Marek A. Perkowski, Malgorzata Chrzanowska-Jeske, and Yang Xu, Portland State University. Portland, OR Abstract

Transcription

1 LATTICE DIAGRAMS USING REEDMULLER LOGIC Mrek A. Perkowski, Mlgorzt ChrznowskJeske, nd Yng Xu, Deprtment of Eletril Engineering Portlnd Stte University Portlnd, OR 9727 Astrt Universl Akers Arrys (UAA) llow to relize ritrry Boolen funtion diretly in ellulr lyout ut re very reineient. This pper presents n extension of UAAs, lled \Lttie Digrms" in whih Shnnon, Positive nd Negtive Dvio expnsions re usedinnodes. An eient method of mppig ritrry multioutput inompletely speied funtions to them is presented. We prove tht with these extensions, our onept ofregulr lyout eomes not only fesile ut lso eient. Regulr lyout is fundmentl onept in VLSI design whih n hve pplitions to sumiron design nd designing new negrin FPGAs. INTRODUCTION. Akers dened Universl Akers Arrys (UAA) to relize ritrry Boolen funtions in regulr nd plnr lyout []. UAA is retngulr rry ofidentil ells, eh of them eing multiplexer, where every ell otins signls from South nd Est nd gives its output to North nd West. All ells on digonl re onneted to the sme (ontrol) vrile (see Fig. ). In generl, vriles hve to e repeted to ensure relizility of n ritrry (singleoutput, ompletely speied) funtion. Akers showed universl method of seleting nd repeting vriles for suh struture so tht ll funtions re relizle in it. His pper did not get due ttention euse the tehnology ws t too erly development stge then. Perkowski, Pierzhl, nd Grygiel generlized the Akers pproh [] to inry, multivlued, fuzzy, nd ontinuous funtions nd more generl regulr nonplnr lyout geometries [9, 2]. Aritrry nonsingulr expnsions were used in the frmework of linerly independent logi. Sso nd Butler onentrted on plnr funtions for multivlued logi [24]. ChrznowskJeske [3, 4] introdued the plnr PseudoSymmetri Binry Deision Digrms (PSBDDs), whih hd the sme ells nd si onnetion ptterns s UAAs, ut the rry ws not neessrily retngle, nd ws not lulted lwys for the worst se s in [], ut heuristi deisiondigrm sed methods were used to nd good order of expnsion vriles. This produed dierent shpes, usully tringulr, nd in most ses muh smller thn the upper ound solution of Akers. To generte PSBDDs for ritrry funtions new opertion on verties ws introdued, lled joining. For the rst time it ws shown in [4] tht for rellife funtions it

2 sum=3 level 2,,2,3,4,5 2, 2,2 2,3 2,4 2,5 3, 3,2 3,3 3,4 3,5 4, 4,2 4,3 4,4 4,5 5, 5,2 sum=2 level 5,3 5,4 sum=4 level 3 5,5 () () (d) (e) x y z v r s RULE(pD,pD) x y z v r s x y z v r r s s RULE(S,S) RULE(nD,nD) RULE(S,pD) x x x r r s r y z y z s r y z y z y z s s v v v () (f) x y z v r s RULE(pD,S) x y r z y z s v x y z v x y z v Figure : () Arry to explin Lttie onepts. () (f) Joining Rules to rete Kroneker Lttie Digrms nd relted digrms. Left side efore joining nonisomorphi nodes, right side fter joining nodes nd possily, propgting orretion to the right predeessor of node s. Corretions re propgted only in rules (), (d), nd (e). ws possile to generte regulr symmetri rrys sed on vrile repetition with fesile numer of suh repetitions. In this pper we generlize nd extend seleted ides from [, 3,4,9,2]. Our struture generlizes the known swith reliztions of symmetri inry funtions, ut it llows for Shnnon (S), Positive (pd) nd Negtive Dvio (nd) expnsions in nodes, s well s for onstnt dt input funtions. Although in this pper the lttie is still plnr nd sed on retngulr grid, the solution spe is muh expnded nd etter designs re otined. Kronekerlike, PseudoKronekerlike ndfolded Lttie Digrms n e relized for every funtion. The remining of this pper is strutured s follows. Setion 2 presents kground on ellulr logi reliztions. Setion 3 introdues our extensions of Universl Akers Arrys: the Lttie Digrms. The si onepts nd types of Ltties re dened. In setion 4 we present how to rete Ordered Shnnon Lttie Digrms. Setion 5 extends these methods for Kroneker Lttie Digrms. Setion 6 presents the entire design methodology. New experimentl results for Shnnonlike ltties re given in setion 7, nd setion 8 desries our urrent extensions to this reserh. Setion 9 onludes the pper. 2 FROM AKERS ARRAYS TO KRONECKER LATTICES. The pproh ofakers from 972 ws the only one in the literture tht proposed ltties s fundment of ellulr strutures. It n e treted s n ttempt to omine the properties of PLAlike nd treelike ellulr strutures. Although sed on retngulr grid similr to PLAs, the UAAs used multiplexer ell, llowing to use Shnnon expnsions, nd thus UAAs were similr to tree expnsions. The universl onstrution of Akers repeted vriles onseutively. Thus, sed on lws = nd =, the UAA silly reted trees inside the rry

