An Enhanced Local Covering Approach for Minimization of Multiple-Valued Input Binary-Valued Output Functions

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1 Proceedgs of the 10th WSEAS Iteratoal Coferece o COMPUTERS, Voulagme, Athes, Greece, July 13-15, 2006 (pp63-68) A Ehaced Local Coverg Approach for Mmzato of Multple-Valued Iput Bary-Valued Output Fuctos MOINUL ISLAM ZABER 1, HAFIZ MD. HASAN BABU 2 Departmet of Computer Scece ad Egeerg, BRAC Uversty 1, Uversty of Dhaka 2, Dhaka BANGLADESH Abstract:- Local coverg techque of Multple-Valued Iput Bary-Valued Output Fuctos (MVIBVO)s heretly a dvde ad coquer techque where the mmzato process works expaso ad selecto phases. I order to get cosderably mmzed output proper base mterms should be chose frst. I ths paper we propose a ehacemet of the exstg techque that successfully fds out the essetal prmes wth least umber of passes. Result of the expermet shows superor result tha the exstg techques. Key-Words: - MVL, Local Cover, Caocal Cube, SOP 1 Itroducto: Smplfcato of sum of products expressos s of great mportace logc sythess. I total computato tme for logc sythess, the rato of the tme spet for smplfcatos for SOPs s drectly related to the smplfcato of PLAs. I ths cotext the ecessty for the use of multplevalued logc (MVL) s gag ts valued mportace day by day. The tercoecto complexty of two-valued fuctos both chp ad betwee chps s reduced effectvely by the adept use of MVL. These fuctos ca be of great use to mmze decoded PLA s ad the realm of sequetal crcuts ad etworks. Amog the heurstc methods to fd out the mmum cover of the MVITVO fuctos MINI ad ESPRESSO-IIC [4] are very well kow. I these methods a ear optmum cover of the fucto F to be mmzed s acheved through teratve mprovemet by reshapg ad reducg the tal cover of t. A slghtly dfferet approach to these heurstcs s the approach of the local coverg techque, where the whole process starts from a properly chose base mterm. A mproved verso of ths techque due to Caruso, [5] s as follows: Frst a set of sub fuctos of the fucto to be mmzed s bult (expaso process). The oe or more prmes are selected from the oes of each sub-fucto (selecto process). I the ed a uo of all the selected prmes s doe whch forms a cover of the fucto F. I ths research we have tred to hold dow the racg computatoal tme by emphaszg fdg out the best mterm (the mterm whch has the fewest umber of adjacet mterms) ad at the same tme ehaced the probablty of detectg ad selectg the essetal prmes whle expadg. Ths ew algorthm preserves the oto of the prevous oes ad we are made aware of the lower boud of prmes the mmum cover of the gve fucto. Here we have worked two phases frstly fdg effcet groupg of the Boolea varables ad secodly by fdg fast the vable cubes wth sutable mterms. I cocse our algorthm s fast computato ad prudet keepg the fuctos as mmzed as possble. 2 Basc Deftos: A multple valued put ad bary valued output fucto s a mappg F(X 1,, X 2,, X 3,, X ): P 1 P 2 P 3 P B, where X s a multple-valued varable, B=0, 1, *, ad P =0, 1,, p -1, ( p 2) s a set of values that ths varable may assume. Symbol * deotes the value 1 or 0. A product of costats a 1 x a 2 x...x a wth a P s a mterm. The ON set of F s formed by all the mterms for whch the fucto takes the value 1. Smlarly, the OFF set s formed by all the mterms for whch the fucto takes the value 0. Ad the Do t Care

