve caocal orm) s deed as ollows Deto: The NN-reresetatve o a class { X ) X ) )} m X K o uctos s a ucto X ) whch s mamal wth resect to the order relato e { m} X ) X ) V roertes o the aocal orm I ths secto we wll observe some mortat roertes o the roosed caocal orm that are used or the urose o comutg the caocal orm Theorem 4: Let X ) be the caocal orm o a NN-euvalece class E X ) s greater tha or eual to that o ts comlemet X ) whch may also be deoted by X ) ) e X ) X ) roo: Obvously X ) ad X ) are NN-euvalet Thereore they belog to the same NN-euvalece class e X ) E X ) X ) X ) E Sce X ) s the NN-reresetatve o class E t s mamal e X ) X ) learly the eualty ot ossble; hece X ) X ) orollary: X ) X ) roo: learly X ) X ) V V where V ad V are the sgature vectors o X ) ad X ) resectvely ad X ) s the zeroth sgature ad the rst etry o sgature vector V ad ) X s the rst etry o V Sce V V s based o lecograhc comarso V V X ) X ) Let deote the coactor o ucto X ) wth resect to where X ) s regarded as a ucto o X ) ) Sgatures the sgature vector V K + o are the orm o where cube always clude lteral We wll use the

22 otato to dcate that lteral s cluded cube ) Obvously V All sgatures that est the sgature vector o wll also est the sgature vector o e V V It s mortat to ot out that the order whch sgatures aear V s mataed V To state ths act ormally let s deote the locato o V by L V ) We have L V ) {3 } or eamle L V ) + ad L V ) sce s the + st etry o V ad the rst etry o V Recall that the revous chater whe deg the sgature vector we deed a orderg betwee cubes o a gve ucto X ) Oe ca very that ths orderg has the ollowg mortat roerty or ay two cubes ad that cota lteral e ad we have I the cubes are ordered based o the orderg the L ) wll be locato o that orderg rom ths argumet t ca be resulted that L V ) < L V ) Thereore two sgatures ad est both V ad V e ad ) the the order whch they aear V ad V s the same e aears beore V the wll aear beore also < V ad vce versa e L V ) < L V ) L V ) L V ) Theorem 5: Let X ) be the caocal orm o a NN-euvalece class E The coactor o X ) wth resect to the ostve hase lterals s greater tha or eual to that o egatve lterals e { K } roo: The roo s by cotradcto Assumg that or some { } we rove that X ) caot be the caocal orm We wll show that the egatg wll trasorm X ) to aother ucto X ) such that whch s clearly a cotradcto osder the N trasormato: T K + ) + )

23 We create a ew ucto X ) TX ) learly X ) ad X ) are NN-euvalet thus X ) E sce X ) E revously we deed the N trasormato o cubes Usg ths deto X ) TX ) Based o the deto o T ) trasormato T a cube does ot clude e the T ) Thereore T ) T ) T ) We made the assumto that The sgatures o V are o the orm where s a ostve cube ot cludg ) Let be the rst cube ot cludg e ) or sgature vectors V ad V that < Based o the revous Theorem or sgature vectors V ad V wll also be the rst cube cludg ) such that < ad or all cubes beore e or euvaletly ) Notce that sce X ) ad X ) are deed over the same varable vector X or all cubes L V ) L V ) ) Sgatures V are o the orm I addto X ) TX ) T ) I a cube does ot clude the T ) ; Thus Now a sgature T ) rom V s deret rom the corresodg sgature rom V the must clude e Based o these argumets or sgature vectors V ad V wll be the rst cube amog all cubes cludg ad ot cludg ) that < or all cubes beore e ) Ths codto results whch s a cotradcto sce X ) s the NN-reresetatve o ts class E that also cotas X ) the relato must be satsed) Hece the assumto must be wrog whch roves that However the X ) wll be deedet o Thereore orollary: roo: Sce ad are the rst etres o the sgature vectors o ad 3

