QMDD and Spectral Transformation of Binary and Multiple-Valued Functions *

Transcription

1 QMDD ad Spectral Trasformatio of Biary ad Multiple-Valued Fuctios * D. Michael Miller Uiversity of Victoria Victoria, BC, Caada mmiller@uvic.ca Mitchell A. Thorto Souther Methodist Uiversity Dallas, TX, USA mitch@egr.smu.edu Abstract The use of decisio diagrams (DD) for the computatio ad represetatio of biary fuctio spectra has bee well studied [2,3,5]. Computatios ad represetatio of spectra for multiplevalued logic (MVL) fuctios have also bee cosidered [6]. For biary fuctios, this approach ca be implemeted usig oe of a umber of the highly efficiet publicly available biary decisio diagram (BDD) packages, e.g. CUDD [3]. Work with MVL fuctios requires a package suited to the MVL case, e.g. [7,8]. Quatum multiple-valued decisio diagrams (QMDD) were itroduced i [9-2] as a meas to represet ad maipulate the matrices required for biary or multiple-valued reversible ad quatum gates ad circuits. I this paper, we show how QMDD ca also be applied to the computatio of spectral trasformatios of biary ad multiple-valued fuctios. A major motivatio for this work is that it itroduces a approach to the spectral aalysis of reversible ad quatum circuits i a represetatio that is applicable to simulatio ad sythesis of such circuits. It is also of iterest i that it is a cosistet approach for a variety of trasformatios of biary ad multiple-valued fuctios.. Spectral Trasformatio We oly cosider issues here that are ivolved i computig spectral trasformatios ad ot the reasos for doig so or how spectral represetatios assist i aalysis ad sythesis problems. The reader iterested i those matters should cosult the literature, e.g. [4,4]. It is also importat to ote that we oly show three represetative spectral trasformatios. The approach is directly applicable to may other trasformatios foud i the literature. Cosider a biary or MVL -variable fuctio represeted by a truth colum vector F. We are iterested i spectral trasformatios defied as i eq. where T is a r! r trasformatio matrix ad r is the radix of the fuctio. The trasformatio matrices of iterest are defied by the Kroecker product show i eq. 2 where T is a r! r base matrix defiig the trasformatio. S " T F () T i" "# T (2) Rademacher-Walsh: Reed-Muller: Chresteso: $ % T " ( ) * $ % T " ) * $ % ( 2+ i 2 T " a a, a e r ", r " 3 2 ) a a * (3a) (3b) (3c) * This research was supported i part by a Discovery Grat from the Natural Scieces ad Egieerig Research Coucil of Caada.

2 Eq. 3 shows three well studied trasformatios. The Rademacher-Walsh trasformatio is computed over the itegers; the Reed-Muller trasformatio is computed over GF(2); ad the Chresteso trasformatio is computed over the complex umbers. The Chresteso trasform show i eq. 3c is for the case r=3. It is readily exteded to higher values of r ad reduces to the Rademacher-Walsh case for r=2. For the Rademacher-Walsh case, the fuctio vector F may be i (,) or (+,-) codig [4]. It is always (,) for the Reed-Muller case where the computatio is doe over GF(2). I the Chresteso case, F is coded with logic value p,, p - r, represeted by a p. 2. Spectral Trasformatio Computatio Computatio of a spectral trasformatio by matrix multiplicatio as per eq. requires r! r multiplicatios ad r!( r ( ) additios so the complexity is O(r 2 ). This becomes impractical for of ay sigificat. Fast trasform techiques aalogous to the fast discrete Fourier trasform are well kow [4] ad are much more efficiet tha direct matrix multiplicatio. For example, Fig. shows a C program implemetig a fast Rademacher-Walsh trasform for biary fuctios. The complexity ca be see from this code to be O ( 2 ) where is the umber of fuctio variables. void fht(it f[], it ) { it i,j,k,t,m,p; for (m=;m<(<<);m=m<<) { for (i=;i<(<<);i+=m<<) { for (j=i,p=k=i+m;j<p;j++,k++) { t=f[j]; // lie a f[j]=(f[j]+f[k]); // lie b f[k]=(t-f[k]); // lie c } } } } Figure : Fast biary Rademacher-Walsh Trasform. The approach illustrated i Fig. ca be adapted to other trasforms. For example, the Reed- Muller trasform is implemeted by replacig lies a, b ad c by the sigle assigmet f[k]=(f[j] +f[k]) where is implemetig the mod-2 operatio (this is more efficiet tha %2). While fast trasform techiques are much more efficiet tha direct matrix multiplicatio, they share a major drawback i that the iitial truth vector ad the resultig spectrum are vectors of legth r. Not oly ca such a vector be of prohibitive, a vector represetatio does ot idetify or exploit ay structure that might be iheret i the spectrum. The use of DD techiques for computig spectral trasformatios has bee studied for both biary ad MVL fuctios. As oted above, this work has bee somewhat separate sice differet DD packages have bee used for the biary ad MVL situatios. The approach described below is uified i that it employs the QMDD package as described i [9-2] for all cases. Relatively mior extesios are required icludig the use of modular arithmetic for the Reed-Muller trasform. The advatages of DD approaches, icludig the approach described below, lie i the fact that the of the DD represetig the origial fuctio ad the spectrum is ot fixed ad i fact exploits the structure that might be preset i each. This has space implicatios ad ca directly affect the complexity of computig a trasformatio. Just as importat, the DD represetig a spectrum ofte makes the uderlyig structure of the spectrum apparet. 3. Quatum Multiple-valued Decisio Diagrams We assume the reader is familiar with the fudametals of DD techiques [,4]. As oted above, we are i geeral cocered with trasformatio matrices of dimesio r! r where r is the radix ad is the umber of variables. Such a matrix ca be partitioed as show i the followig equatio:

3 $ M M! M r( % M r M r M. 2r( M! " " " # " M M! M ) r ( r r ( r. r ( * (4) ( (. where each M i is a matrix of dimesio r! r Each of the M i ca be similarly partitioed ad the process repeated util scalars are reached. This repeated partitioig leads to the fudametal QMDD structure. Defiitio : A quatum multiple-valued decisio diagram (QMDD) is a directed acyclic graph with the followig properties:. There is a sigle termial vertex with associated value ad o outgoig edges. 2. There are some umber of o-termial vertices each labeled by a r 2 -valued selectio variable. Each o-termial vertex has r 2 outgoig edges desigated e, e,..., e 2 (. r 3. Oe vertex is the start vertex ad has a sigle icomig edge that itself has o source vertex. 4. Every edge i the QMDD, icludig the oe leadig to the start vertex, has a associated complex-valued weight. A edge with weight of must poit to the termial vertex. 5. The selectio variables are ordered (assume with o loss of geerality the orderig x $ x $%$ x ( ) ad the QMDD satisfies the followig two rules: i) Each selectio variable appears at most oce o each path from the start vertex to the termial vertex. ii) A edge from a o-termial vertex labeled x i poits to a o-termial vertex labeled x j, j < i or to the termial vertex. Hece x is closest to the termial ad x - labels the start vertex. 6. No o-termial vertex is redudat, i.e. o o-termial vertex has its r 2 outgoig edges all with the same weight ad poitig to a commo vertex Each otermial ode is ormalized such that for some j, e,, j, r (, has weight ad all ei, i - j, have weight. Note, such a j must exist or the vertex would be redudat (all weights ). 8. No-termial vertices are uique, i.e. o two o-termial vertices labeled by the same x i ca have the same set of outgoig edges (destiatios ad weights). As is commo for DD represetatios, a key property of QMDD is that the represetatio for ay give matrix is uique. A proof is available from the authors. A key feature of this proof is the ormalizatio process that is applied durig the costructio of a QMDD. The ormalizatio rule used here is as itroduced i [9]. This ormalizatio procedure extracts commo multiplicative factors as each vertex is created ad thus extracts commo factors for each subgraph of the QMDD ad the QMDD itself. Skipped variables (see Def. 2) must be cosidered i DD based algorithms. Defiitio 2: Give the orderig x $ x $%$ x ( a edge from a vertex labeled xi, i /, skips a variable, or variables, if it poits to the termial vertex or it poits to a vertex labeled x, j - i (. For reversible circuits, skipped variables oly appear whe a edge has weight ad poits to the termial vertex. However, this is ot the case for circuits composed of geeral quatum gates. Skipped variables also occur whe QMDD are used for spectral trasformatio ad must be take ito accout as described i the ext sectio. 4. QMDD-Based Spectral Trasformatio Fig. 2 shows the QMDD represetatio for the 3-variable Rademacher-Walsh trasform matrix. Note that the umbers o the edges are multiplicative weights ad do ot idicate variable value idetificatios as is usually the case for DD figures. The edges values represet trasformatio matrix quadrats,, 2, ad 3 from left to right out of each vertex. The QMDD has a sigle vertex for each variable ad thus grows liearly as icreases. This is a importat property which is true for ay trasformatio matrix defied as show i eq. 2 ad is particularly importat with regard to efficiecy i spectral computatios. j j