2 Proceedgs of the 10th WSEAS Iteratoal Coferece o COMPUTERS, Voulagme, Athes, Greece, July 13-15, 2006 (pp63-68) set s the set of mterms for whch the fucto ca dfferetly take the value 1 or 0. Let X be a varable, whch takes oe of the values P= 0,1,, p -1. For ay subset S P, X s s a lteral such that 1 f X S X s = 0 f X S A example for MVITVO fucto ca be expressed as: F(X 1, X 2, X 3 )= X 0 1 X 1 2 X 1 3 V X 1 0,2 1 X 2 X 1 3 V X 2 0,3 1 X 2 X 3,0 3 V X X 2 X 3 Here the products of lterals are termed as cubes. A MVITVO fucto s sad to be mmum f t has the mmum umber of cubes. Let two products be T = 1 T T c 2 1 X X... X 1 2 = c X S1 S S X 2 X The the dstace betwee c 1 ad c 2 s defed as follows: dstace( c 1,c 2 )= [The umber of s such that ( S T )] Two mterms of F are adjacet f they dffer the value of oly oe varable. Two mterms c 1 =X 0 1 X 1 2 X 1 3 ad c 2 =X 0 1 X X 3 are adjacet. We call the cube S1 S S X X 2... X caocal f S =1,for =1,2,, ad S s maxmal covers all the mterms of F that are adjacet oe aother ad dffer the value of the last varable. A example of such cube s c 3 =X 1 0 X 1 2 X 0,1,2 3. A expresso formed exclusvely by such cubes s uque [4]. If C 1 ad C 2 are two cubes the C1 s sad to mply C f C C A prme mplcat, p s a product term whch mples o other cube of the fucto. A essetal prme mplcat deotes a prme mplcat that covers at least oe stadard product term (ot a Do t Care term) that caot be covered by ay other prme mplcat. The essetal prme mplcats, therefore must be cluded to get a mmum cover of the fucto. A dstgushed mterm s a mterm that s covered by oly oe prme mplcats.the essetal prmes cover oe or more dstgushed mterms. F=A 1, A 2, A 3, A 4, A 5, A 6, A 7 U=0,1,2,3 0,1,2,3 0,1,2,3 0,1,2 A 1 = ,1,2 A 2 = ,1,2 A 3 = A 4 = ,1,2 A 5 = ,1,2 A 6 = ,1 A 7 = ,1,2 Fg. 1. Multple-valued fucto example all cubes are the form of caocal cubes. The supercube of the cubes S 1, S 2,..., S h s defed as follows Supercube(S 1, S 2,..., S h 1 ) = (S 1 S 2 1 S h 1 ) S 2 S h ). (S 1 The mmzato procedure for MVITVO fuctos, cossts of the followg steps: (a) The determato of all prme mplcats of the fucto. (b) Fdg out the essetal prme mplcats. (c) From the remag prme mplcats a mmum set s chose so that together wth the essetal prme mplcats they cover the fucto. 3 Algorthms for Mmzg the Multple-Valued Fuctos: I the local cover approach, the mmzato process has two steps amely expaso ad selecto. The expaso process creates a set of sub fuctos of the fucto F to be mmzed. Each sub fucto cossts of the mterms of F covered by all the prmes that cover a base mterm properly chose. I fact the objectve of ths step s to expad the base mterm so to result the SOP where o cubes ca be expaded ay more. The cubes obtaed ths way represet a prme mplcat. I the secod step, amely the selecto step prme mplcats are chose from each sub-fucto so that the uo of all the chose prmes form a rredudat cover of F. The success of the techque depeds largely o the umber of sub-fuctos. The larger the set of sub-fuctos the closer the foud cover s to the mmum. Moreover the detecto of essetal prmes as early as possble plays a vtal part decreasg the computatoal tme. Both of these two essetal crtera ca be acheved by choosg base mterms from the mterms wth the smallest umber of adjacet mterms. I ths process we ca