24 Let s use Q to deote the set all ostve cubes created rom coucto o some lterals amog K Also let Q deote the set o cubes that do ot clude e Q { Q } The set Q s deed as the set o cubes that do ot clude ad e Q Q { Q } It s easy to see that Q Q I Q Smlarly Q s deed as Q I Q I Q Sce we clude the sgature vector o ucto X ) we dee the cube {} 0 whch cotas o lterals Notce that 0 ad Q 0 I the ollowg we study the symmetry o varables the caocal orm Theorem 6: Let X ) be the caocal orm o a NN-euvalece class E ad rereset a cube that does ot clude lterals ad ) or every such cube Q Q ad are eual ad oly ad are symmetrc e ) Q roo: We wll rove that sgatures ad are eual the swag ad wll ot chage the value o the sgature vector o X ) ad vce versa Let s deote the N trasormato that swas ad by T e T K ) ) urthermore let s deote X ) TX ) We wll vestgate the relato betwee sgature vectors o uctos X ) ad X ) The cubes aearg the sgature vector o X ) or X ) ) ca be classed to our tyes or Q the ollowg relatos are satsed: T ) T ) T ) ad T ) As we ca see cubes o the orm ad are varat uder N trasormato T Recall that or a cube X ) TX ) T ) ; thereore the sgatures o the sgature vectors V ad V that corresod to cubes o the orm ad are always eual The sgature vectors V ad V are eual ad oly or every cube ; thereore V V & However rom T ) t ca be cocluded that ad whch combed wth the revous relato results 4

25 Sce X ) TX ) ad T oly swas ad ad are symmetrc Thereore meas that Theorem 7: Let X ) be the caocal orm o a NN-euvalece class E Assume that ad wth < ) are ot symmetrc There ests a cube Q such that > ad or every cube Q beore e ) ad are eual e Q > ad Q roo: Let Q be the rst cube Q that rom the revous Theorem sce ad are ot symmetrc such a cube ests We wll rove that > The roo s by cotradcto We wll rove that < the swag ad X ) wll result aother ucto X ) E wth whch s a cotradcto sce X ) s the NN-reresetatve o E Let s deote the N trasormato that swas ad by T K ) ) Also let s deote X ) TX ) Now we comare sgatures ad o sgature vectors V ad V Assumg that geerally deotes a cube ot cludg ad e sgatures ad are Q eual or cubes o the orm o or e ad T ) Hece T ) V ad V may oly be deret sgatures wth cubes o the orm or Notce that ad T ) Hece < < ad T ) Q whch roves that V V Sce V V results a cotradcto; Thus > must be correct orollary: I or the caocal orm X ) ad < ) are ot symmetrc the roo: Sce 0 {} s the rst cube Q I addto sce 0 t ca be see that ad 5

26 orollary: or the case that let { } be the rst umber that ; such ests > roo: Notce that cubes o the orm where { } belog to Q or l { } < l l Aother way o descrbg ths result s V ) V ) where V ) s a artal sgature vector deed as V ) K K + + ) Smlarly V ) s deed as V ) K K + + ) Theorem 8: Let X ) be the caocal orm o a NN-euvalece class E I or < ad are symmetrc ucto X ) the all other where < < are also symmetrc to ad e + L roo: The roo s by cotradcto Assume that there est where < < ) that s ot symmetrc to ad Let s deote the N trasormato that swas ad by T K ) ) Sce ad are symmetrc TX ) X ) Let Q be the rst cube Q such that Based o the revous Theorem sce < > ad Q I addto let Q be the rst cube Q that Based o the revous Theorem sce < > ad Q ube Q does ot clude ad However t may or may ot clude Smlarly cube Q may or may ot clude Hece our ossbltes est We wll vestgate all cases rst let s assume that ad e Q whch results T ) ad T ) sce Q Now sce TX ) X ) T rom the deto o ) > > Thereore based o the deto o ad sce Q 6

27 should satsy the relato because the ad must be eual Thereore aga based o the deto o Q However T sce Q ) ad thereore T whch combed wth the ) revous relato results Ths s clearly a cotradcto sce based o the deto o > Hece our last assumto e ad ) must be correct Net we vestgate the case ad learly Q rom the deto o > > Sce Q T ) T whch combed wth the revous ) relato results > Thereore based o the deto o ad sce Q should satsy the relato Sce whch combed wth results Notce that ; hece Thereore aga based o the deto o Q However T ) sce Q ad thereore T whch combed wth the revous relato ) results Ths s a cotradcto sce based o the deto o > Hece the assumto e ad ) s also correct Smlarly oe ca rove that ad leads to cotradcto Hece the oly remag oto s ad whch the ollowg we wll show that results a cotradcto learly Q ad Q rom the deto o > > Sce Q T ) T ) ad thereore whch combed wth the revous relato results > Thereore based o the deto o ad sce Q should satsy the relato because the ad must be eual Thereore sce Based o oe o mortat roertes o the order relato that we deed betwee cubes whch 7

