Hybrid Fuzzy Direct Cover Algorithm for Synthesis of Multiple-Valued Logic Functions

Transcription

1 IJCSI Internatonal Journal of Computer Scence Issues, Vol. 8, Issue 3, No., May 011 ISSN (Onlne): Hybrd Fuzzy Drect Cover Algorthm for Synthess of Multple-Valued Logc Functons Mostafa Abd-El-Barr Informaton Scence Department, Kuwat Unversty, Safat 13060, P.O. Box 5969, Kuwat Abstract Drect Cover (DC) based technques for synthess of multple-valued logc (MVL) functons have been reported n the lterature. In ths paper, we propose Fuzzy-based DC algorthm for synthess of MVL functons. The proposed algorthm uses the prncples of Fuzzy Logc n selectng the set of mnterms and the approprate set of mplcants needed to synthesze a gven MVL functon. The proposed Fuzzfed-Drect-Cover (FZDC) heurstc s tested n comparson wth fve other DC-based algorthms reported n the lterature. The benchmark used n our comparson conssts of varable 4-valued randomly generated functons. The bass for comparson s the chp area consumed n terms of the average number of mplcants needed to synthesze a gven MVL functon. It s shown that on average the FZDC heurstc requres less number of mplcants to synthesze a gven MVL functon as compared to those requred by any of the other fve DCbased heurstcs. Keywords: Mult-Valued Logc (MVL) synthess, Heurstc algorthms, Drect Cover (DC) algorthms, Weghted drect cover algorthm (WDC), Ordered DC algorthm (ODC), Fuzzfed-Drect-Cover algorthm (FZDC). 1. Introducton Sgnal processng usng multple-valued logc (MVL) s carred out usng more than two logc levels [11][14][17][19]. Successful hardware realzaton of MVL crcuts has been reported n the lterature. Examples of MVL crcuts reported n the lterature usng bnary CMOS (Complementary Metal Oxde Semconductor) crcuts [45]-[47] nclude arthmetc crcuts [4], [39], [48], and [49], memory (ROM, RAM, and Flash) [38] and [51]-[5], and machne learnng [4][7]. Determnstc synthess of MVL functons s more complex than ts bnary counterpart. Exact mnmzaton of MVL functons s prohbtvely expensve and the use of heurstc algorthms n MVL synthess s emnent [33], [50]. A number of heurstc algorthms for producng near mnmal sum-of-products realzaton of MVL functons have been ntroduced n the lterature [9] [31] [34][36]. These algorthms can be categorzed as drect cover-based [][8][10][15], algebrac-based [1][16][18][6][9], decomposton-based [1][4]-[7] [1] [5] [35], decson dagram-based [1], [], [7], [8], [3], and [37], and teratve-based [9]. Among these, the drect cover-based technques have shown promsng results and therefore have receved ncreasng attenton by MVL systems desgners. In the context of MVL functonal synthess, the followng defntons are relevant to present. Defnton 1: An n -varable r -valued functon, f (X ), s a mappng f : R n R, where R 0,1,, r 1 s a set of r logc values wth r and X x, x, 1, x n s a set of r -valued n varables. Defnton : A wndow lteral of a MVL varable x s defned as: ( r 1) f ( a x b) a x b 0 otherwse where a, b R and a b. Defnton 3: A tsum (truncated sum) operator s defned as n a1 a... a n f a r 1 tsuma ( 1, a,..., a n ) a1 a... a n 1 r 1 otherwse Where a R and represents the truncated sum operaton. Defnton 4: The maxmum (MAX) of two MVL varables s defned as: x1 f x1 x MAX ( x1, x ) x otherwse Defnton 5: The mnmum (MIN) of two MVL varables s defned as:

2 IJCSI Internatonal Journal of Computer Scence Issues, Vol. 8, Issue 3, No., May 011 ISSN (Onlne): x1 f x1 x MIN ( x1, x ) x otherwse A functonally complete set of operators s the set capable of realzng all possble functons. A number of functonally complete sets have been used n syntheszng MVL functons. In terms of the set of MVL operators dscussed above, the set consstng of {Lteral, MIN, TSUM, Constant} s used n ths paper. Defnton 6: A product term (PT), P ( x1, x,..., ), s defned as the mnmum of a set of wndow lterals such that P( x, x 1,..., x n ) c x a1, b1 a, b an, bn a1, b1 a, b an, bn 