3 nd the ells were often used only s "extenders". This ws, however, very wsteful method, leding to lrge rrys in ll ses. A more fruitful pproh is to derive UAAs nd Shnnonsed ltties from the wellknown plnr BDD of symmetri funtion. Suh BDD leds diretly to Universl Akers Arry onnetion struture nd lyout, ut not neessrily to Akers' wy of repeting vriles nd synthesizing funtions. Our pproh diers from Akers in the wy how the vriles re repeted, we rete only the minimum neessry numers of repetitions. The reted y us lgorithms for vrile ordering nd repeting re lso quite dierent [7, 2]. The shpe of our ltties is pproximtely tringle or trpezoid of vrious sizes nd shpes, while UAA is lwys squre of the lrgest size for given numer of vriles. The rrys of Akers were universl in sense tht it ws proven tht every inry funtion n e relized with suh struture, ut n exponentil numer of levels ws neessry (whih mens, the ontrol vriles in digonl uses were repeted very mny times). So, they were unneessrily lrge, euse they were lulted one for ll for the worst se funtions. No eient proedures for nding order of (repeted) vriles were given, nd it is esy to show simple funtions tht hve very lrge UAAs. Nevertheless, the ide of the Akers Arry is very ptivting from the point of view of sumiron tehnologies, euse: () ll onnetions other thn input uses re lol nd short, (2) delys re equl nd preditle, (3) lterriving vriles n e given loser to the output, nd (4) logi synthesis n e omined with lyout, so tht no speil stge of plement nd routing is neessry (similrly s in PLAs). Beuse of the progress in hrdwre nd softwre tehnologies sine 972, our pproh is quite dierent from tht of Akers. Wedonotwnt to design universl rry forll funtions, euse they would e very ineientfornerly ll funtions. Insted we rete lyoutdriven logi funtion genertor tht gives eient results for mny rellife funtions, not only symmetril ones. We rgue tht there is no need to relize the "worstse" funtions, sine it ws shown in [22] tht, in ontrst to the rndomly generted "worstse" funtions, 98% of funtions from rellife re AshenhurstCurtis deomposle [6]. Therefore, the "other" funtions re either deomposle to the esy relizle funtions, or they do not exist in prtie [6, 22]. In ddition, numer of heuristis for vrile ordering nd lttierelted logi prtitioning ws proposed [3, 4, 5, 7, 2], nd it ws shown tht with good order sustntil minimiztion of the digrms n e hieved. Bsed on the nlysis of vrious reliztions of rithmeti, symmetri, unte, "tough", nd stndrd enhmrk funtions nd new tehnologies [2, 8], we hve sustntilly generlized the onepts from [, 3, 4, 5] in the following wys:. We strt from tree expnsion nd, levelylevel, we omine together nonisomorphi nodes t the sme level, thus reting Direted Ayli Grphs in similr wy sin [4]. This leds in turn to the requirement of vriles repetition. There is no onstrint on repeting vriles onseutively (s in UAAs) or not repeting them onseutively s in [4]. Here, the methods for ordering nd repeting vriles re totlly unonstrined, whih leds to etter results. The ompleted lttie digrm with onseutive repetitions in loks, n e next modied to regulr rry tht hs less nodes with repeted vriles. Therefore, in oth ses, the onept of repeting vriles is useful strting point of lyout design. 2. Insted of ssuming only Shnnon expnsion s in UAAs nd PSBDDs, we useny

4 suset of S, pd nd nd expnsions. We llow to use them s in single polrity digrms, Kroneker type digrms or PseudoKroneker type digrms. So, mny ides from the re of \ReedMuller" logi n e now orrowed nd expnded. 3. Our pproh is for ritrry multioutput, inompletely speied funtions. Lttie Digrms re ounterprts of the known trees nd Deision Digrms of respetive nmes [23]. For ny type of Deision Digrm known from the literture one n design n equivlent Lttie. In this pper we hve spe to demonstrte this property for only few simpler representtions. We demonstrte tht for every funtion n rry ofthe new type n e reted tht is never worse (nd in most ses, it is muh etter) thn those formulted in [3,4,5,9]. 3 Denitions of Lttie Digrms. In this setion, we will rst dene preisely \wht re" the Lttie Digrms, nd only then in next setions, we will desrie \how to rete" them for funtions. Roughly, Lttie Digrms re dt strutures tht desrie oth regulr geometry of onnetions, nd logi of iruit. Given is retngulr rry L, see Fig., with rows nd olumns enumerted strting from. Fig. shows the enumertion of entries. Eh nonzero entry L[i j] in rry L is lled node nd inludes dt struture tht desries logi pled in this entry. L[ ] is the root node of the lttie. In gures, the rrys will e rotted lokwise 45 degrees to mke the digonls horizontl. Denition. A digonl of the mtrix is set of entries tht hve the sme sum of indies. The sum in the rst digonl is 2, in the seond digonl is 3, nd so on. A digonl orresponds to the level of the lttie. Levels re enumerted strting from. A symmetri funtion of n vriles tht hs no vuous vriles, hs n levels in its orresponding lttie (mtrix). Denition 2. For every entry (node) L[i j] the following entries (nodes) re dened (if entries with these indies exist): (geometril) left predeessor (LP) of L[i j] istheentry (node) L[i + j], (geometril) right predeessor (RP) of L[i j] is the entry (node) L[i j + ], (geometril) left suessor (LS) of L[i j] istheentry (node) L[i j ; ], (geometril) right suessor (RS) of L[i j] istheentry (node) L[i ; j], (geometril) left neighor (LN) of L[i j] istheentry (node) L[i + j ; ], (geometril) right neighor (RN) of L[i j] istheentry (node) L[i ; j + ]. If x is predeessor of y, then y is suessor of x. Every nonterminl node in L relizes funtion S, pd or nd of: () its two geometri predeessors nd the ontrol vrile vr or (2) one of its two geometri predeessors nd the ontrol vrile vr (the other dt input is ssigned to onstnt or.) The ontrol vrile is tken from the us in the digonl. The root L[ ] orresponds to the output of the funtion f. Theterminl node (lef) is funtion of only the ontrol vrile (the vlue of the ontrol vrile is tken from the signl orresponding to the digonl. Both its dt inputs re onstnts.) Evlution of node funtions do is done s follows:

5 do = vrdi LP vrdi RP,forStype of node expnsion, do = di LP vrdi RP, for pd type of node expnsion, do = di LP vrdi RP, for nd type of node expnsion. where: If di LP is not onstnt, its vlue omes from LP. If di RP is not onstnt, its vlue omes from RP. If the vlues re onstnts, they re used in the evlution. Every nonoutput node gives its vlue to two geometril suessors, ut one of them n hve onstnt dt input, so logilly, every nonoutput node gives its vlue to single or to oth of its logi suessors. Whether the geometri left suessor eomes logi suessor of the right predeessor node, depends on the vlue set for the right dt input in this suessor. If it is or, then it is disonneted from the predeessor. Denition 3. A Lttie Digrm for singleoutput funtion is represented y mtrix L in whih: () Nonzero entries L[i j] orrespond to logi nodes nd re represented y reords [expnsion type vr LEF T RIGHT ]. where: (A) expnsion type is the expnsion type pplied in this node, S, pd, nd, LE nd RE where LE is the left extender, nd RE is the left extender. The extenders represent wires going from left nd from right, respetively. (B) vr is the ontrol vrile in this expnsion. Its vlue is irrelevnt for extenders. (C) LEF T is onstnt or reord (pointer to ON LP,pointer to OF F LP ), (D) RIGHT is onstnt or reord (pointer to ON RP,pointer to OF F RP ). (2) Every terminl node hs no logi predeessors. (3) Every nonterminl node hs one or two logi predeessors. (4) Every nonterminl node hs one or two logi suessors. (5) For every lef node there exists logi pth to the output. (6) All other entries, tht do not represent logi nodes in the mtrix, hve vlue. Denition 4. A Lttie Digrm relizes funtion F when the funtion f otined y its nlysis forms n \inomplete tutology" with funtion F. Grphilly, n esy method to nlyse the lttie is to nd EXOR of ll produt terms of literls on the pths leding to onstnts. Denition 5. An Ordered Lttie Digrm is lttie digrm in whih there is one vrile on digonl. Denition 6. An Ordered Lttie Digrm with Repeted Vriles is one in whih the sme vrile my pperonvrious levels, ut only one vrile in level. Denition 7. A Free Lttie Digrm is lttie digrm in whih there re dierent orders of vriles in the pths leding from lefs to the root. Denition 8. An Ordered Shnnon Lttie Digrm (OSLD) is n ordered lttie digrm in whih ll expnsions re Shnnon. It is ounterprt of the Shnnon Trees nd the Binry Deision Digrms.

6 A simplied wy of drwing OSLDs is shown in Fig. 3. Denition 9. A Funtionl Lttie Digrm is n ordered lttie digrm in whih ll expnsions re Positive Dvio. It is ounterprt of the Positive Dvio Trees nd the Funtionl Deision Digrms. Denition. A Negtive Funtionl Lttie Digrm is n ordered lttie digrm in whih ll expnsions re Negtive Dvio. It is ounterprt of the Negtive Dvio Trees nd the Negtive Funtionl Deision Digrms. Denition. A ReedMuller Lttie Digrm is n ordered lttie digrm in whih inevery level ll expnsions re of the sme type, either Positive or Negtive Dvio. It is ounterprt of the ReedMuller Trees nd the ReedMuller Deision Digrms. For symmetri funtion of n vriles nd given ordering, there re 2 n dierent ReedMuller Lttie Digrms. In generl, for symmetrized funtion with N vriles (in N the repeted vriles re ounted seprtely), there re 2 N dierent ReedMuller Lttie Digrms (it n e shown tht every funtion my esymmetrized y repeting its vriles). Denition 2. An Ordered Kroneker Lttie Digrm (OKLD) is n ordered lttie digrm in whih inevery level ll expnsions re of the sme type, Shnnon, Positive or Negtive Dvio. It is ounterprt of the Kroneker Trees (lled lso KronekerReedMuller) nd the Ordered Kroneker (Funtionl) Deision Digrms. For symmetri funtion of n vriles nd given ordering, there re 3 n dierent OKLDs. Eh OKLD is desried, efore or during its retion, y the list: f(vr exp ) (vr exp 2 ) (vr 2 exp 3 ) (vr 3 exp 2 ) (vr exp 2 ) :::: (vr n exp m 2)g. whih speies the order of (repeted) vriles nd expnsion types orresponding to them. Eh element of this list orresponds to level of the lttie. Denition 3. A Pseudo ReedMuller Lttie Digrm (PRMLD) is n ordered lttie digrm in whihinevery level ll expnsions re either PositiveDvio or NegtiveDvio. It is ounterprt of the Pseudo ReedMuller Trees nd the Pseudo ReedMuller Deision Digrms. Denition 4. A Pseudo S/pD Lttie Digrm is n ordered lttie digrm in whih in every level ll expnsions re either Shnnon or Positive Dvio. Denition 5. A Pseudo S/nD Lttie Digrm is n ordered lttie digrm in whih in every level ll expnsions re either Shnnon or Negtive Dvio. Denition 6. A Pseudo Kroneker Lttie Digrm is n ordered lttie digrm in whih in every level ll expnsions re either Shnnon, Positive Dvio, or Negtive Dvio. In other words, there re no onstrints on expnsion types S, pd, nd in levels. It is ounterprt of the Pseudo Kroneker Trees nd the Pseudo Kroneker Deision Digrms. Denition 7. A Free Kroneker Lttie Digrm is lttie digrm in whih there re no onstrints on orders of vriles in rnhes nd on expnsion types S, pd, nd in levels. It is ounterprt of the Free Kroneker Trees nd the Free Kroneker Deision Digrms. Denition 8. A Folded Kroneker Lttie Digrm is lttie digrm in whih there re no onstrints on expnsion types S, pd, nd in levels nd on the numer of vriles in level, ut the order of vriles in levels must e the sme, with some vriles possily missing. Thus mny vriles my pper in level, whih we ll folded level. Thus, in one rnh the order my e d nd in nother e. However it is