28 combed wth results Thereore aga based o the deto o Q However ; also T ) sce Q Thereore T whch combed wth the revous ) relato results Ths s clearly a cotradcto sce based o the deto o > Hece the statemet that there est where < < that s ot symmetrc to ad s correct whch roves that + L Ths Theorem dcates that or the caocal orm X ) symmetrc varables are ostoed cosecutvely X ) e varables wll be arraged as: L + + L + L + L where K are symmetry classes ad VI + omutg the caocal orm Gve a ucto X ) the obectve s to d ts caocal orm X ) ad the corresodg set o trasormatos { T Γ TX) X )} Euvaletly the set o trasormatos ca be eressed as: { T Γ T Γ TX) T X )} revous Theorems mose some codtos o the caocal orm X ) or a trasormato T such that T X ) X ) we roect the codtos o X ) to codtos o T These codtos sgcatly lmt the search sace or ) sce T X ) X ) X ) T X ) Let s deote the verse o T by T T where ) Let s also deote the varable ater hase assgmet by X X e ) or smlcty Wth ths otato T ) ) X X X K ) Based o the relato X ) T X ) the ) ) ) codtos o revous Theorems are traslated as ollows The rst codto s o the outut hase assgmet Sce should satsy the codto Aga or smlcty we deote hase assgmet The secod 8

29 codto s o hase assgmet whch ollows rom or ad traslates to ) The et codto s as ollows I or < ) ad ) are symmetrc ucto X ) or X ) ) the all other ) where < < are also symmetrc to ) ad + L ) e ) ) ) ) ) Aother mortat costrat s that ) ad ) are symmetrc ucto X ) or X ) ) the ) V V ) where V ) s a artal sgature vector deed as V V ) ) K ) ) ) ) ) ) + ) ) ) ) + ) ) ) ) K ) ) ) ) ) ) ) K ) ) ) ) K + + ) ) ) Symmetry classes o varables wll be arraged as: ) L L L ) ) ) ) L ) ) where wth members s a mamal symmetry class or ucto X ) The rst order sgatures are sorted ocreasgly L ) > ) L ) > L > ) L ) ) Oe ca use ths roerty to reduce the comlety o detyg symmetry classes sce a ecessary codto or symmetry o ) ad ) s the eualty o ad e ) ) ) ) ) ) These codtos o T are ecessary codtos or T ) Hece we dee T verse o as { T Γ T } the these are ecessary codtos or T A mortat roerty o the trasormato set s as ollows I or two N trasormatos T ad T satsy the eualty TX ) T X ) the T s detely ot a trasormato e TX ) T X ) T These costrats are eectvely used to dety trasormatos as descrbed the ollowg 9

30 VI The omute_ Algorthm The roosed algorthm called omute_ uses sgatures o ucto X ) to comute the trasormatos o uts ad corresodg outut hase assgmets However at the begg oly 0 th st ad ecessary d sgatures are used I most cases st -sgatures aloe determe a cosderable orto o the eualtes reured to dety the desred N trasormato Otherwse as wll be elaed later the remag comarsos are erormed usg d -sgatures Smlarly hgher order sgatures are used oly they become ecessary Eermetal evdece shows that the great maorty o cases a sgature eualty occurs or the low order sgatures 0 th st ad d sgatures) Itutvely the reaso s that the lower order sgatures deed o hgher umber o mterms o the ucto ad thus cota more ormato about the ucto or eamle a st -sgature deeds o - mterms whch s oe hal o the whole Boolea sace mterms) whereas a d -sgature deeds o oe-orth o all mterms - mterms) Hece the st -sgatures are the most owerul ad eectve sgatures The d -sgatures are the et most eectve sgatures ad so o The reader wll observe that ths arragemet o the roosed sgature vector mmzes the comutatoal comlety The rst ste o the omute_ algorthm s to dety the outut hase assgmet I the the outut hase ca be uuely determed rom the relato > However the outut hase caot be determed by usg oly 0 th sgatures I ths case ) the outut hase s cosdered udecded whch s deoted by u Thus we have eteded the set o accetable hases { 0} to { 0 u }) I the outut hase s udecded t wll be determed subseuet stes o the algorthm I the ed the algorthm wll retur a value { 0} or the hase or smlcty we wll deote I the et ste ut hase assgmet s erormed by usg st -sgatures rst assume that the outut hase s decded e u The ooste case wll be dscussed later or varable the the hase o ca be uuely determed by usg the relato > However caot be determed by usg oly st -sgatures whch case the hase o s cosdered udecded whch s aga deoted by u ) 30

