Subset Family of AND/OR/XOR Canonical Forms. Portland State University Fremont, CA Portland State University

Transcription

1 Universl XOR Cnonicl Forms of Boolen Functions nd its Subset Fmily of AND/OR/XOR Cnonicl Forms Mrek A. Perkowski Andisheh Sri F. Rudolf Beyl Dept. of Electricl Engineering Viewlogic Systems, Inc. Dept. of Mthemticl Sciences Portlnd Stte University Fremont, CA Portlnd Stte University Abstrct In this pper new concept of Universl XOR Cnonicl Forms is presented. Such forms include ll well-known fmilies of AND/XOR cnonicl forms s specil cses. A generl mthemticl tretment of these forms is presented. It is shown tht utilizing liner group theory, mny properties nd clsses of these cnonicl forms cn be studied. By this pproch, the number of possible XOR cnonicl forms is shown to be enormous. Severl opertors to crete vrious AND/OR/XOR cnonicl forms re lso introduced. Such opertors, which generlize the Kronecker tensor product, limit these cnonicl forms to the ones nding pplictions in most technologies. Introduction The XOR logic is nding more interest due to its inherent chrcteristics, vilility of new synthesis tools, nd the new technologies which mke ecient reliztion of this logic possible. In terms of inherent eciency of XOR logic, it hs been shown tht AND/XOR PLAs on verge re more compct thn AND/OR PLAs []. Other studies hve shown this compctness in prcticl circuits [8, 0, ]. The circuit reliztions in this logic cn lso be esily testle []. The lrge size of XOR gtes in CMOS nd their delys hve historiclly been mjor reson for underutiliztion of XORs in design. This is despite the fct tht reliztion of rithmetic, encoding, telecommuniction, nd liner circuits with XOR logic hs been well known to be more compct thn other reliztions. The recent dvnces in PLD nd FPGA technologies hve hd big impct on utiliztion of XORs in circuit design. Mny PLDs include XOR gtes s one of their min components which mke the utiliztion of XOR logic much more prcticl. Furthermore, mny FPGA rchitectures remove ny distinction in reliztion mong dierent logics. While this is true for LUT-type FPGAs, mny ne grin FPGAs hve XOR s one of their min blocks. With new dvnces in VLSI nd deep submicron technologies, the delys This reserch ws prtilly supported by the NSF grnt MIP-90. due to XORs would be not min issue s the routing will be the min source of re nd dely. An dvntge tht XOR logic cn provide in this spect is the regulrity thtitprovides which cn contribute to lyout driven synthesis []. A mjor other chrcteristic of the XOR logic is the numerous possible cnonicl representtions of switching functions it provides. Vrious decision digrms re bsiclly multi-level representtions of the functions bsed on this logic []. The cnonicity provided in this logic hs found pplictions not only in the representtion of the functions but s n exmple in Boolen mtching techniques []. While mny AND/XOR cnonicl representtions hve been know in literture, it is the purpose of this pper to provide systemtic wy of studying ll possible cnonicl representtions in XOR logic. It further provides n insight tothemultitude of these representtions compred to other forms such s AND/OR. The AND/XOR cnonicl forms hve been the subject of mny studies [, 8, 0, 9]. The rst AND/XOR cnonicl form introduced in the literture is known s Reed-Muller Cnonicl (RMC) form [, 9]. In this form, which ws introduced erlier by Zheglkin [], ll literls occur in positive polrity only. A lrger clss of AND/XOR cnonicl forms, known s Consistent Generlized Reed-Muller (CGRM) Cnonicl forms [, ] ws introduced by Akers []. These forms hve beenlsoknown s xed polrity Reed-Muller (FPRM) Cnonicl forms [9]. CGRMs re members of severl other lrger clsses of AND/XOR cnonicl forms. Generlized Reed-Muller (GRM) Cnonicl forms [], lso termed s Cnonicl Restricted Mixed Polrity (CRMP) [, ] re one such clss. A dierent superclss of CGRMs rethekronecker Reed-Muller (KRM) cnonicl forms [, ]. KRMs re members of lrger clss of Pseudo-Kronecker Reed- Muller (PKRM) cnonicl forms which themselves re included in the clss of Qusi-Kronecker Reed-Muller (QKRM) cnonicl forms. Other extensions to these forms cn be found in [8]. The extensions to the multivlued cses cn be found in [0]. The lrgest clss of AND/XOR expressions is clled the Exclusive Sum of Products (ESOP) expressions. This pper ttempts to provide tool to generte nd investigte ll XOR cnonicl forms. These forms in generl re Exclusive Sum of generl terms which include ny possible opertions on literls rther thn just their products. This, obviously will include the