1, x,..., mn( c, x1, x,..., wth a,b R, a b and c (the value of the PT) {1,,, r-1}. Defnton 7: An assgnment of values to varables such that x1 a1, x a..., x n an, where a {0, 1,, r-1}, n an MVL functon f ( x1, x,..., ) s called a mnterm, ff: f ( x1, x,..., ) 0. A mnterm s a specal case of a PT for whch a1 b1, a b..., a n bn. If the value of a mnterm s r, then t s consdered as don t care and s represented as d. Defnton 8: An mplcant of a functon f ( x1, x,..., ), s a PT, I ( x1, x,..., ), such that f ( x1, x,..., ) I ( x1, x,..., ) for all assgnments of x s. Fg. 1 shows an example of a -varable 4-valued functon. Some of the mnterms n the fgure are 1 0 X X 3, 0 X X and 3 0 X 0 1 X whle 0 X X 1 and 0 X X are examples for mplcants. Fg. 1. A Tabular Representaton of f(x 1, X ). The paper s organzed as follows. In Secton, we brefly present related publshed work. In Secton 3 we ntroduce the hybrd Fuzzy based Drect Cover heurstc (FZDC). The results obtaned usng the proposed and the related ) DC-based heurstcs usng a benchmarks consstng of varable 4-valued randomly generated functons are presented n Secton 4. In Secton 5, we show a case study for the applcaton of both the ntroduced and the related heurstcs. The presented case study s that for a 1- bt 4-valued adder block. Concludng remarks are drawn n Secton 6. Ths document s set n 10-pont Tmes New Roman. If absolutely necessary, we suggest the use of condensed lne spacng rather than smaller pont szes. Some techncal formattng software prnt mathematcal formulas n talc type, wth subscrpts and superscrpts n a slghtly smaller font sze. Ths s acceptable.. Related Work In ths secton, we brefly cover the related work publshed n the lterature..1 The Drect Cover Algorthm The Drect Cover (DC) technques for synthess of MVL functons consst of the followng man steps: 1. Choose a mnterm (see Defnton 7),. Identfy a sutable mplcant (see Defnton 8) that covers the chosen mnterm, 3. Obtan a reduced functon by subtractng the dentfed mplcant from the (remanng part of the) functon, and 4. Repeat steps 1 to 3 untl no more mnterms reman uncovered. The method used n the selecton of mnterms and mplcants s crucal n obtanng reduced number of product terms to cover a gven functon. The Drect Cover approaches reported n the lterature dffer n the way mnterms are selected and the way accordng to whch mplcants are dentfed. Dfferent mnterm selecton metrcs have been reported n the lterature. These nclude usng the Isolaton Weght (IW) [3] as a measure of the degree to whch other mnterms cluster around a gven mnterm, the Isolaton Factor (IF) [10] whch measures the number of drectons n whch a gven mnterm can be combned wth a nonzero number of other mnterms, and the Clusterng Factor (CFN) [8] whch measures the number of mnterms that are connected to a gven mnterm. Smlarly, a number of metrcs were used n selectng the approprate mplcant to cover a gven mnterm. These nclude the Largest Number of Mnterms Reduced to Zero (LRZ) [3] [15] whch counts the number of 0 (or don t care) that resulted from removng the selected mplcant from the gven MVL functon, the Relatve Break Count (RBC) [10] whch a measure of the degree to whch a functon s smplfed f a specfc

3 IJCSI Internatonal Journal of Computer Scence Issues, Vol. 8, Issue 3, No., May 011 ISSN (Onlne): mplcant s chosen, and the Neghborhood Relatve Count (NRC) [8] whch measures the degree of the strength to whch a gven mplcant couples wth ts neghbors. It should be noted that there s no general agreement on whch set of crtera s the best to use for a gven MVL functon. An analyss of a lmted subset of the dfferent suggested crtera has been reported n [3]. Consderng the dfferent mnterm/mplcant selecton crtera, t s possble to combne all metrcs to create a new selecton crteron. One possble way to acheve ths s by usng the weghted sum approach. However, the dfferent scale and behavor of each crteron makes t dffcult to determne the perfect weghts for each