31 Udecded ut hases wll be determed subseuet stes o the algorthm I the outut hase s udecded e u the st -sgatures are used as ollows to determe outut ad ut hases The are two ossbltes or the al value o e 0 or Each ossblty or results ut hase assgmets ) where the arethess dcates the deedece o o ) based o the ollowg relato > ad ) ) ) u Notce that eve ) u the value o wll be uue e ) ) u ) Net we sort sgatures a o-creasg order Let ) be a ermutato that results such a orderg e L where ) ) ) ) or smlcty Let V ) deote the vector comrsg o the ordered st -sgatures e V ) ) Assumg that s the outut hase assgmet that ) ) ) results the caocal orm oe ca easly see that V ) cossts o d to + ) st etres o the sgature vector o the caocal orm o X ) The rst etry s ) Now V ) V ) the must satsy the ollowg relato: V ) V ) sce the caocal orm has the mamal sgature vector Ths relato uuely determes the outut hase assgmet uless V ) V ) whch case the comutato o s aga ostoed to the subseuet stes o the algorthm Let s deote a varable ater hase assgmet as I the et ste symmetry classes o varables are determed A ecessary codto or symmetry chec s erormed or ad are udecded we wll determe ad s ; thereore the ad oly I the hases o varables to be symmetrc oly they are symmetrc deedet o ther hases e ad A eamle o ths stuato s whe X ) deeds o Based o these symmetry relatos we orm the mamal symmetry classes o varables m ucto X ) wll rema varat uder ermutatos sde a symmetry 3

32 class Based o ths act ad sce symmetrc varables are ostoed cosecutvely stead o dg N trasormatos o the varables t s sucet to search or N trasormatos o classes m whch wll tur greatly reduce the sze o search sace I act ths s a maor advatage o the roosed techue comared to revous aroaches The obectve s to determe all members o member o ad Obvously a N trasormato T s a T S the the trasormato T T s also a member o Recall that S s the set o S trasormatos e T S X ) T X ) ) Let s deote the subgrou o S trasormatos that corresod to smle symmetry o varables by W A eamle o a member o W s the N trasormato that swas varables ad Notce that W also cludes the cascades o N trasormatos that corresod to smle symmetry o varables e W s closed wth resect to the cascade oerato: T W T W TT W Obvously W S ad also T TW Thus to avod comutatoal redudacy the roosed algorthm wll retur oly oe member o TW sce gve a N trasormato T oe ca easly obta all members o TW To ela ths matter ormally we dee a relato betwee T T where T T meas that there est a T W such that T T T Ths relato breas to euvalece classes The roosed algorthm returs the set whch cotas eactly oe member o each such class e T T T W TT or euvaletly T T T Now we wll dscuss the cocet o N trasormatos o classes The hases o classes that cota varables wth decded hases are deed as The hase assgmet or classes that cota varables wth udecded hases s deed as: { } where s to be determed A N trasormato o classes m sges a trasormato o varables X ) The coactor o ucto X ) wth resect to ay member o a class s a uue ucto; hece the coactor o X ) wth resect to the class ca be deed as or 3