2 AND/XOR cnonicl forms s prticulr cse nd encompsses much lrger set of cnonicl forms. The concept of bsis functions llows for extending these studies nd provides forml method of investigtion for ll the forms in systemtic wy. Centrl to this presenttion is the concept of representtion of the Boolen functions s vectors in the vector spce of ll Boolen functions. It hs long been known tht the set of n-vrile Boolen functions under ddition mod- forms n - dimensionl vector spce over the Glois eld of two elements, GF() [, ]. Utilizing this concept, it will be possible to investigte ll the ove cnonicl forms s well s ll other possible XOR cnonicl forms which hve not been studied yet. An exmple of utiliztion of these representtions in physicl driven synthesis will lso be provided s possible ppliction. In section, certin liner lgebr concepts re reviewed which provide the generl frmework of this presenttion. In section, the concept of Universl XOR cnonicl form will be presented nd the number of such forms will be clculted. In section, the utiliztion of UXFs in generlized PLA reliztion of functions will be presented. In section, subset of these forms will be given tht hve suitle reliztion in these generlized PLA congurtions. Bckground We use the following bsic concepts of liner lgebr: Denition Let V be vector spce over F. A subset B = f i ji Ig of V is bsis for V over F if ech vector V cn be expressed s = i + i + ::: + n in with unique coecients j F where i ::: in B. A vector of the form P n j= j ij is clled liner combintion of the ij. The vectors nd liner trnsformtions, which re functions of vectors tht preserve ll liner combintions, hve mening independent ofny prticulr bsis, but their representtions re entirely dependent on the bses chosen. Indeed, mny such representtions my be simplied by choosing new bsis. To this end, we describe how bses cn be represented in terms of ech other. Speciclly, let ::: m be one bsis of the vector spce V nd ::: m be nother. Then ech bsis vector cn be P expressed s liner combintion of the 's, n i = j= P ij j with unique P ij F. The mtrix thus dened, P =[P ij ], is clled the trnsition mtrix from the bsis A to the bsis B. Denition A squre mtrix A of eld elements is clled nonsingulr if deta is nonzero. It is the nonsingulr mtrices tht re of interest here, becuse mtrix is the trnsition mtrix of suitle bsis chnge precisely when it is nonsingulr []. Universl XOR Cnonicl Forms nd their Number Ech bsis in the vector spce over GF() formed by the set of n-vrile Boolen functions under ddition mod-, hs n vectors in it. The ddition mod- is obviously the XOR opertion. Once bsis hs been chosen, its vectors re clled bsis functions. Thus every Boolen function cn be represented uniquely s liner combintion of the bsis functions, or in other words, their Exclusive Sum. The tsk of the identiction of ll cnonicl forms of the Boolen functions in this eld thus entils the identiction of ll possible bses of the n -dimensionl vector spce. In the following, systemtic method of identifying these bses will be presented. Awell known cnonicl form of the Boolen functions is the sum of minterms form. In the notion of the vector spces, the n minterms of the function provide bsis for the vector spce nd ech minterm is bsis function. Any Boolen function then cn be uniquely represented s liner combintion of the minterms. Two of the most well-known AND/XOR cnonicl forms re the Reed-Muller nd the Consistent Generlized Reed-Muller Cnonicl forms. The RMC representtion consists of only positive product terms nd is given s: M n ; f(x x ::: x n )= i i () i=0 where Q i f0 g nd i = x en n x n; en; :::x e x e n = j= x j ej where e j f0 g such tht e n e n;:::ee is the binry representtion of the number i. Moreover x 0 i =nd xi = xi. The symbol denotes the summtion over GF(). If the restriction tht ll the vriles should tke positive polrity is removed nd they re lso llowed to tke negtive polrities, one hs Generlized Reed- Muller (GRM) cnonicl form. If the vriles re, however, restricted to retin the sme polrity, either positive or negtive, in ll product terms, the cnonicl form will be tht of the Consistent Generlized Reed-Muller (CGRM) form. Ech term in the ove cnonicl forms is bsis function. These bsis functions cn be expressed in terms of the minterm bsis using trnsition mtrix. These trnsitions will be shown for two bses in Exmples nd for the cse of functions of two vriles. Exmple In the vector spce of two-vrile Boolen functions, the Reed-Muller bsis functions re,, b, nd, while the minterms re,,, nd. Thetrnsition from the minterm bsis to the Reed- Muller bsis is given by: b = ()