objectve and the perfect way of aggregatng these nto a sngle functon. We have reported n the lterature two approaches to deal wth such ssue. These are the weghted DC-based (WDC) and the Ordered DC-based (ODC) heurstc algorthms [53]. These are brefly covered below.. The Weghted Drect Cover (WDC) [53] In the WDC, the term weght pattern s used to specfy the weght for each selecton crteron. Assumng 4-valued logc, we assume that the weght should be nteger n the set {0, 1,, 3}. Consder the mnterm selecton process. In ths case, a weght pattern of (11) means that the followng weghts are used: w CF =1, w CFN =1, and w IW =. The number of dfferent weght patterns for mnterms (CF, CFN and IW) combned wth the dfferent weght patterns for mplcants (LRZ, RBC, and NRC) s calculated as We have shown n [53] that t s possble to reduce the number of weght combnatons to ( 4 ) 576. It has also been shown n [53] that the best weghted combnaton of crtera s as follows: (a) Mnterm Selecton: w CF 0, w CFN 1, w IW, weght pattern (01) (b) Implcant Selecton: w RBC 1, w NRC 0, W LRZ 0, weght patter (100). Usng the patterns the patterns shown above, a combned weght patter s wrtten shortly as Example 1: Consder the SUM output functon of the 1- bt 4-valued adder buldng block. Ths buldng block receves two 4-valued nputs x and y and produces two outputs S and C. The SUM output S s gven by the table shown n Fg.. We have used the WDC algorthm to synthess the SUM functon. The total number of mplcants needed to synthesze the SUM functon s eght. Ths s shown below usng the MVL operators ntroduced n Secton S ( x, y ) 1. x y 1. x y 1. x y 1. x y x y. x y 3. x y 3. x y A benchmark consstng of randomly generated 4- valued -varable functons has been used to test the performance of the WDC. It was found that the (best) average number of product terms requred to cover a gven 4-valued -varable functon s More detals about the obtaned results are ncluded n Secton 4, n whch we hold a comparson between the DC-based heurstcs. x y Fg.. The SUM output S of a 1-bt 4-valued adder block. The Ordered Drect Cover (ODC) [53] We have observed that dfferent orderng of the Mnterm and the Implcant crtera can result n dfferent number of product terms for a gven MVL functon. Hence, n the ODC the orderng of the selecton crtera for both mnterm and mplcant selecton s taken nto consderaton whle syntheszng a gven MVL functon. It should be noted that the mnterm selecton crtera are: Smallest CF, Smallest CFN, and Smallest IW. The mplcant selecton crtera are: Smallest RBC, Smallest NRC, and Largest LRZ. A number of dfferent orderngs of these crtera have been expermented wth usng the same benchmark consstng of randomly generated 4-valued -varable functons. It was found that the followng crtera orderng: mnterm selecton: Smallest IW; mplcant selecton: Smallest RBC followed by Largest LRZ. Example : We have appled the ODC n syntheszng the SUM output functon of the 1-bt 4-valued adder buldng block (as explaned n Example 1 above). The total number of mplcants needed to synthesze the SUM functon s eght (the same as n the case of the WDC). Ths s shown below usng the MVL operators ntroduced n Secton S ( x, y ) 1. x y 1. x y 1. x y 1. x y x y. x y 3. x y 3. x y The same benchmark consstng of randomly generated 4-valued -varable functons whch has been used n the case of WDC has been used to test the performance of the ODC. It was found that the (best) average number of product terms requred to cover a gven 4-valued -varable functon s Ths s somewhat better than the results obtaned usng the WDC. 01