3 Proceedgs of the 10th WSEAS Iteratoal Coferece o COMPUTERS, Voulagme, Athes, Greece, July 13-15, 2006 (pp63-68) easly detect the essetal prmes from the subfuctos cotag oly oe prme as each essetal prme s uque ad base mterms wth smallest umber of adjacet are dstgushed mterms. Our proposal to the mprovemet of the local cover techque s addto of aother procedure called cube-rearragemet whch helps to fd the ext best mterm to be chose as a base mterm from the potetal caocal cubes (PCC) so that the set of sub fuctos wll crease ad essetal prmes are detected the earlest phase of the computato. I order to realze ths procedure the gve expresso s preferred to be caocal form. The expaso process ths algorthm s doe by crcular shftg the cubes. I case of caocal cubes t geerates a set of mterms adjacet to the orgal cube. Our motvato to the ew procedure les the fact that the mterm wth the smallest umber of adjacets would resde the caocal cubes wth smallest umber of adjacets. So f we ca arrage the gve caocal cubes wth respect to ther adjacecy the t would be less tme cosumg whe searchg for the mterms wth the smallest umber of adjacets. Our algorthm uses a table of dces for all the caocal cubes ad rearrages t accordg to ther umber of adjacets (fg. 4). The procedure frst checks out the X 1 s (1<<m, m = umber of dfferet multple values) ad couts the umber of dstct values (.e. 0, 1, 2, 3 s) for each X 1 s ( [0, -1], s the umber of lterals) of all the caocal cubes. The we update the dex table of the caocal cubes accordg to the weght of dstct values( a weght here s the umber of occurreces of that value). We perform the same procedure for each X s (1<j<-1, =umber of lterals). I j ths way we come to a pot whe the table caot be updated aymore ad ths s whe we have succeeded rearragg properly the cubes such that we ca get the mterms wth the smallest adjacets just by cosultg ths table sequetally. Algorthm 1: Rearrage_caocal_cubes() table=has the tal arragemet of the caocal cubes; pass =1, start =1, ed = table Rearrage (pass, start, ed) a[ ] be a array, It s talzed to zero every pass. For (start=1 to ed) do Cout the umber of dstct values X pass ad put them ther correspodg postos a. If (for all elemets k of a[ ], a[k] s ot 0 ) Rearrage correspodg dces of table wth respect to the cout values accordg to the least occurred values of a[ ]. ed = start + K rearrage (pass+1, start, ed) start = ed Example 1: Let, 5 caocal cubes of F (X 1, X 2, X 3,, X ): A 1 A 2 A 3 A B, A 1 =X 1 1 X 0 2 X 1 3 1, X 2 4, A 2 =X 0 1 X 1 2 X 0 3 1, X 2 4., A 3 =X 0 1 X 0 2 X 3 3 1, X 3 4, A 4 =X 2 1 X 2 2 X 1 3 X 1 4, A 5 =X 1 1 X 0 2 X 0 3 1, 2 X 4, If we choose base mterm a= X 1 1 X X 3 1 X 4 from the caocal cube A 1 we get a Adjacet cube A 5 2 but f we choose A 4 ad a = X 1 X 2 2 X X 4 the we get o adjacet. I order to fd the mterms wth the smallest adjacet we have to rearrage the caocal cubes. The rearragg procedure of caocal cubes dfferet passes of the Algorthm 2 s show the fgure 2. Each sub-fucto cossts of the caocal cubes whch are adjacet to the base mterm properly chose from the fucto F. Let, F= A 1, A 2, A 3 where,a 1 =X 0 1 X X 3 0,1, X 2 4, A 2 =X 3 1 X 1 2 X 2 3 X 2 4, A 3 =X X 2 X 3 3 0,1, 2 X 4 ad base mterm a= X 0 1 X X 3 0 X 4 the after performg the expaso procedure, we get two sub-fuctos P ad Q of F, where P has A 1, A 3 ad Q has A 2. I ths techque we frst geerate a supercube S, by computg from the base mterm a ad all the

4 Proceedgs of the 10th WSEAS Iteratoal Coferece o COMPUTERS, Voulagme, Athes, Greece, July 13-15, 2006 (pp63-68) mterms of F that are adjacet to a, usg crcular shft operato. Now the usg base mterm a ad the supercube S, we geerate the caocal cubes for the sub-fuctos to be bult. The selecto procedure processes oe at a tme the sub-fuctos of a rredudat set. Hece the Algorthm for the Local cover techque ca be descrbed as follows: Algorthm 2: Local cover () F s the Multple-Valued fucto to be mmzed. rearrage_caocal_cubes () /* Rearrage dces for the caocal cubes of F the dex table (algorthm 2). */ Expaso () /* creates sub- fuctos wth prmes coverg the base mterm (algorthm 3). */ Selecto () /* selects prmes from each subfucto ad forms a rredudat cover of F (algorthm 7). */ */perform the uo of all the chose prmes from each sub-fucto so that a rredudat set of prmes s foud. */ Pass Pass Pass Pass A 4 A 4 A 4 A 4 A 1 A 1 A 5 A 5 A 5 A 5 A 1 A 1 A 2 A 2 A 3 A 3 A 3 A 3 A 2 A 2 A 6 A 6 A 6 A 6 Fg. 2. The dfferet arragemets of the dex table the dfferet passes. Algorthm 3: Expaso () a s the base mterm. do If (there s more tha oe mterms left, whch has ot yet bee cluded the subfucto) Look up the dex table to get the cube for the base mterm a, whch has the smallest umber of adjacet mterms. geerate_subfucto(a, F)/*Create subfucto assocated wth the base mterm utl (false) chose (algorthm 4)*/ Algorthm 4: geerate_subfucto(a, F) /*P=P 1, P 2,, P s a subfucto of F ad P,0<< are caocal cubes of P. S s a supercube of P. */ P = A /*A s the caocal cube of F coverg a */ K=1; R= Θ, I0; S=geerate_supercube (a,f). /* (algorthm 5) */ Whle(k<= P ) for = k +1to -1 for j=1 to S -1 k, j P B /* produce B by crcular shftg Pk ; */ (B,B ) =Check_B(B,a,R,F); f (B Θ ) P=P+B, I= ; f (B Θ ) P=P+B ; k=k+1; retur (P, R); Algorthm 5: geerate_supercube(a, F) for =1 to -1 S = a, j for j= 1 to U -1 /* U = U A */ k = 1 a b /* produce b by crcular shftg a */ f (b F ) S = S + b S = A /* where A s the caocal cube of F coverg a */. Example 2: Costructo of Supercube.: If we select the base mterm a= from the cube A 1 = ,1,2 of Fg 1; the the supercube we get from the algorthm 7 ad the terato processes are descrbed the table Fg 2. Iterato of S 1 0,1 2 0,2,3 3 1,3 Here S 4 =A 4 =0,1,2,therefore the supercube s S= 0,1 0,2,3 1,3 0,1,2 Fg. 3. Costructo of Supercube Example 3: Geerato of sub-fucto: k

5 Proceedgs of the 10th WSEAS Iteratoal Coferece o COMPUTERS, Voulagme, Athes, Greece, July 13-15, 2006 (pp63-68) The Algorthm sub_fucto() bulds the subfucto assocated wth a base mterm a= ad therefore we get the o multple-valued ON set cube P ad OFF set cube R after rug ths algorthm we obta P ad R as follows Algorthm6: Check_B (B, a R, F) /* B s a caocal cube of F */ D= SuperCube (a,b); For each r R such that r D Θ I D=D# r; B = D ; B = Θ ;B =B; f a B f there s a C F such that C I B Θ ad a C B = CI B; B = B# B; retur B, B ; The route Check_B reduces D so that DI P OFF = Θ. Here D = Sepercube (a, B) =P ad P s the offset of P. Sce P s ot OFF OFF avalable, t uses a subset of t R bult durg the gereato of P. C =C + C k ; Else For = 1 to h f (a r ) C =C + C k # r h ; Delete each cube of C mplyg a cube of C ; C=C +C ; retur C; Example 4: Dervg the prmes of a sub-fuctos: Algorthm 8 [2] geerate the set of cubes C that ca be buld by computg the set S # R ad deletg every cube that mples oe of the other yelded cubes or does ot cover the base mterm. Itally C cotas oly S. The er loop processes oe at a tme the cubes of C. Let C k be the cube uder processg. If C k does ot tersect R h, t s serted C, other wse, each cube C k # R h that covers the base mterm s serted C. The each cube of C mplyg a cube of C s deleted ad a ew C s formed by the resdual C ad C. Ths process s repeated for such a C the ext terato of the outer loop.after rug ths algorthm we get the prmes whch are as follows Fg.4. geereate_sub_fucto() produce P,R. Algorthm 7: Selecto (S, a, R) /*Select prmes from each sub-fucto.