7 not possile to hve rnhes with orders nd. This would e possile in Free Kroneker Lttie Digrms. Oserve tht Folded Ltties re speil se of Free Ltties. Free Ltties will e not disussed here. 4 Methods to rete Ordered Shnnon Ltties. The Ordered Shnnon Lttie for funtion is expnded levelylevel, strting from the root level ( single node orresponding to the funtion), nd from left to rightinevery level (see Fig 2). First oftors of nodes re reted using Shnnon expnsion, nd next the joining opertions re exeuted on some oftors y, z from the lowest level refer to Fig. f. Nonjoined oftors re onverted to nodes. Joining opertion \RULE(S,S)" from Fig. is pplied to Shnnon nodes. It refers to ny two geometri neighor nodes r nd s when oth oftors y nd z of them re nononstnt (y is the positive oftor of r, ndz is the negtive oftor of s, in Figure, negtion of is denoted y ). In ontrst to stndrd BDDs, the joining opertion omines lso nonisomorphi nodes of trees. If tutologil funtions (i.e. isomorphi nodes) hppen to e neighor oftors y, z for joining, these oftors re omined using stndrd \joining opertion", in this se, z z = y y = z, nd the se of isomorphi oftors is not espeilly distinguished y our methods. If funtion is not symmetri, using joining opertions leds, in generl, to the neessity of repeting some vriles in ltties. The lookhed vrile seleting heuristis tht we use, serve tovoid too mny repetitions, nd lso to rete s few s possile rnhes of the lttie. The eorts of our vrious heuristis is to omplete s soon s possile every rnh of the lttie, thus mking node terminting it onstnt. Mximizing the numer of logi onstnts re then the \gol funtions" towrds whih lookhed seletions of vriles nd expnsions re exeuted. Our exmples show, tht for rellife enhmrk funtions, nd strting from the Curtis deompositionl hierrhy of prtitioning vriles [6], the overhed of vrile repeting is not exessive ineh lttierelized lok otined from the Curtis deomposition. The vrile ordering/repeting nd Curtis deomposition spets will e not disussed here. Exmple. Figure 2 presents the retion proess of the Ordered Shnnon Lttie Digrm otined using the method for singleoutput ompletely speied funtions sed on rules from Fig.. The rrows point from suessors to predeessors, to emphsize the order of reting the lttie from outputs to inputs, in ontrst to OSLD dt struture in whih the diretion of rrows is reversed nd orresponds to the diretion of informtion ow (lso in the orresponding iruit from Fig. 2 the ow is from inputs, t the ottom, to the output). A funtion relized in every node of Fig. 2 is written inside the ovl orresponding to the node. The rrow to the left predeessor is for negted vrile nd, efore joining, leds to the negted oftor. Right rrow is for the positive oftor nd leds to the positive oftor efore joining. Thus, strting from level, the Shnnon expnsion for seleted vrile X is pplied whih leds to two oftors orresponding to two nodes of the seond level. Now vrile X2 is seleted for the seond level. Negtive oftor is lulted for node (X2 X3). Itis. Positive oftor of node (X2 X3) is R = (X3). Negtive oftor of node (X2' X4' + X2 X4 + X2 X3) is L = (X4') nd its positive oftor is (X4 + X3). Now the joining opertion RULE(S,S) is pplied to R nd L euse prent nodes re neighors nd oth oftors R nd L re not onstnts. This retes the seond level node (X2 X3 + X2' X4')

8 Level x x2 x4 + x2 x3 + x x2 x4 x x X x2 x2 x2 x2 x2 x3 x2 x4 x2 x4 x4 R x2 x4 + x2 x4 + x2 x3 L x2 x2 x3 + x2 x4 x3 x3 R2 x3(x2+x2 x4 )+x3 x4 x3x2+x3x2 =x3 x2 x4 x4 x4 + x3 x3x2+x3 x2 x2 x2 x2 x3 x3 L2 x3 x3 X3 X4 X () () Figure 2: Method for retion of SingleOutput Shnnon Lttie for ompletely speied funtion represented y ON ues, () the iruit orresponding to ) efore the propgtion of onstnts. whih is pled into the lttie. This wy, three top levels of the lttie were reted. Vrile X3 is seleted for expnsions in the third level (nd for joinings in the fourth level). Negtive oftor of node (X2 X3 + X2' X4') is (X2' X4') nd its positive oftor is R2 = (X2 + X2' X4'). Negtive oftor of node(x4 + X3) is L2 = (X4), nd positive oftor is onstnt. After the joining opertion for L2 nd R2, level 4 is ompleted. Vrile X4 is seleted for level 4, nd negtive nd positive oftors of node (X2' X4') re lulted. Beuse the positive oftor is onstnt, the joining opertion for it will e not exeuted (oserve tht nodes (X2' X4') nd (X3 (X2+X2'X4') + X3' X4) re neighors.) Two oftors of node (X3 (X2+X2'X4') + X3' X4) re then lulted nd pled. In level 5 vrile X2 is now seleted, for the seond time. Both oftors of node (X2') re onstnts, so joining is not exeuted for the positive oftor Both oftors of node (X3) re the sme, so node (X3) in this level eomes the left extender (whih mens tht its iruit interprettion is only wire going to left, see Fig. 2). Coftors of node (X3 X2 + X3') re lulted. No joinings will e exeuted euse the left neighor ws left extender. Vrile X3 is gin seleted in level 6. Now ll oftors of nodes re onstnts, whih ompletes the lttie retion proess. The regulr lyout of the orresponding iruit is shown in Fig. 2. Oserve, tht this iruit n e further simplied y the propgtion of onstnts. For instne, the left node in the seond level hnges to n AND, the right node in the third level hnges to n OR, the left node in the fourth level hnges to n AND, nd the node in the fth level hnges to n OR. But the regulr onnetion pttern in the lyout remins unhnged. This exmple illustrtes, tht lwys, the generl lyout pln otined during lttie retion is uneted, nd it is only

9 () f2 f join f f f2 f2 join = join join f3 = f3 f3 = () f f2 f3 RE = () f f2 f3 Figure 3: The method to rete the MultiOutput Ordered Shnnon Lttie Digrm for n inompletely speied funtion of three outputs, () the method to rete the expnsions nd joining oftors, () the MultiOutput Ordered Shnnon Lttie Digrm derived using method from (), () the Folded Shnnon Lttie Digrm otined fter logi/lyout simplition of the Ordered Shnnon Lttie Digrm from ).