4 IJCSI Internatonal Journal of Computer Scence Issues, Vol. 8, Issue 3, No., May 011 ISSN (Onlne): More detals about the obtaned results are ncluded n Secton 4, n whch we hold a comparson between the DC-based heurstcs. 3. Fuzzy based Drect Cover Algorthm In DC-based algorthms, the mplcants that should be ncluded n the fnal synthess of the gven functon are the ones that best cover the chosen set of mnterms and such that all mnterms are covered. The selecton of a mnterm s based on the qualty of ts CF, CFN, IW or a combnaton thereof. We observe that such condton can be translated nto a Fuzzy Logc Rule of the form: IF a gven mnterm has a good CF crteron OR a good CFN crteron OR a good IW crteron, THEN t s a potental mnterm to select. If a desgner wants to emphasze a gven crteron he/she wll have to use a set of Preference Rules n order to gve more sgnfcance to that partcular crteron. In order to formulate the (PRs), preference terms need to be defned. These terms wll be assocated wth the man lngustc terms. In the fuzzy rules, the lngustc terms low, hgh, and good have been used. A number of approaches to fnd preference terms and preference rules based on Membershp Functons has been reported n the lterature [0][40][41][43][44]. These approaches map a PR relaton P to a fuzzy membershp functon P n the range [0, 1] wth 1(0) means the crteron s defntely preferred (not preferred). There are two basc types of fuzzy operators. The operators for the ntersecton, nterpreted as the logcal AND", and the operators for the unon, nterpreted as the logcal OR" of fuzzy sets. The ntersecton operators are known as trangular norms (t-norms), and unon operator as trangular co-norms (t-co-norms or s-norms). In mult crtera decson problem, such as ours, ths translates to two extremes whch lead to the formaton of overall functons. One extreme s the case requrng all crtera to be satsfed, whch leads to the pure-and-ng operaton. On the other hand, when t s requred to satsfy any of the crtera, the pure OR-ng operaton s used. The formulaton of mult crteron decson functons nether requres the pure AND-ng" of t-norm nor the pure ORng" of s-norm. The reason for ths s the complete lack of compensaton of t-norm for any partal fulfllment and complete submsson of s-norm to fulfllment of any crtera.the use of ordered weghted averagng OWA operator has been used by a number of researchers n the soluton of mult-objectve problems [4]. The OWA category of operators allows easy adjustment of the degree of ANDng and ORng embedded n the aggregaton. The OR-lke and AND-lke OWA for two fuzzy sets A and B are expressed n the followng equatons. 1 A B ( x ) Max ( A, B ) (1 ) ( A B ) A B 1 ( x ) Mn ( A, B ) (1 ) ( A B ) In the above equatons represents the membershp value n a fuzzy functon; [0, 1] s a constant parameter, whch represents the degree to whch the OWA operator resembles the pure OR or pure AND, respectvely. Our proposed Fuzzy-based Drect Cover (FZDC) algorthm employs fuzzy rules (along wth preferences) to select the best set of mnterm and the most approprate mplcant coverng each such that the whole functon s covered. The goodness of a mnterm (mplcant) s examned usng the abovementoned fuzzy rules and preferences. Lookng at these rules, t s easy to deduce that we can use the OR- lke operator to aggregate all decson crtera. Tables 1 shows the mathematcal formulae we ntroduced for each membershp functon n the mnterm selecton. Table shows the same for mplcant selecton. Table 1: Membershp functons n Mn terms selecton. Mnterm Selecton Technque Crteron Our Formulated Membershp Functon DM [10] CF 1 f CF 0 ND [8] BS [3] CFN IW Max CF μ CF CF Max CF 0 f 0 CF f other wse Max CF 1 f CFN 0 Max CFN μ CFN CFN f 0 CFN MaxCFN MaxCFN 0 otherwse 1 f IW Mn IW Max IW IW IW Max IW Mn IW f Mn IW IW Max IW 0 otherwse Table : Membershp functons n mplcant selecton. Implcant Selecton Technque Crteron Our Formulated Membershp Functon DM [10] RBC 1 ND [8] BS [3] NRC LRZ f RBC Mn RBC MAX RBC RBC RBC f MIN RBC RBC Max RBC Max RBC Mn RBC otherwse 0 1 f NRC Mn NRC MAX NRC NRC NRC Max NRC Mn NRC f MIN NRC NRC Max NRC 0 otherwse 1 f LRZ Max LRZ LRZ LRZ Max LRZ f 0 LRZ Max LRZ 0 otherwse