*/ C=S; For h =1 to R C =C = Θ ; For k=1 to C f (C k I r h == Θ ) Espresso ad MINI eed the prelmary geerato of the OFF set of the gve fucto. Ufortuately, there exst fuctos for whch such a set s exdeedgly large. I some cases, t s possble to overcome such a drawback by usg a reduced OFF set [8]. The referred work s focused o mmzato of bary-valued fuctos wth sgle output. Our local cover algorthm uses a subset of the OFF set of a subfucto both to buld the subfucto tself ad extract prmes from t. However, such a subset does ot cocde wth the reduced OFF set troduced [8]. Cosder for stace, the followg example draw from[8] F ON = a b cd+ a b c d, F OFF = ab +a b+ac +cd, F DC = a b c d+abcd. The reduced OFF set assocated wth a b c d s a+b+c. Whereas the set R yelded by gererate_sub_fucto() by expadg a b c d s empty. I fact, the yelded subfucto holds oly

6 Proceedgs of the 10th WSEAS Iteratoal Coferece o COMPUTERS, Voulagme, Athes, Greece, July 13-15, 2006 (pp63-68) oe essetal prme of F;.e. a b c. a b c d s a dstgushed mterm of such a prme. 4 Expermetal Results: Our proposed algorthms for groupg ad mmzato processes has bee wrtte usg the laguage C ad tested extesvely o Wdows Workstatos. The expermetal results gve below have bee receved from a Itel Petum III Moble 1000 MHz uder Mcrosoft Wdows XP professoal edto. The table 1 shows the superorty of our groupg scheme usg the ehaced assgmet graph (EAG). Ad the results of the comparso of the algorthm proposed by [2] ad our proposed oe s show table 2. It ca be uderstood that most of the cases our mprovemet resulted decrease the umber of products ad computato tme. 5 Cocluso: New Boolea varable assgmet algorthm ad mmzato techques have bee proposed, so that both the total computato tme ad umber of products decreases. Our algorthmc exteso to [2] has bee prove to be effcet detectg ad selectg the essetal prme mpcats as well as furshg the lower boud o the umber of prme mplcats the frst phase of the computato process. The ew cocepts of ehaced assgmet graph, use of Hamltoa path fdg out the best pars, ad the techque of cube rearragemet are prove to be effcet step-by-step mmzato process. Alog wth these some heurstcs used dfferet phases of expaso ad selecto has evertheless mproved the qualty of the whole techque. The comparso of our proposed techque wth the algorthm proposed by Caruso [2] has proved ts superorty. Refereces [1] H. M. H. Babu, M. Zaber, R. Islam, ad M. Rahma, O the mmzato of Multple Valued Iput Bary Valued Output Fuctos, Iteratoal Symposum o Multple Valued Logc (ISMVL 2004), May 18-21, 2004, Toroto Uversty, Toroto, Caada. [2] Caruso, G., 1996, A local Cover Techque for Mmzato of Multple-Valued Iput Bary- Valued Output Fuctos. IEICE Tras, Fudametals, Vol. E79 A. [3]Sasao, T., 1984, Iput varable assgmet ad output hase optmzato of PLA s. IEEE Tras, Comput, vol. C-33, pp [4]Brayto, R. K., Hatchel, G. D., McMulle, C. T., ad Sagova-Vcetell, A., 1984, Logc Mmzato Algorthms for VLSI Sythess. Hgham, MA: Kulver Academc. [5] Caruso, G., 1984, A local selecto algorthm for swtchg mmzato. IEEE Tras, Comput, Vol c-33, pp [6]Sasao, T., 1998, Multple-Valued Logc ad Optmzato of Programmable Logc Arrays. IEEE Tras. [7]Brayto, R. K, Watabe, Y,Oct. 1993, Heurstc mmzato of multple-valued relatos,computer-aded Desg of Itegrated Crcuts ad Systems, IEEE Trasacto o, Vol: 12, Issue: 10. [8] A.A. Malk, R.K. Brayto, A.R.Netwo ad A.Sagova-Vcetell, Reduced offset for mmzato of bary-valued fuctos, IEEE tras.computer Aded Desg, vol. 10, pp , Aprl Fuc To Stadard PLA Proposed algorthm p tme Algorthm [2] p tme Decoded PLA Proposed algorthm Np tme Algorthm [2] p tme Clp Co sym Rd Rd Bw Z5xp p: umber of products Table 1. Comparso of the proposed algorthm wth the algorthm [2].