3 The opertions re in GF(). Exmple A CGRM form of two-vrile Boolen function is represented by bsis functions b. Similrly, the trnsition is given by: b = The opertions re gin in GF(). () As it cn be observed, the rows of the trnsition mtrices re linerly independent. In generl, in the spce of Boolen functions, ll nonsingulr mtrices of dimension n provide the trnsition mtrices for ll possible bses. These re the bses of Universl XOR forms (UXF). Among the UXFs, there exist ll AND/XOR cnonicl forms, XOR of products of literls which include ll the previously mentioned cnonicl forms. There exist lso other XOR forms which their bsis functions cn not be relized by product of the literls lone. Denition Let be vector spce ofn-vrile Boolen functions over GF (). A Universl XOR form (UXF) is bsis in this vector spce. If bsis function in UXF cn be relized sproduct of literls, it is clled monoterm. In generl, term in UXF is clled uxf-term of f. The Exclusive Sum of the uxf-terms is the UXF cnonicl representtion of the Boolen function f. The monoterms re only subset of ll possible bsis functions. Hence the number of ll UXFs is much more thn the number of ll possible AND/XOR cnonicl forms. The number of ll possible XOR cnonicl forms cn be given in terms of the number of ll nonsingulr mtrices of given dimension. This is relted to the notion of generl liner group. Denition The group of ll nonsingulr m-by-m mtrices with entries in the eld k is clled the generl liner group nd denoted bygl m (k) []. The number of such mtrices is given in the following Lemm: Lemm Let k = GF (q) be the Glois eld with q elements. The order of GL m (k) is Y q m(m;)= m (q i ; ): () i= Proof: See Theorem. in []. Theorem Let f(x x ::: x n ) be Boolen function of n vriles. The number of ll possible XOR cnonicl representtions of the function is given by: (n; )( n ;) Y n ( i n ; ): ()! i= Proof: Substituting q =ndm = n in Lemm, the number of such mtrices cn be seen to be Y (n; )( n ;) n ( i ; ) i= for this specil cse. This is the number of ll ordered bses. As the order of bsis functions is not relevnt to the cnonicity of the expnsion, it is the number of unordered bses tht is of interest here. Hence, the number of cnonicl forms is given by the number of unordered bses which is the ove quntity divided by m! = n!. QED By Theorem, there exist 00/! = 80 different XOR cnonicl forms for -vrile function lone. This number for -vrile function is round : 0. As it is evident, the number of cnonicl forms grows stronomiclly with the number of vriles in this eld. Up until now, only AND/XOR cnonicl forms hve been studied in the literture. Here, the most generl XOR cnonicl forms re introduced which utilize gtes other thn AND nd NOT for their reliztion nd could provide more compct reliztions thn ESOPs. Since there re mny more of these thn AND/XOR forms, the probility of nding the miniml circuit mong them is much higher. Utiliztion of UXFs in Generlized PLA Reliztions As there re huge number of possible UXFs, there will be two problems of prcticl interest. The rst is to nd such fmilies of forms which hve esy circuit reliztion. The second is to nd the best form mong ll forms of ech fmily i.e. the one with the miniml number of uxf-terms. This pper ddresses the rst problem nd the second will be the subject of nother pper. UXFs cn be utilized s Boolen pproch tothe logic optimiztion stge of the Complex Mitr Logic Arry, CMLA [] reliztions. CMLA reliztion of Boolen functions is combined logic synthesis nd physicl design pproch which is comprised of relizing functions in two distinct plnes: the complex (input) plne nd the collecting (output) plne. The input vriles of the input plne run in verticl buses. These inputs re AND, OR, or XORed together in the rows of the complex plne resulting in Mitr terms. In other words, the Mitr terms, which re the generliztion of product terms, re relized in the rows of the complex plne. The outputs of the Mitr terms re plced on the horizontl buses. The Mitr terms re then collected in the output plne composed of the two-dimensionl rry with OR or XOR gtes. The CMLA concept, shown in Figure, is powerful generliztion of PLAs. An exmple of complex plne of CMLA is shown in Figure.