10 rened in the next stges of the entire \lyout driven logi synthesis proess" tht we herey propose. Theorem. The proedure outlined ove termintes for ritrry order of expnsion vriles. Sketh of proof. Fig. 2 is helpful. Oserve rst, tht to every node orresponds \pth funtion" f T speied y ll produts of literls on ll pths leding to this node from the root. (This pth funtion n e, for instne in the middle third level node with expnsions vriles in two top levels =. In suh se, this node serves only to mke more spe in the lttie y llowing to rete trees on its sides. We will ll suh node sper). In generl, node rings the ontriution f T f N to funtion f, where f N is the node sufuntion relized in it from inputs. Thus ontriution of the node to f is the sme s output funtion f in the Kmp res inluded in f T, nd is don't re outside them. In every level, these \pth funtions" f T s well s funtions f T f N re disjoint. It mens, oth their ON nd OFF sets re disjoint, nd they n e represented y disjoint res in Kmps for illustrtion. These disjoint pth funtions in susequent levels hve smller numers of minterms. If node sufuntion f N is onstnt, the orresponding re in the mp is the sme onstnt. This re is no further eted y expnsions in next levels, euse onstnt nodes terminte the expnsion proess. For instne, nding node to whih pth X'X2' leds from the root, to e onstnt see Figure, termintes this rnh nd will exlude further serhing nd modifying the re X'X2' in the Kmp there is no other pth interseting this re in the lttie. Thus, reting every new level of the lttie with onstnts, dds more of these onstnt res, whih mens tht sets ON nd OFF in nodes still to e onsidered for expnsions, re shrinking with levels. Thus the proedure lwys termintes. It remins only to prove tht onstnt nodes will our in levels. Although vriles re repeted ut the numer of the produts of their literls is nite. Oserve tht every level with new vrile, mkes the produts smller nd every level of repeted vrile, seprtes sums of produts to smller sums of produts. Thus for every new level dded, the pth funtions eome sums of smller nd smller mount of minterms Every minterm of f must e then ultimtely rehed s seprte onstnt node (if it hs een not found erlier in sum of lrger produts). This sketh shows si onvergene proving priniples for ny lttiereting proedures. Oserve tht the inresed eieny omes from the ft tht on the sides of the lttie nd lose to onstnts, the nodes sooner eome minterms euse no joinings re exeuted there, nd the prts of the lttie eome trees. The sper nodes mke levels wider, thus llow to rete minterms nd smller sums on their sides. These properties re used to rete good serh heuristis. Of ourse, the numer of repetitions of vriles nd the numer of nodes will depend muh on the vrile ordering, ut the ft of the onvergene itself will not. We will ll the produted oftor the produt of oftor with its own literls, for instne: f is produted oftor of oftor f. Let us ssume top of lttie with vriles nd in levels nd the four oftors lulted. It n e derived, tht out of four produted oftors of vriles nd, f, f, f, f, in the node reted y joining the left nd the right middle oftors, the produted oftors f,ndf re don't res. For remining two produted oftors, the funtion is the sme s the originl funtion. Exmple 2. Figure 3 presents grphilly the method to lulte the Ordered Shnnon Lttie Digrm for multioutput, inomplete funtion. In Fig. 3 Kmps re used to illustrte lu

11 d Figure 4: The method to rete Funtionl Lttive Digrm. Positive Dvio expnsions re used nd (pd,pd) joinings re pplied to the funtion represented in RM form. lting the produted oftors nd joining opertions for oftors. The joining opertion is just ppending ON nd OFF sets, whih is illustrted grphilly y settheoretil unions of mps. In the rst two levels the mps of oftors re shown. Level 4 shows how two oftors from level 3 re unioned to new mp in level 4. Let us oserve, tht this method retes inomplete funtions in ltties, even strting from omplete funtions. Every order of vriles leds to solution. A Theorem nlogous to Theorem n e proven, nd the serh heuristis of the progrm serve only to selet good order of vriles. Fig. 3 n e lso helpful to understnd the onvergene proof. As we see, t every level, more nd more don't res re introdued, whih inreses proility of nding onstnts, nd improves the qulity of results with respet to the proedure outlined in Exmple. Of ourse, Kmps re used only for explntion, ON nd OFF sets or BDDs re used to represent ON nd OFF funtions. Figure 3 presents the reted lttie digrm. Oserve thtyllowing of folding the vriles in levels, s well tking into ount the rules = = the solution from Fig. 3 is simplied to the Folded Shnnon Lttie Digrm from Figure 3. 5 Creting Ordered Kroneker Lttie Digrms. Funtionl Lttie Digrms. Exmple 3. An exmple of reting Funtionl Lttie Digrm for singleoutput funtion is shown in Figure 4. It is like for OSLDDs in Exmple, ut PositiveDvio expnsions re used insted of Shnnon, nd the (pd,pd) joining rules insted of the (S,S) joining rules. denotes opertion in the Figure. We use Positive Polrity ReedMuller forms in nodes to represent the funtions for simplition, ut ny representtion n e used. First, positive Dvio expnsion is pplied to the node in the rst level. The seond