5 IJCSI Internatonal Journal of Computer Scence Issues, Vol. 8, Issue 3, No., May 011 ISSN (Onlne): Effcency of the proposed fuzzy selecton process s nfluenced by the value of n OWA operator (see the equatons above). In order to assess such nfluence, we have expermented wth the effect of on the results obtaned usng the FZDC algorthm. Table 3 summarzes such effect n terms of the average number of product terms needed to synthesze a gven functon from the benchmark functons. Table 3: Effect of # PT In addton to the mpact of there s also the mpact of the fuzzy preference rules. Snce there are three crtera for each of the mnterm and mplcant selecton, there wll be addtonal 6 parameters that need to be fne tuned to get the best performance of the algorthm. In order to obtan the best result usng the proposed fuzzy-based selecton crtera, the followng set of experments are conducted: 1. Experments wth dfferent fuzzy operators. Experments wth dfferent parameter values n the fuzzy operators For the frst experment, the followng fuzzy operators are used: 1. Max operator. Max operator wth fuzzy preference rules 3. The Ordered Weghted Averagng (OWA) operator 4. OWA operator wth fuzzy preference rules 5. Weghted Averagng (usng fuzzy preference as the weght) The fuzzy preference rules can be obtaned by lookng at the results of WDC and ODC algorthms. Usng the nformaton obtaned from our work on these two approaches, we adopted some general rules: (A) Accordng to the WDC: 1. mnterm selecton process PM1a. IW s preferred as compared to other crtera PM1b. Ether IW or CFN s preferred as compared to CF PM1c. If CF s preferred more than IW, then the preference of CFN should be greater than or equal to that of CF.. mplcant selecton process PL1a. RBC s strongly preferred as compared to other crtera PL1b. LRZ s slghtly preferred as compared to NRC (B) Accordng to the ODC: 1. mnterm selecton process PMa. IW s strongly preferred as compared to other crtera PMb. CF s slghtly preferred as compared to CFN. mplcant selecton process PLa. RBC s strongly preferred as compared to other crtera PLb. LRZ s slghtly preferred as compared to NRC As can be seen, the preference rules for mplcant s selecton are consstent n both WDC and ODC. However, ths s not the case wth mnterm s selecton. Although we can also deduce a general rule that IW s the strongly preferred crteron from PM1a and PMb, there s some nconsstency n PM1b, PM1c and PMb. However, snce results of ODC are superor to those of WDC, we wll use the preference rules deduced from ODC experments. Table 4 shows the fuzzy preference used for the frst experment. It should be noted that we have used = 0.5 for the OWA operator n the obtanng the results reported n Table 4. Table 4: Fuzzy preference for mnterm and mplcant crtera Mnterm Implcant Crtera Fuzzy Preference Crtera Fuzzy Preference IW 0.9 RBC 0.9 CF 0. LRZ 0. CFN 0.1 NRC 0.1 Usng the abovementoned fuzzy preferences, we tested the proposed fuzzy selecton crtera aganst the randomly generated 4-valued -varable benchmark MVL functons. Fve dfferent fuzzy operators are used for ths purpose. Table 5 shows the results of the experment. Table 5: Performance of dfferent fuzzy operators n FZDC. Operator # PT No CMV # PT Wth CMV Max Max wth pref OWA OWA wth pref Weghted average It should be noted that we lst the results obtaned n two cases: not consderng mnterm values n any order (No CMV) and takng mnterm values n ascendng orders; lower to hgher values (Wth CMV). The results show that takng the mnterm value nto consderaton whle selectng the next mnterm to cover produces on average better results n terms of the average number of mplcants needed to cover a gven MVL functon. Ths can be attrbuted to the use of the tsum operator as a connectng operator for the obtaned mplcants. Ths result s consstent wth the observaton made n [10]. Example 3: We have appled the FZDC n syntheszng the SUM output functon of the 1-bt 4-valued adder buldng block (as explaned n Example above). The total number of mplcants needed to synthesze the SUM functon s eght (the same as n the case of the ODC).