33 Let s deote the classes ater hase assgmet as m I the et ste classes are ordered based o ther rst sgatures Let be a ermutato o classes that resects the orderg: L eve whe the hase o a class ) ) s udecded m ) the st -sgature ca be deed sce ) I the st -sgatures are dstct values or m the a uue orderg ca be acheved sce > > L > whch case the algorthm termates ) ) m ) returg the N trasormato resultg rom ad the corresodg hase assgmet as a member o { T Γ T } Otherwse the classes are laced grous such that all classes sde a grou have the same st -sgature: G G G ) L ) ) L ) L L ) m) where L L L L ) ) + ) + ) + ) We reer to a grou as uresolved t cotas more tha oe class or the hases o classes that grou are udecded I all the grous are resolved a uue orderg has bee obtaed ad the algorthm termates The obectve o et stes s to resolve all uresolved grous Let G { ) + ) l) } be the rst uresolved grou Sce all grous G G K G are m ) resolved e they cota a sgle class wth decded hase) the orderg o classes u to deted The case that G s uresolved s dscussed later) G s Now the d -sgatures are used to secy the orderg sde the uresolved grous startg wth G Sce G s resolved G { )} d -sgatures wth resect to ) ad ) or l e ) ca be used or hase assgmet eeded) ad orderg classes ) ) ) + ) l) later o ths ste wll be reerred to as terato ) 33

34 I hases o classes ) G s udecded the d -sgatures ) ) ad are ) ) comared ad hase or ) s decded based o > I case o eualty o ) ) ) ) the d -sgatures the hase o ) remas udecded Net ew values o d -sgatures ater hase assgmet are used to order classes ) ) ) + ) l) ad subseuetly regrou these classes e the values o ) + ) l) are udated to resect the ollowg orderg: L Subseuetly G ) ) ) + ) ) s slt to smaller grous such that l ) sde each grou the d -sgatures are eual The same rocedure hase ) ) assgmet orderg ad regroug based o ) s aled to all other uresolved grous ) ) ally the dces o ew grous are roerly udated I ater these stes there stll ests some uresolved grou G l a smlar rocedure called terato ) s aled based o d - sgatures wth resect to ) ad ) Gl e assumg l > ) I eeded ) ) teratos 34 K are aled I at terato there stll ests some uresolved grous ad G tsel s also uresolved the rocedure descrbed below wll be used Ths case cludes the case where G s uresolved) At ths ot the values o ) ) K ) have bee alzed so s the value o or ay other grou wth oly oe class) However the values o ) + ) l) grou G ) s ot al sce st ad d sgature have ot made them dstct) The al value or ) ca be ay value amog ) + ) l) alzg the value o ) to oe o ) + ) l) s euvalet to slttg G { ) + ) l) } to grous { )} ad { ) K ) + ) l) } ad udatg ad dces o grous so these two grous are rereseted by G } ad G } Thereore there are l + { ) + { + ) + ) l) ways to do ths slt l + ways to secy ew G ) ad the hase o ) s udecded 34

35 the there wll be two ways to resolve the grou ew G two hases ad 0) oseuetly there are r l + or r l + ) ) ways to secy ad resolve the ew grou G All these r cases eed to be traced sce t s uow whch oes) wll result a mamal trasormato or each case the d -sgatures are used to rst order ) ) classes sde the uresolved grous amog G + G ad the slt them based o the G + K m outcome o orderg Ths rocess wll cotue or all r cases recursvely c the recursve-resolve algorthm) utl all grous are resolved All resultg N trasormatos cases are stored where geeral result o the recurso rocess) T T K Ts resultg rom deret s r sce each returs more tha oe trasormato as a Because o the way they have bee costructed the verses o trasormatos are amog T T K T } or euvaletly { T T K T } Hece by usg st ad d sgatures we { s s have lmted the search or a amog all! trasormatos o Γ to search amog { T T K T s } whch s a sgcatly smaller sze sace tha Γ Ideed our eermetal results corm that most o the tme all members o { T T K } are trasormatos s e { T T K T } whch mles that most o the tme oly st ad d sgatures are caable o determg the caocal orm Members o amog { T T K T s } are deted based o the act that T ad oly T X ) T X ) or s Ths tas reures the erormg comarso T X ) T X ) reeatedly The comarso s doe based o ther sgature vectors However beore usg the sgature vectors the ossblty euvalecy o T ad T should be cosdered e rst the relato T X ) T X ) should be checed sce case o eualty ther sgature vectors wll be eual as well O the other had T X ) T X ) as we roved a theorem earler ther sgature vectors are deret e ether T X ) T X ) or T X ) T X ) T s 35