12 3 * 3 7 * 6 e d h f g i d e f g () i h () Figure 5: Are omprison of folded nd ordered Ltties for the sme funtion: () new pproh of Folded Lttie Digrm with every input ville t every node (more omplex routing), () PSBDD nd Ordered Lttie reliztion with the sme vriles in digonl uses. level is lulted s follows. f = d d d d d. f = d d d, whih is the left node of the seond level. f = d d d. f f = d d, whih is the right node of the seond level. Vrile d is seleted for the seond level. The right oftor of node ( d d d) is ( ). The left oftor of node ( d d) with respet to vrile d is ( ). The joining opertion on these oftors is: d( ) d( ) = d d d d d = ( d). Similrly, the omplete lttie from Fig. 4 is reted. Ordered Kroneker Lttie Digrms nd their speil ses. The methods to rete Kroneker Lttie Digrms re quite similr in priniple to the methods from setion 4, ut slightly more omplex in lultions. Oserve tht in Ordered Kroneker Lttie Digrms in every level the edges re for the sme vrile nd the expnsions re of the sme type. For every node of level, oftors f, f nd \logi derivtive" f f re lulted, s in ternry digrms [3,25] (from now on, for simplition, the nme \oftors" will e used for ll f, f,ndf f ). Next, ny two out of these three \oftors" re seleted, onsistently in level for OKLDs. For ompletely speied funtions, the pirs of nonempty oftors re joined using (S,S) rule for Shnnon level, using rule (pd,pd) for Positive Dvio level, nd using rule (nd,nd) for Negtive Dvio level. For inompletely speied funtions, the pirs of nonempty oftors re joined using rules similr to the ove, ut whih use oth ON nd OFF funtions to represent the funtions. They still use propgtion orretion only to the right node. Proedures for OKLDs for inomplete multioutput funtions re very similr to the two methods shown ove for OSLDs. (Both methods n e pplied to multioutput

13 funtions). The root nodes of every prtiulr output funtion n e loted in the strting moment in ritrry orders nd mutul distnes (the distnes nd order f f2 f3were used in Exmple 2 see Fig. 3). It n e heked on funtion from Exmple 2 with vrious orders nd distnes, tht the strting distnes nd orders hve ig inuene on the nl numers of nodes nd numers of levels in the lttie. Similrly s in Theorem, it n e proved, tht every proedure of the types shown, termintes while reting orresponding Kronekertype lttie (n OKLD, Funtionl Lttie Digrm, ReedMuller Lttie Digrm, et.), for n (in)ompletely speied single/multipleoutput funtion. Folded Kroneker Lttie Digrms. As illustrted in Exmple 2, ig dvntge is otined from using Folded Ltties. This is seen espeilly when mixed expnsions re used in the nodes of PseudoKronekertype ltties. Figure 5 illustrtes shemtilly the priniple of this dvntge. By llowing folded lttie, the retngulr envelope re hs een redued from 7 *6=42to3*3=9,thus nerly 5 times. PseudoKroneker Lttie Digrms nd their speil ses. Unfortuntely, retion of PseudoKroneker Ltties is more diult thn tht of the OKLDs, euse the rules like those given ove in Figures ()(f), tht trnsform lwys from left to right, nnot e reted for omintions of expnsion nodes (pd,nd) nd (nd,s). Therefore, more omplex methods to rete ltties hve een developed, whih will e not presented here. It is n open prolem whether reting of PseudoKroneker Lttie Digrms for (S,pD,nD) n e solved nlogously to the methods from setions 4 nd 5, ut with some other set of xed joining rules. However, for omintion of nodes (pd,s) the pseudolttie n e reted, we ll it the Pseudo S/pD Kroneker Lttie Digrm. It n hve only the mixture of S nd pd nodes in level. Besides, the method is the sme s for Kroneker ltties from this setion, the proess of expnding nd joining goes from left to right. 6 The omplete design methodology for Ordered Kroneker Lttie Digrms. The design methodology hs two phses. In the rst phse the funtion is deomposed using very powerful Ashenhurst/Curtis deomposer of multioutput reltions nd funtions [6]. So, our method n strt from Boolen reltion. This wy, funtion of mny vriles is splitted to smller loks, nd eh lok is dense funtion of few vriles. The size of the loks n e userontrolled, nd we pln to experiment with vrious sizes of loks to evlute sizes nd numers of levels of the resulting iruits. We proved [9] tht every totlly symmetri funtion of more thn 4 inputs is deomposle, nd our deomposition method works in suh wy tht it deomposes to predeessor funtions tht re often symmetri (our deomposition is not disjoint, whih mens tht input vriles n e repeted in ound nd free sets, whih inreses gretly the numer of deomposle funtions). If it is possile, symmetri predeessor funtion is found. This wy, lredy our preproessing stge dereses somewht the \symmetriztion oeient" of the funtion loks. Symmetriztion oeient is the minimum numer of vriles tht must e repeted to mke nonsymmetri funtion symmetri. Symmetriztion oeient of totlly symmetri funtion is. In the seond stge every lok is relized seprtely s lttie, using the inomplete, multi