6 IJCSI Internatonal Journal of Computer Scence Issues, Vol. 8, Issue 3, No., May 011 ISSN (Onlne): Ths s shown below usng the MVL operators ntroduced n Secton S ( x, y ) 1. xy 1. xy 1. xy 1. x y xy. xy 3. xy 3. xy In addton, we have also appled the three reported DC algorthms,.e. ARM, BS, and MD n syntheszng the SUM output functon of the 1-bt 4-valued adder block. The results obtaned are shown below. Example 4: The results obtaned n the case of ARM [15] S ( x, y ) 1 3 X 3 1 X 1 1 X X 1 0 X X 1 X X 0 X 0 1 X 1 X X 3 X X 1 1 X X 3 X X 3 X X 1 X 1 1 X 3 0 X X Example 5: The results obtaned n the case of BS [3] S ( x, y ) 1 3 X X 1 X X 1 0 X X 1 0 X 0 1 X 1 1 X X 1 X X 1 X X 3 X X 3 X X 0 X X 1 1 X 1 1 X Example 6: The results obtaned n the case of DM [10] S ( x, y ) 1 X X 1 3 X X 1 0 X X 1 1 X X 0 X 1 1 X X X 3 0 X X 3 3 X X As can be seen synthess of the SUM output functon of the 1-bt 4-valued adder usng the ARM [15] requres 11 PTs as compared to 8 PTs f the FZDC s used (a savng of 50%) whle synthess of the same functon usng the BS [3] requres 11 PTs as compared to 8 PTs usng the FZDC (a savng of 37.5%). It s only n the case of DM [10] that the same number of PTs wll be needed as n the case of the FZDC. A number of other experments were conducted n order to further assess the performance of the FZDC. These are explaned below. 4. Comparson In order to assess the performance of the proposed FZDC algorthm wth respect to the performance of exstng DCbased technques, we have tested the fve DC-based algorthms (ARM [15], BS [3], DM [10], WDC [53], and ODC [53]) as well as the proposed FZDC algorthm usng the set of benchmark consstng of valued - varable randomly generated functons. The crteron used for assessng these algorthms s the average number of mplcants requred to cover a gven functon. The results obtaned are shown n Table 6. As can be seen from Table 6 the proposed FZDC algorthm performs on average better than the other fve DC-based algorthms. For more nsght nto the obtaned results, we report n Table 7 the consttuents of the valued -varable functons benchmark classfed accordng to the number of mnterms n each functon generated. The average number of Product Terms (PTs) requred to synthess the functons n each category are lsted n Table 7. Table 6: Overall Comparson among dfferent algorthms Algorthm Average Number of Product Terms needed ARM [15] BS [3] DM [10] WDC [53] ODC [53] FZDC Table 7: Average # PTs usng MVL Functons havng Dfferent Number of Mnterms #Mnterms/ ARM BS DM ODC WDC FZDC #Fuctons [15] [3] [10] [53] [53] 16/ / / / / / / / / / / / / / The results shown n Table 7 reveal that the proposed FZDC algorthm performs better than the other DC-based algorthms n each category of functons wthn the benchmark. In order to analyze further the results obtaned, we have computed the maxmum percentage mprovement acheved by the FZDC algorthm wth respect to any of the other DC-based algorthm. Table 8 provdes such nformaton. From Table 8 we can see that the maxmum percentage of mprovement acheved over all DC-based technques s and ths s acheved n 6589 functons out of the (about 13.% of the benchmark functons). Furthermore, we have categorzed n Table 9 the mprovement acheved by the proposed FZDC. The results shown n Table 9 reveal that for about 6% (31083 functons) of the benchmark functons mprovement greater than or equal to 10% have been acheved usng FZDC. For about 37% (18550 functons) of the benchmark functons mprovement between 5% and 10% has been acheved. Ths ndcated that for