4 b c d e f g h i j 0 0 Figure : Exmple of Complex Mitr Logic Arry The comprehensive pproch to the logic nd lyout synthesis here includes two stges: Logic optimiztion which tkes the geometry nd lyout constrints into ccount to crete CMLA in which every output function is n OR or XOR of Mitr terms. Technology-folding which mps the CMLA representtion of the function to the trget rchitecture, such tht the re of the lyout is minimized. A distinct feture of this pproch is minimiztion of routing resources s well s the logic. Furthermore, the plcement is lredy gured out by the end of the two stges. Hence, it cn prove useful not only in cellulr rry type FPGAs with limited routing resources but lso in deep submicron technologies in generl. In the CMLA pproch, the UXFs cn be utilized in the logic optimiztion stge where the Boolen functions re relized s XOR of Mitr terms. In this cse, the bsis elements tht re comprised of AND, negtion, nd OR of literls will be the most useful ones, s they exist in most technologies. Hence, developing methods for genertion of such terms would be of most prcticl interest. In the next section, methods for systemtic genertion of such bses will be presented. Genertion of Dierent Fmilies of Bses In the following, certin opertionl trnsforms on mtrices to generte dierent product nd AND/OR terms will be described. The terms with positive polrities will be discussed rst with more generl terms, incorporting NOT gtes, following. The signicnce of these opertions is in tht the huge numberofuxfs will be reduced to fewer mngele numbers. Furthermore, the underlying structures bring in simple methodology to hndle the UXFs which nd more pplictions in circuit design. b c (+b)c c +c Figure : An Exmple of Complex Plne of CMLA. Positive Polrity Fmily of Bses Dierent positive polrity AND/OR bses cn be generted by ppliction of two bsic opertions in vrious orders. These two opertions re clled the Reed-Muller nd the AND/OR opertors. From now on, the bsis of reference consists of the minterms in reverse binry order with reversed bits, similr to the order presented in the previous exmples. Denition Let R be nonsingulr mtrix. Reed-Muller Opertor on R is: R R The () where 0 stnds for squre mtrix of the size of R with ll entries 0. Denition Let R be nonsingulr mtrix. AND-OR Opertor on R is: R The () where stnds for squre mtrix of the size of R with ll entries nd 0 hs the sme mening s in previous denition. Theorem Reed-Muller nd AND-OR opertors result in nonsingulr mtrices of higher dimension. Proof: This follows from the fct tht both (R)nd (R) re block tringulr mtrices of the form (8) R Although the fmily is the generliztion of Reed-Muller fmily of expressions, we do not cll them \Generlized Reed- Muller" forms since this nme is lredy reserved nd lso these forms re n order of mgnitude more generl.