14 output methods from previous setions, euse in Curtis deomposition most loks re multioutput inomplete funtions. The funtion of every lok n e totlly symmetri, prtilly symmetri, pseudosymmetri or not symmetri. The type of the funtion is found from the nlysis of oftors nd their negtions. The seletion of good order of vriles nd expnsion nodes is sed on generlized prtil symmetries of oftors. These re lyout symmetries sed on positions of nodes representing sufuntions. Eh suh symmetry leds to the possiility of joining together two oftors, or oftor nd negtion of nother oftor. Fortuntely, thenumer of suh symmetries is very lrge. For instne, there re s mny s8 \polrized Kroneker" symmetries for S, pd nd nd expnsions [7]. It n e esily shown tht inverting the ontrol vriles or inverting the dt funtions, dditionlly inreses the numer of usle symmetries nd thus redues the lyout s ompred to those shown here. If the funtion is symmetri nd omplete, it is relized with lttie of ritrry order of vriles without repetitions. If the funtion is symmetri nd inomplete, or prtilly symmetri, the symmetrivriles go on top, nd dditionl vrile ordering nlysis is performed in the proess of mpping to lttie, in order to derese the re. In other ses, the lgorithm is pplied tht performs lookhed nlysis of vriles nd selets vriles tht est seprte the true nd flse minterms. In se of folded nd free ltties, if etter vrile ordering is found lolly, itnoverome the ordering found in the glol nlysis [7]. 7 EVALUATION OF EXPERIMENTAL RESULTS We hve implemented set of lgorithms for generting ltties in the C lnguge whih run in the UNIX environment on SPARC worksttions. In Tle the results for set of funtions from the MCNC enhmrks re presented. As n e seen from the lttie desription, the re oupied y the lttie is proportionl to the numer of nodes nd n e esily estimted y multiplying the numer of nodes y the re of single ell. The nl lyout reted with ltties is very ompt nd no unused loks re left in the middle of the designs. In Tle we present omprison etween results otined y dierent vrileordering heuristis, nd results presented in [4]. Two prmeters, the numer of levels nd the numer of nodes, were used for our urrent results. Funtion nmes re given in the rst olumn. For etter nlysis the results presented here re for singleoutput funtions, therefore, the numer next to the funtion nme indites whih output form the multioutput funtion ws used. The numer of input vriles (numer of vrile for the spei output is given in prenthesis if dierent) is given in the seond olumn nd numer of produt in Espresso generted SOP is given in olumn three. All heuristis re sed on lookhed pproh nd dier only in the priorities ssigned to suh inditors s numer of nodes, numer of literls et. For eh of the heuristis numer of levels ( totl numer of vriles inluding repetitions) nd totl numer of nodes in generted lttie is given. CPU time is very similr for three presented heuristis, therefore it is given for heuristi III only. For omprison, in the lst setion of the tle the results from [4] re given with dditionl prmeter eing numer of loop. In [4] the vrile ordering ws done in very strutured form. A loop of vriles ws dened s set of levels in lttie whih is reted y using n ordered set of expnsion vriles, where eh expnsion vrile n pper t most one. So numer of loops indites the mximum numer of time vrile ppers in lttie in pth from root to lef. In [4] the order of

15 Funtion Heuristi I Heuristi II Heuristi III from [4] Nme #of #of #of #of #of #of #of #of CPU Best # #of #of Inputs SOP Produts Levels Nodes Levels Nodes Levels Nodes Time of Nodes Levels Loops 5x.esp* x7.esp* 7(3) w.esp* n n n w3.esp* n n n on.tt* on2.esp* 7(5) ex2.tt* f2.esp* n n n f54.esp* 8(5) n n n mjority.esp* misex5.esp* misex53.esp* misex6.esp* misex6.esp* misex64.esp* 4 X z43.esp* 7(5) n n n z44.esp* 7(3) Tle : Results for the version of the progrm with: () One Polrity, (2) Look Ahed. X mens the proess nnot stop. Heuristis I, II nd III will e desried in detil in forthoming pper. Lst three olumns hs the results from [4] for omprison. vriles in ll loops ws the sme. A the results in [4] were given for dierent rndom orders we hve hosen the est results for eh funtion. In the lst setions "n" mens tht the results for the funtion ws not ville, nd we elieve tht there ws typo in reporting the results for z43.esp. It n e esily seen tht for the rel life funtions we hve generted OSLDs, whih re proly the most restritive representtion from the fmily of the Lttie Digrms we introdued in this pper, with the resonle numer of nodes nd levels. Compring numer of funtion vriles nd numer of levels it n e seen tht for the worst se funtion in this set of enhmrks the verge numer of times vrile is repeted is three. It is even more importnt tht for the mjority of tested funtion the verge repetition is lose to two. We hve demonstrted tht the regulr twodimensionl representtion of the funtion n led to prtil solutions nd tht the size of tht representtion ould e very ttrtive espeilly for tehnologies limited y the interonnetions dely. 8 CURRENT WORK Our urrent work goes in the following diretions:. Improving the lookhed vrileseleting serh heurstis for the existing lttiereting lgorithms. 2. Finding lgoritms for onurrent vrile ordering/repetition nd expnsion type seletion [7]. 3. Finding methods for symmetrizing generl funtions y repeting vriles. As preproessing, they will trnsform nonsymmetril funtion to ompletely speied symmetri one. In this pproh, with repeted vriles renmed, n ritrry existing BDD/KFDD pkge n e used in the next stge to rete the tul lttie. 4. Beuse we omine our pproh with the Ashenhurst/Curtis deomposition, there is prolem for every deomposed lok "when to keep deomposing nd when to symmetrize?" Sometimes, smll symmetri loks of few vriles resulting from deomposi

16 tion, suh s twoinput EXORs, should e gin reomined to lrger symmetril loks. The generl heuristis is: "use ltties for symmetri nd lose to symmetri funtions". 5. Funtionl deomposition of logi s preproessing to reliztion of deomposed loks in ltties [6]. Improvement to the deomposer so tht the sufuntions generted y it will e either lwys symmetril or s lose to symmetril s possile (we wnt to minimize the totl symmetriztion oeient for ll loks). 6. Design using prtitioned multilevel strutures of ltties nd other loks [2]. This leds to severl levels of lyout plnes, suh s in TANT networks [2]. 7. Development of methods for seletion of the est type of lttie or other struture for given funtion. 8. Generliztions of the lttie model. Severl generliztions to the proposed lttie model hve eeninvestigted. For instne, in [9, 7, 8, 5, 2, 7] these methods hve een extended for more then two inputs nd more thn two outputs to ell, more generl expnsions in nodes (lso, nonnonil), pseudo nd free digrms. The ltties n e generlized y using the onepts of Linerly Independent (LI) logi [9,,, 4, 5, 7]. We llow ll LinerlyIndependent expnsions [6, 9, 5], nd the Boolen Ternry expnsions from [3], s well s ll Zheglkin expnsions from [4]. Logi orretions n e now propgted to oth right nd left, whih further extends the serh spe nd n improve the results. Note, tht some of these extensions led lso to nonplnr nd not inry strutures. 9 CONCLUSIONS. We dened the hierrhy of Kroneker Ltties nd their speil ses s ounterprt of the hierrhy of Kroneker Deision Digrms. Our experimentl results demonstrte tht even for Shnnon expnsions only, ut with the order nd repetition priniples dierent thn those in UAAs, very good results n e otined for prtil enhmrk funtions. Next, we showed tht y dding more expnsion types nd introduing Pseudo nd Folded Lttie Digrms muh more power is gined when ompred to the erly pprohes from [, 3, 4]. In ddition, we showed tht the new methods re good for ompletely s well s inompletely speied funtions, nd most importntly, they hndle well multioutput funtions. We elieve tht lttie digrms re fundmentlly new pproh to onstrut ritrry funtions s plnr regulr lyouts in twodimensionl spe, nd tht new methodologies tht will omine intimtely funtionl deomposition, symmetriztion, lttie digrms, nd lyout genertors sed on them, should e extensively investigted for sumiron tehnologies nd new genertions of FPGAs. Referenes [] S.B. Akers, \A retngulr logi rry," Trns. IEEE Comp., Vol. C2, pp , August 972.