4 x 2 x 2 - x x x - x - x Figure 2: QMDD for 3-variable Rademacher- Walsh trasform matrix. Figure 3: QMDD for 3-variable majority fuctio. QMDD were developed for represetig ad maipulatig r! r matrices. To apply QMDD to the spectral trasform computatio problem requires a represetatio for colum vectors. Rather tha itroducig a separate structure, we adapt the QMDD structure. I particular, a colum vector ca be cosidered to be a matrix with empty submatrices which ca be represeted by ull poiters i the QMDD structures. This is illustrated i Eq. 5 where deotes a empty submatrix. $ V! % Vr V! " (5) " " # " V 2 )! r ( r * This structure is repeatedly applied as the QMDD is formed. For example, the QMDD i Fig. 3, represets the colum vector for the 3-variable majority fuctio. The QMDD structure is similar to may DD structures the reader will be familiar with. However, we ote each ode has multiple outgoig edges (ot two as i a BDD), that the umbers o the edges are multipliers, ad there is a sigle termial vertex. The diagram i Fig. 3 is i effect a BDD but agai ote that the use of weighted edges is differet from BDD. Our QMDD-based spectral trasformatio method ivolves implemetig the multiplicatio of a matrix ad colum vector represeted as a QMDD resultig i a colum vector. The implemetatio is modeled upo Bryat s APPLY algorithm for BDD [], ad is based o the fact that matrix multiplicatio ca be decomposed as show i eq. 6. The required QMDD matrix additio is also discussed i [9] ad is agai based o Bryat s APPLY. $ A A! Ar( % $ B B! Br( % Ar Ar A 2r Br Br B.! (.! 2r(! " " # " " " # " ) A A! A B B B r ( r r ( r. r ( * )! r ( r r ( r. r ( * (6) $ A B. A Br.... Ar ( B 2 A B. A Br... Ar B 2 r ( r... (! " r ( r. % Ar B. Ar. Br.... A2 r( B 2 "! " r ( r " " " # " A 2 Br A 2 B )!!! r r (. r r r(.. A B ( (. r ( r ( * Three issues must be cosidered whe maipulatig colum vectors represeted as QMDD. First, ull poiters must be take ito accout. I multiplicatio, oe of the argumets beig a ull poiter results i a ull poiter i the product. Additio whe a argumet is a ull poiter results i a aswer equal to the other argumet.

5 Secod, QMDD are defied with complex-valued edge weights ad the package thus supports arithmetic operatios o complex values. The Rademacher-Walsh spectrum is computed over the itegers which is ot a issue sice the itegers are a subset of the complex umbers. Reed-Muller trasforms are computed over GF(2) which requires a mior modificatio to the complex value arithmetic packages origially implemeted for QMDD. The third, ad most ivolved issue, is the hadlig of skipped variables. For a matrix, a skipped variable meas the matrix is such that all the submatrices i its decompositio are idetical so the selectio variable ca be skipped. For a colum vector, the iterpretatio is the same but oly for the left-most submatrices the others must be uderstood to be empty. Both situatios are readily take ito accout i the matrix multiplicatio algorithm. The mior complicatio is that a differet iterpretatio is required depedig o whether the QMDD represets a matrix or a colum vector. The iterested reader ca fid details o the implemetatio of the QMDD package i [,] icludig a variety of stadard BDD ad QMDD specific techiques used to ehace the speed of the implemetatio. Much of the speed comes from usig stadard DD compute table techiques as well as computatio tables for cachig the results of complex umber operatios. 5. Experimetal Results The QMDD package is implemeted i C. Extedig it to compute spectral trasforms was relatively straightforward ad ivolved:. Implemetig a routie to build a QMDD give the truth vector for a fuctio. This is a simple recursive costructio. 2. Implemetig a routie to build the QMDD for the required trasformatio matrix. This is a simple iterative routie as a result of the compact liear structure illustrated i Fig Modificatios to the matrix multiplicatio ad additio routies to hadle ull poiters ad geeral skipped variables separately for matrices ad colum vectors. 4. Modificatio to the complex value operatio routies to do computatio over GF(2) whe required for the trasformatio. The results reported are for experimets ru o a Toshiba Protégé laptop with a 75 MHz Petium III ad 52 MB Ram. The Metrowerks compiler V.4 was used with the highest level of global optimizatio. 5. Biary Examples Table shows results for the AND of a icreasig umber of variables. The results show that the QMDD method becomes icreasigly better i terms of umber of operatios as icreases, although it should be oted that the operatios for the QMDD approach are cosiderably more complex. The of the QMDD for the spectrum grows liearly. The space required for the QMDD is iitially much greater tha for the FT spectrum but the relative decreases as icreases ad will cotiue to do so for higher sice while the QMDD is growig liearly, the of the FT vector grows expoetially. Note that each QMDD ode occupies 44 bytes while each FT value is 4 bytes (a simple iteger which is sufficiet up to =3). Tables 2 ad 3 show similar performace for the OR ad the XOR fuctio. This is ot surprisig sice these are highly regular fuctios. Table 4 is similar to the earlier tables but for radomly geerated fuctios. This shows that the QMDD techique is ot very effective for radom fuctios ad geerally requires more storage tha the fast trasform approach. This is ot uexpected sice QMDD exploit structure. Table 5 shows a umber of situatios comparig Rademacher-Walsh (+,-) codig (S), Rademacher-Walsh (,) codig (R), ad Reed-Muller (RM) trasform results. The data verifies that, as expected, the RM spectrum is most efficiet i terms of QMDD. For the 2 variable radom fuctio, the user timig routie provided i C showed a executio time of.,.3 ad.2 sec. for the S, RM ad R spectra computatios. The timigs for all other examples were egligible (less tha sec.) It is oteworthy that the performace for the structured fuctios is much better i terms of time ad space for this higher (5) value of. The performace for radom fuctios is improved i terms of umber of operatios but still requires more space tha the FT approach. The results give must be cosidered i the cotext that the QMDD structure ca illumiate the structure of a fuctio spectrum, whereas the FT represetatio is by defiitio flat ad idicates othig directly about structure.