5 with determinnt det(r). The vlue of the mtrix denoted by * is irrelevnt. QED The strting mtrix for single vrile, used in the genertion of the positive polrity fmily of bses, is: 0 T = : (9) This mtrix essentilly gives the bsis : (0) A specil cse of pplying the opertor is the genertion of the Reed-Muller Trnsform. In this cse the repetitive ppliction of the opertor is the sme s the Kronecker tensor product of the generted nonsingulr mtrices. Tle shows the bsis functions of the Reed- Muller, AND/OR, nd Reed-Muller/AND/OR expnsions for three input vriles. The Reed-Muller/AND/OR expnsion is constructed by ((T)). Other similr constructs re possible - incorporting dierent orders of ppliction of nd opertors - which give rise to vrious AND/OR/wire connections in the complex plne of CMLA. While the order of vriles is irrelevnt for Reed-Muller bsis, for AND/OR nd ll other combintions of the nd opertors, it gives rise to new bsis. Denition The fmily of bses generted by pplictions of nd opertors in ll possible orders nd ll possible permuttions of the vriles is the positive polrity fmily of bses. Exmple The nonsingulr trnsition mtrix of AND/OR expnsion for -input function is given s: [T] = T 0 T = () The trnsition mtrix ove results in the following bsis functions: b = b + b : (). Consistent Generlized Fmily of Bses The trnsformtion of the Reed-Muller bsis to ny other polrity nd genertion of n Consistent Generlized Reed-Muller bses is well known [, ]. Here, Reed-Muller AND/OR Reed-Muller/AND/OR c ( + b)c c c c c + c + c b b+c b+c +b+c +c Tle : Exmples of Bses for -Input Functions every vrile is llowed to tke either positive or negtive polrity. Hence there re n possible forms for n n vrile function. It is similrly possible to generlize ll members of the fmily of bses to n dierent xed polrities. This fmily will be clled the Consistent Generlized (CG) fmily of forms. In terms of the Complex Mitr Logic Arrys, this would led to the inclusion of n dditionl bus for ech inverted input signl. Alterntively, the inverter on the inputs cn be tken s nother possible gte in the CMLA. In order to introduce negtion of vriles, two negtion opertions nd new strting trnsformtion mtrix need to be introduced. Notice tht similr to Eqution (9), it is possible to dene negtive polrity bsis of single element. This is given s T below: 0 T = () which gives essentilly the bsis : () Now, corresponding to the nd opertors, nd opertors re dened s the following: Denition 8 Let R be nonsingulr mtrix. The Negtive Reed-Muller Opertor on R is: 0 R R R : () Denition 9 Let R be nonsingulr mtrix. The Negtive AND-OR Opertor on R is: 0 R : R () Theorem The nd opertors result in nonsingulr mtrices of higher dimension. Proof: As in the proof of Theorem, the determinnts of both (R) nd (R) re det(r), thus nonzero. To this end, column exchnges will trnsform ech of the ove mtrices into block tringulr mtrix of the form (8). Moreover, since ; =+

6 in GF (), column exchnges do not lter the determinnt. QED Agin, ech vrile cn tke either positive or negtive opertion nd thus there exist n possible Consistent Generlized forms for ech positive fmily of bses. Exmple In the following, CG bsis of three vriles will be shown. Here, the order nd the polrity of the vriles is given s:, where the \nturl" order of vriles is ssumed to be. First, the trnsition mtrix for the nturl order is generted nd then the corresponding trnsition mtrix for the given order will be shown. (T) = (T ) 0 (T) (T) (T) = () The trnsition mtrix ove results in the following bsis functions: = ( + b)c c b + b (8) : (9) The corresponding trnsition mtrix for the \" ordering will be: = c ( b +c) c b +c (0) : () The comprison between Equtions (8) nd (0) shows tht dierent orderings re chieved by swpping of pproprite columns in the trnsition mtrix.. Generlized Fmily of Bses The members of the fmily of bses need not be conned to xed polrities in order to provide new bses. The polrity of the literls cn be \inconsistently" vried nd still result in new bsis. This will be shown by the following exmple: Exmple The bsis generted in Exmple cn now be inconsistently chnged for polrity of literls to give the following new bsis: c ( b +c) c b +c : () As it cn be observed, the literls, b, nd c tke dierent polrities in dierent bsis functions. This lrger fmily of bses will be termed Generlized (G).. Fmily of Bses A dierent generliztion of the fmily of bses is possible through the introduction of third opertor clled the Shnnon opertor,. First the strting trnsformtion mtrix for this extension will be introduced. Agin similr to Eqution (0), the bsis for single element isgiven s T: 0 T = 0 : () Denition 0 Let R be nonsingulr mtrix. The Shnnon Opertor on R is: 0 R : () Theorem The opertor results in nonsingulr mtrix of higher dimension. Proof: As in the proof of Theorem, the determinnt of(r) isdet(r),thus nonzero. QED Notice tht T, T, nd T re three possible nonsingulr mtrices for single vrile function. All other nonsingulr mtrices for single vrile function cn be constructed from swpping the rows of these three mtrices nd would not result in ny new