17 [2] Conurrent Logi In., \CLI 6 Series Field Progrmmle Gte Arrys," Prelim. Inf., De., 99, Rev..3. [3] M. ChrznowskJeske, nd Z. Wng, \Mpping of Symmetri nd PrtillySymmetri Funtions to CAType FPGAs," Pro. Midwest'95, 995, pp [4] M. ChrznowskJeske, Z. Wng nd Y. Xu, \A Regulr Representtion for Mpping to FineGrin, LollyConneted FPGAs," Pro. ISCAS'97, 997. [5] M. ChrznowskJeske nd J. Zhou, "AND/EXORsed Regulr Funtion Representtion," Pro. Midwest Symp. Cir. Syst., 997. [6] K.M. Dill, K. Gnguly, R. Sfrnek, nd M.A. Perkowski, \A New Zheglkin Glois Logi," Pro. RM'97. [7] B. Druker, C. Files, M.A. Perkowski, nd M. ChrznowskJeske, \Polrized Pseudo Kroneker Symmetry with nd Applition to the Synthesis of Lttie Deision Digrms," sum. ICCIMA onf., 997. [8] Motorol MPAXX Dt Sheet, 994. [9] M.A. Perkowski, nd E. Pierzhl, \New Cnonil Forms for Fourvlued Logi", Internl Report, Deprtment of Eletril Engineering, Portlnd Stte University, 993. [] M. Perkowski, \A Fundmentl Theorem for Exor Ciruits," Pro. RM'93, pp [] M. Perkowski, A.Sri, F. Beyl, \XOR Cnonil Forms of Swithing Funtions," Pro. of RM'93, pp [2] M. Perkowski, nd M. ChrznowskJeske, \MultipleVlued TANT Networks," Pro. IS MVL'94, My 2527, 994, pp [3] M. Perkowski, M. ChrznowskJeske, A. Sri, nd I. Shefer, "MultiLevel Logi Synthesis Bsed on Kroneker nd Boolen Ternry Deision Digrms for Inompletely Speied Funtions," VLSI Design, Vol. 3, Nos. 34, pp. 333, 995. [4] M. A. Perkowski, L. Jozwik, nd R.Drehsler, \ A Cnonil AND/EXOR Form tht inludes oth the Generlized ReedMuller Forms nd Kroneker ReedMuller Forms, " Pro. ReedMuller'97. [5] M. Perkowski, L. Jozwik, R. Drehsler, nd B. Flkowski, \Ordered nd Shred, Linerly Independent, VrilePir Deision Digrms," Pro. RM'97. [6] M. Perkowski, M. MrekSdowsk, L. Jozwik, T. Lu, S. Grygiel, M. Nowik, R. Mlvi, Z. Wng, nd J. Zhng, \Deomposition of MultiVlued Reltions," Pro. ISMVL'97, Nov Soti, My 997, pp. 38. [7] M.A. Perkowski, E. Pierzhl, nd R. Drehsler, \Lyout Driven Synthesis for Sumiron Tehnology: Mpping Expnsions to Regulr Ltties," Pro. ISIC'97, 92 Sept., Singpure 997. [8] M.A. Perkowski, E. Pierzhl, nd R. Drehsler, \Ternry nd Quternry Lttie Digrms for LinerlyIndependent Logi, MultipleVlued Logi, nd Anlog Synthesis," Pro. ICICS97, Sept. 2, Singpure 997. [9] M.A. Perkowski, L. Jozwik, nd R. Drehsler, \New hierrhies of Generlized Kroneker Trees, Forms, Deision Digrms, nd Regulr Lyouts," Pro. RM'97.

18 [2] E. Pierzhl, M.A. Perkowski, nd S. Grygiel, \A Field Progrmmle Anlog Arry for Continuous, Fuzzy, nd MultiVlued Logi Applitions," Pro. 24th ISMVL, pp. 4855, Boston, My 2527, 994. [2] M.A. Perkowski, M. ChrznowskJeske, nd Y. Xu, \MultiLevel Progrmmle Arrys for SuMiron Tehnology Bsed on Symmetries," of Lttie Deision Digrms," sum. to ICCIMA'98. [22] T. D. Ross, M.J. Noviskey, T.N. Tylor, D.A. Gdd, \Pttern Theory: An Engineering Prdigm for Algorithm Design," Finl Tehnil Report WLTR96, Wright Lortories, USAF, WL/AART/WPAFB, OH , August 99. [23] T.Sso (ed.), \Logi Synthesis nd Optimistion,"Kluwer Ad. Pul., 993. [24] T. Sso, nd J.T. Butler, \Plnr MultipleVlued Deision Digrms," Pro. ISMVL'95, pp [25] I. Shefer, nd M. Perkowski, \Synthesis of MultiLevel Multiplexer Ciruits for Inompletely Speied MultiOutput Boolen Funtions with Mpping Multiplexer Bsed FP GAs," IEEE Trns. on CAD,, Vol. 2, No., Novemer 993. pp [26] N. Song, M. A. Perkowski, M. ChrznowskJeske, A. Sri, \A New Design Methodology for TwoDimensionl Logi Arrys," VLSI Design, (L. Jozwik ed)., Vol. 3, No. 34, 995.