7 IJCSI Internatonal Journal of Computer Scence Issues, Vol. 8, Issue 3, No., May 011 ISSN (Onlne): % (49985 functons) of the benchmark functons an mprovement greater than or equal to 5% have been possble wth the proposed FZDC algorthm. It s only n 15 functons out of the (about 0.003%) there has been no mprovement. Table 8: Maxmum mprovement acheved by FZDC #Functons ARM [15] BS [3] DM [10] FZDC Max.% Improvement Table 9: The percentage of mprovement acheved by FZDC Functons wth Number of functons % age Improve. 1% % 10% mprove. <1% ( ) = % < 10% mprove. 5% ( ) 37.1% = < 5% mprove. (77+75)= % no mprovement % 5. Conclusons In ths paper, a Fuzzy-DC-based algorthm (FZDC) for synthess of MVL functons has been proposed. The algorthm uses the prncples of Fuzzy logc n order to make the selecton of mnterms and the approprate mplcants needed to cover them n syntheszng a gven MVL functon. The performance of the proposed FZDC algorthm n terms of the average number of product terms (mplcants) requred to synthesze a gven MVL functon from a set of benchmark consstng of bvalued - varable functons has been assessed. In makng such assessment we have compared the proposed FZDC algorthm wth fve other exstng DC-based algorthms. The results obtaned usng the ntroduced FZDC algorthm were compared to those obtaned usng exstng DC-based technques. It has been shown that the proposed algorthm performs better than the other exstng DC-based algorthms. In partcular t has been found that the proposed algorthm acheved a maxmum of about 1% mprovement n the number of mplcants requred n syntheszng 6589 functons (about13.% of the benchmark functons). The FZDC acheved mprovement of greater than or equal to 10% n 3183 functons (about 6% of the benchmark functons). It has also been shown that n functons (about 99% of the benchmark functons) an mprovement of greater than or equal to 5% s possble usng the proposed FZDC algorthm. We have also shown the applcablty of the FZDC algorthm n syntheszng the SUM output of a 1-bt 4-valued adder crcut. In ths later case, t was found that the FZDC algorthm produces results that are as good as those produced by our prevously proposed ODC & WDC [53] and the algorthm reported n [10]. Acknowledgment The author would lke to acknowledge the contrbuton made by Mr. Bambang Sarf n an earler draft of the paper. References [1] Mshchenko, B. Stenbach, M. Perkowsk, Bdecomposton of mult-valued relatons. Proc. Internatonal Workshop on Logc and Synthess 001, pp [] Mshchenko and R. Brayton, Smplfcaton of nondetermnstc mult-valued networks, Proc. Internatonal Conference on Computer Aded Desgn 00, pp [3] Besslch P. W. "Heurstc Mnmzaton of MVL functons: A Drect Cover Approach." IEEE Transactons on computer, Feb. 1986, pp [4] Zupan, B., Machne Learnng Based on Functonal Decomposton. PhD thess, Unversty of Ljubljana, Slovena, [5] Fles, C., R. Drechsler, and M. Perkowsk. Functonal decomposton of MVL functons usng mult-valued decson dagram. Internatonal Symposum on Mult- Valued Logc (ISMVL), May 1997, pages 7 3. [6] Fles, C. and M.A Perkowsk, New mult-valued functonal decomposton algorthms based on MDDs. IEEE Trans. CAD. Vol. 19(9), Sept. 000, pp [7] Fles, C. and M. A Perkowsk, Mult-valued functonal decomposton as a machne learnng method, Proc. ISMVL '98, pp [8] Yang, C. and Y.