7 bses. As n exmple, similr to nd opertors, it is possible to dene negtive opertor s shown below: Denition Let R be nonsingulr mtrix. The Negtive Shnnon Opertor on R is: 0 R The bsis for single element here is: 0 T = 0 : () () which gives essentilly the bsis : () It cn be observed tht T nd T dene essentilly the sme bsis nd no new bses will be generted by negtive Shnnon opertor when the corresponding positive opertor is present. Introduction of the still extends the possibilities of generting new AND/OR bses. Certin other generliztions hve been known for the AND/XOR bses in the literture. Those generliztions cn lso be extended to the AND/OR bses resulting in even lrger clsses of AND/OR/XOR cnonicl forms. Conclusions In this pper, concept of cnonicl Universl XOR Forms ws formulted. It ws shown tht through dening vrious bses in the vector spce of Boolen functions, it is possible to generte ll possible XOR cnonicl forms. AND/XOR cnonicl forms, both well-known s well s lesser-known ones, were shown to be specil clsses of Universl XOR Forms. Since these new cnonicl forms include ll known nd not yet known AND/XOR cnonicl forms, they cn never be worse in terms of the number of uxf-terms nd therefore re very much worthy of further investigtions. While the number of these forms is huge, the mpping constrints limit the number of cnonicl forms tht cn be utilized in given technology. Twomtrix opertions of Reed-Muller nd AND/OR long with their negtions were used to generte the Consistent Generlized fmily of forms. They were further generlized by introduction of the positive nd negtive Shnnon opertors. These new cnonicl forms require dierent AND/OR/wire congurtions for their reliztions nd cn nd specil pplictions in the Complex Mitr Logic Arry reliztion of Boolen functions []. The liner group theoretic pproch introduced for the study of XOR cnonicl forms opens up new res for reserch nd study. It provides the systemtic tretment of ll these forms nd llows the utiliztion of liner lgebr in the study of these logicl forms. It lso illumintes the underlying structure of these forms. It ws shown through this pproch tht the number of these forms is huge compred to AND/OR cnonicl forms. The sme pproch cn be utilized for the identiction of miniml XOR cnonicl representtion of the functions. References [] S. B. Akers, \On Theory of Boolen Functions", Journl of SIAM, Vol., pp. 8-98, December 99. [] E. Artin, Geometric Algebr, Interscience Publishers, Inc., 9. [] Ph. W. Besslich, \Ecient Computer method for ExOR logic design", Proceedings of IEE Pt. E, Vol. 0, pp. 0-0, 98. [] L. Csnky, M. A. Perkowski nd I. Schfer, \Cnonicl Restricted Mixed Polrity Exclusive- OR Sums of Products nd the Ecient Algorithm for Their Minimiztion", Proceedings of IEE Pt. E, Vol. 0, No., pp. 9-, October 99. [] M. Dvio, J. P. Deschmps, nd A. Thyse, Discrete nd Switching Functions, McGrw-Hill, 98. [] R. Drechsler, A. Sri, M. Theobld, B. Becker nd M. A. Perkowski, \Ecient Representtion nd Mnipultion of Switching Functions Bsed on Ordered Kronecker Functionl Decision Digrms", Proceedings of DAC '9, pp. -, Sn Diego, CA, June 99. [] D. H. Green, \Reed-Muller Cnonicl Forms With Mixed Polrity nd Their Mnipultions", Proceedings of IEE Pt. E,Vol., No., pp. 0-, Jnury 990. [8] D. H. Green, \Fmilies of Reed-Muller cnonicl forms", Interntionl Journl of Electronics, pp. 9-80, Februry 99. [9] D. E. Muller, \Appliction of Boolen Algebr to Switching Circuit Design nd to Error Detection", IRE Trnsctions on Electronic Computers, Vol. EC-, pp. -, September 9. [0] M. A. Perkowski, \The Generlized Orthonorml Expnsion of Functions With Multiple-Vlued Inputs nd Some of its Applictions", Proceedings of the nd IEEE Interntionl Symposium on Multiple-Vlued Logic, pp. -0, June 99. [] M. A. Perkowski, L. Csnky, A. Sri, nd I. Schfer, \Fst Minimiztion of Mixed-Polrity AND/XOR Cnonicl Networks", Proceedings of the IEEE Interntionl Conference on Computer Design, pp. -, Cmbridge, MA, October 99.