6 5.2 Terary Examples Tables 6, 7 ad 8 show the results for computig the Chresteso spectra for the MIN, MAX ad MOD-SUM fuctios for r=3 for =2,,. The data show is labeled as i the biary case. Note that i this case each QMDD vertex occupies 84 bytes while each FT value, which i this case is a complex value occupies 6 bytes. The performace for these structured fuctios is as expected. The QMDD represetatio gais efficiecy o the FT approach more quickly i the terary case as compared to the biary case. Ad, oce agai, it should be emphad that the QMDD method better represets the uderlyig trasformatio matrix structure. The CPU timigs were agai egligible. 6. Cocludig Remarks We empha that while this paper has used the Rademacher-Walsh, Reed-Muller, ad Chresteso spectral trasformatios for illustratio, the approach is directly applicable to ay trasformatio that ca be expressed as show i eq. with arithmetic over the complex umbers or some subset thereof. It is most efficiet for trasformatios defied as the iterative Kroeker product of a r! r matrix as i Eq. 2 sice that leads to a compact QMDD for the trasformatio matrix. We have implemeted a QMDD-based implemetatio of explicit matrix multiplicatio. It is also possible to implemet a form of fast trasform o a DD but we do ot aticipate that this will lead to a sigificatly faster implemetatio sice the QMDD for the trasformatio matrices are liear ad essetially direct the computatio i effectively the same way a fast implemetatio would. The use of a compute table avoids duplicate computatio. We will however examie this i depth to esure our ituitio is correct. Formal aalysis of the computatioal complexity ad memory use of our approach is ogoig particularly with respect to BDD approaches for the biary case. The aalysis is complicated by the use of compute ad computatio table techiques which are highly fuctio depedet. The examples preseted here are prelimiary. Our ogoig work will ivolve testig the approach o appropriate bechmark fuctios. Also, we have yet to cosider the effect of variable orderig o QMDD i the represetatio of spectra. It is possible, for example, that the radomly geerated results here are very pessimistic ad could be improved through variable reorderig of the QMDD. Refereces [] R.E. Bryat. Graph-Based Algorithms for Boolea Fuctio Maipulatio. IEEE Trasactios o Computers, C-35(8): , 986. [2] E.M. Clarke, K. McMilla, X. Zhao, M. Fujita, ad J. Yag. Spectral Trasforms for Large Boolea Fuctios with Applicatios to Techology Mappig. Formal Methods i System Desig: A Iteratioal Joural, (2-3):37 48, 997. [3] M. Fujita, P.C. McGeer, ad J.C.-Y. Yag. Multi-termial Biary Decisio Diagrams: A Efficiet Data Structure for Matrix Represetatio. Formal Methods i System Desig: A Iteratioal Joural, (2-3):49 69, 997. [4] S. L. Hurst, D.M. Miller ad J.C. Muzio. Spectral Techiques i Digital Logic. Academic Press, New York Lodo, 985. [5] D.M. Miller. Graph Algorithms for the Maipulatio of Boolea Fuctios ad their Spectra. Cogressus Numeratium, 57:77-99, 987. [6] D.M. Miller. Spectral Trasformatio of Multiple-valued Decisio Diagrams. Proc. IEEE Iteratioal Symposium o Multiple-Valued Logic, pp , May 994. [7] D.M. Miller ad R. Drechsler. Implemetig a Multiple-valued Decisio Diagram Package. Proc. IEEE Iteratioal Symposium o Multiple-Valued Logic, pp , 998. [8] D.M. Miller ad R. Drechsler. Augmeted Siftig of Multiple-valued Decisio Diagram. Proc. IEEE Iteratioal Symposium o Multiple-Valued Logic, pages , 23. [9] D.M. Miller ad M.A. Thorto. QMDD: A Decisio Diagram Structure for Reversible ad Quatum Circuits. Proc. IEEE Iteratioal Symposium o Multiple-Valued Logic, 6 p. o CD, 26. [] D.M. Miller, M.A. Thorto, ad D. Goodma. A Decisio Diagram Package for Reversible ad Quatum Circuit Simulatio. Proc. IEEE World Cogress o Computatioal Itelligece, 6 p. o CD, July 26. [] D.M. Miller, D.Y. Feistei ad M.A. Thorto. Variable Reorderig ad Siftig for QMDD. Proc. IEEE Iteratioal Symposium o Multiple-Valued Logic, 7 p. o CD, May 27. [2] D. Michael Miller ad Mitchell A. Thorto. Multiple-Valued Logic: Cocepts ad Represetatios. MorgaClaypool Publishers, 28. [3] F. Somezi. CUDD: CU Decisio Diagram Package vlsi.colorado.edu/~fabio/cudd/cudditro.html. [4] S. N. Yaushkevitch, D.M. Miller, V.P. Shmerko ad R.S. Stakovic, Decisio Diagram Techiques for Micro- ad Naoelectroic Desig, CRC Taylor ad Fracis, 26.