-M. Wang. A neghborhood decouplng algorthm for truncated sum mnmzaton. Proceedngs of the 0 th ISMVL, 1990, May 1990, pp [9] Yldrm, C., J. T. Butler and C. Yang, "Multple-valued PLA mnmzaton by concurrent multple and mxed smulated annealng," Proceedngs of the 3 rd ISMVL, May 1993, pp [10] Dueck, G. and Mller, D. M., "A Drect Cover MVL Mnmzaton Usng the Truncated Sum." Proceedng of the 17 th ISMVL, May 1987, pp [11] Olmsted, D., The Hstory and Prncples of Mult-valued Logc, [1] Mller, D., Decomposton n many valued desgn. PhD thess, Unversty of Mantoba, May [13] Dubrova, E., Y. Jang, R. Brayton, Mnmzaton of multvalued functons n Post algebra, Proc. Internatonal Workshop on Logc and Synthess 001, pp

8 IJCSI Internatonal Journal of Computer Scence Issues, Vol. 8, Issue 3, No., May 011 ISSN (Onlne): [14] Post, E., Introducton to a general theory of elementary propostons, Amercan Journal of Mathematcs, Vol. 43, 190, pp [15] Promper, G. and J. A. Armstrong. Representaton of multple valued functons usng the drect cover method. IEEE transactons on Computers, September 1981, pp [16] Dueck, G. and G. H. J. van Rees, On the Maxmum Number of Implcants Needed to Cover a Multple-Valued Logc Functon Usng Wndow Lterals, Proceedngs of the 1 st ISMVL, May 1991, pp [17] Epsten, G., G. Freder and D. C. Rne, The development of Multple-valued logc as related to computer scence, Computer, Vol. 7 (9), 1974, pp [18] Goa, J.-H. and R. Brayton. Optmzaton of mult-valued mult-level networks. Proceedngs of the Internatonal Symposum on Multple-Valued Logc, Tahoe Cty, June 00. [19] Lukasewcz, J. englsh translaton: On three valued-logc, n L. 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Avalable at

9 IJCSI Internatonal Journal of Computer Scence Issues, Vol. 8, Issue 3, No., May 011 ISSN (Onlne): [5] Okuda, T. and Murotan, T., A four-level storage 4GB DRAM, IEEE Journal of Sold State Crcuts, vol. 3, no. 11, 1997, pp [53] Abd-El-Barr, M. and Bambang Sarf, Weghted and Ordered Drect Cover Algorthms for Mnmzaton of MVL Functons, Proceedngs of the IEEE 37 th Internatonal Symposum on Multple-Valued Logc (ISMVL07), Oslo, Norway, May 007, pp Mostafa Abd-El-Barr (Senor Member of IEEE) has receved hs B.Sc. and M.SC. degrees n Electrcal Engneerng (Computer Track), Caro Unversty, Egypt n 1973 and 1979, respectvely. He receved hs PhD degree n Computer Engneerng, Unversty of Toronto, Ontaro, Canada n In July 1986 he joned the faculty of the Department of Computer Scence, Unversty of Saskatchewan (U. of S.), Saskatoon, Canada, where he was promoted to the rank of full professor n July He was also an assocate member of the Department of Electrcal Engneerng, U. of S. In September 1996, he joned the Department of Computer Engneerng at Kng Fahd Unversty of Petroleum and Mnerals (KFUPM), Saud Araba as a full professor. In September 003 he was apponted as the foundng charman of the Informaton Scence Department, Kuwat Unversty (KU). In September 005 he was apponted as the Vce Dean for Academc Affars, CFW, KU. He s currently an Adjunct Professor wth the Department of Electrcal and Computer Engneerng, Unversty of Vctora, BC, Canada. Hs research nterests nclude Desgn and Analyss of Relable & Fault-Tolerant Computer Systems, Computer Networks Optmzaton, Parallel Processng/Algorthms, Informaton and Computer Systems Securty, Beyond-Bnary Logc System Desgn & Analyss, VLSI System Desgn & Algorthms, and Dgtal Systems Testng. He s the author and/or co-author of more than 150 scentfc papers publshed n journals and conference proceedngs. He s a certfed professonal engneer n the Provnce of Ontaro, Canada.