8 [] F. P. Preprt, \Stte-Logic Reltions For Autonomous Sequentil Networks", IEEE Trnsctions on Electronic Computers, Vol. EC-, pp , December 9. [] I. S. Reed, \A Clss of Multiple-Error-Correcting Codes nd Their Decoding Scheme", IRE Trnsctions on Informtion Theory, Vol. PGIT-, pp. 8-9, 9. [] A. Sri nd M. A. Perkowski, \Fst Exct nd Qusi-Miniml Minimiztion of Highly Testle Fixed-Polrity AND/XOR Cnonicl Networks", Proceedings of the 9th ACM/IEEE Design Automtion Conference, pp. 0-, Anheim, CA, June 99. [] A. Sri nd M. A. Perkowski, \Design For Testility Properties of AND/XOR Networks", Proceedings of the IFIP WG 0. Workshop on Applictions of the Reed-Muller Expnsion in Circuit Design, Hmburg, Germny, September 99. [] A. Sri, N. Song, M. Chrznowsk-Jeske, nd M. A. Perkowski, \A Comprehensive Approch to Logic Synthesis nd Physicl Design for Two- Dimensionl Logic Arrys", Proceedings of the st ACM/IEEE Design Automtion Conference, pp. -, Sn Diego, CA, June 99. [] T. Sso nd Ph. W. Besslich, \On the Complexity of MOD- Sum PLAs", IEEE Trnsctions on Computers, Vol. 9, No., pp. -, Februry 990. [8] T. Sso, \EXMIN: A simpliction lgorithm for Exclusive-or-Sum-Of-Products expressions for multiple-vlued input two-vlued output function", IEEE Trnsctions on Computer Aided Design, Vol. 9, No., pp. -, 990. [9] T. Sso, \AND-EXOR Expressions nd their Optimiztion", in Sso(ed.), Logic Synthesis nd Optimiztion, Kluwer Acdemic Publishers, pp. 8-, 99. [0] N. Song nd M. A. Perkowski, \EXORCISM- MV-: Minimiztion of Exclusive Sum of Products Expressions for Multiple-Vlued Input Incompletely Specied Functions", Proceedings of the rd IEEE Interntionl Symposium on Multiple- Vlued Logic, pp. -, My 99. [] H. S. Stone nd A. J. Korenjk, \Cnonicl Form nd Synthesis of Two-input Flexible Cells", IRE Trnsctions on Electronic Computers, Vol. EC-, pp. -, 9. [] S. Swmy, \On Generlized Reed-Muller Expnsions", IEEE Trnsctions on Computers, Vol. C-, pp , September 9. [] C. C. Tsi, nd M. Mrek-Sdowsk, \Boolen Mtching Using Generlized Reed-Muller Forms", Proceedings of the st ACM/IEEE Design Automtion Conference, pp. 9-, Sn Diego, CA, June 99. [] I. L. Zheglkin, \Arifmetiztsiy simbolichesko logiki (Arithmetiztion of Symbolic Logic)", Mtemticheski Sbornik, Vol., pp. -, 98 nd Vol., pp. 0-8, 99.