7 Table : Results for AND fuctio Rademacher-Walsh Trasform (+,-) codig. F S Edge Add Mult Total FT Total / (a)qmdd (b)ft (a) / (b) ops FT ops % % % % % % % % % % % % % % Table 2: Results for OR fuctio Rademacher-Walsh Trasform (+,-) codig. F S Edge Add Mult Total FT Total / (a)qmdd (b)ft (a) / (b) ops FT ops % % % % % % % % % % % % % % Table 3: Results for XOR fuctio Rademacher-Walsh Trasform (+,-) codig. F S Edge Add Mult Total FT Total / (a)qmdd (b)ft (a) / (b) ops FT ops % % % % % % % % % % % % % % Table 4: Results for radom fuctios Rademacher-Walsh Trasform (+,-) codig. F S Edge Add Mult Total FT Total / (a)qmdd (b)ft (a) / (b) ops FT ops % % % % % % % % % % % % % % Table Key umber of variables F umber of vertices i QMDD for the fuctio S umber of vertices i QMDD for the spectrum Edge umber of uique edge weights Add umber of add operatios i QMDD trasform Mult umber of multiplicatio operatios i QMDD trasform Total total operatios i QMDD trasform FT ops operatios i fast trasform QMDD of QMDD for spectrum (bytes) FT of spectrum vector for fast trasform (bytes)

8 Table 5: Comparisos for various fuctios ad trasformatios. F S Edge Add Mult Total FT ops (a)qmdd ((b)ft (a) / (b) S AND E+8.% % RM AND E+8.% % R AND E+8.% % S OR E+8.% % Rm OR E+8.% % R OR E+8.% % S XOR E+8.% % RM XOR E+8.% % R XOR E+8.% % S radom % % RM radom % % R radom % % Table 6: Results for MIN fuctio r=3 Chresteso Spectrum. F S Edge Add Mult Total FT ops Total / (a)qmd (b)ft (a) / (b) FT ops D % % % % % % % % % % % % % % % % % % Table 7: Results for MAX fuctio r=3 Chresteso Spectrum. F S Edge Add Mult Total FT ops Total / (a)qmd (b)ft (a) / (b) FT ops D % % % % % % % % % % % % % % % % % % Table 8: Results for MOD-SUM fuctio r=3 Chresteso Spectrum. F S Edge Add Mult Total FT ops Total / (a)qmd (b)ft (a) / (b) FT ops D % % % % % % % % % % % % % %