Exact Minimization of Fixed Polarity. Specied Functions. Debatosh Debnath and Tsutomu Sasao. Kyushu Institute of Technology

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1 Exact Mnmzaton of Fxed Polarty Reed-Muller Expressons for Incompletely Speced Functons Debatosh Debnath and Tsutomu Sasao Department of Computer Scence and Electroncs Kyushu Insttute of Technology Izuka , Japan Abstract Ths paper presents an exact mnmzaton algorthm for fxed polarty Reed-Muller expressons (FPRMs) for ncompletely specfed functons. For an n-varable functon wth unspecfed mnterms there are 2 n+ dstnct FPRMs. A mnmum FPRM s one wth the fewest products. The mnmzaton algorthm s based on the mult-termnal bnary decson dagrams. Expermental results for a set of functons are shown. The algorthm can be extended to obtan exact mnmum Kronecker expressons for ncompletely specfed functons. Index Terms AND-EXOR, Reed-Muller expresson, Kronecker expresson, exact mnmzaton, ncompletely specfed functon. I. Introducton Fxed polarty Reed-Muller expresson (FPRM) s one of the canoncal AND-EXOR expressons [15]. FPRMs are a generalzaton of postve polarty Reed-Muller expressons (PPRMs). A PPRM, whch s unque for a completely speced functon, s an AND-EXOR expresson wth only uncomplemented lterals. PPRMs are also known as Zhegalkn polynomals after the Russan logcan Ivan I. Zhegalkn who rst publshed ths canoncal form [27]. Each varable n an FPRM can appear ether n complemented or uncomplemented form. An n-varable completely speced functon has 2 n dstnct FPRMs. For ncompletely speced functon, the number of FPRMs ncreases exponentally wth the ncrease n the number of unspeced mnterms: 2 n+ dstnct FPRMs exst for an n-varable functon wth unspeced mnterms. An expresson wth the fewest products s a mnmum expresson. FPRMs are mportant because they can be used to desgn easly testable crcuts [14], to detect symmetrc varables of swtchng functons [22], to desgn multlevel crcuts [23], and n Boolean matchng [24]. Moreover, for some classes of practcal functons, FPRMs requre fewer products than sum-of-products expressons (SOPs) [15{18]. For completely speced functons, numerous exact and heurstc mnmzaton algorthms for FPRMs exst [7, 10, 14, 17, 21]. However, lttle research has been done to mnmze FPRMs for ncompletely speced functons. Tran dscussed a graphcal procedure, whch s based on a tral-and-error method, to smplfy FPRMs for ncompletely speced functons [20]. The method can be applcable to functons wth up to sx varables. By usng spectral technques [9], Varma and Trachtenberg developed heurstc algorthms to smplfy PPRMs for ncompletely speced functons [25]. Chang and Falkowsk reported methods to smplfy FPRMs for ncompletely speced functons [3, 4]. Recently, Zlc and Vranesc presented a heurstc scheme to compute multple-valued Reed-Muller transform for ncompletely speced functons [28]. McKenze et al. developed a branch and bound algorthm for the exact mnmzaton of PPRMs for ncompletely speced functons [13]. Green descrbed an exhaustve search method [8]. Zakrevskj formulated the exact mnmzaton of PPRMs for ncompletely spec- ed functons as a soluton of a system of lnear logcal equatons, and presented expermental results for functons wth up to 20 speced mnterms [26]. McKenze et al. [13] and Zakrevskj [26] also consdered heurstc smplcaton methods. In ths paper we present an algorthm to obtan exact mnmum FPRMs for ncompletely speced functons. The method s based on the computaton of extended truth vector and weght vector, whch are also used for the exact mnmzaton of FPRMs for completely spec- ed functons [6, 17]. Every component of these vectors s an nteger-valued functon represented by mult-termnal bnary decson dagram (MTBDD) [5]. Kronecker expressons [6, 12, 18], ntroduced by Boul et al. [1], are a generalzaton of FPRMs. We also dscuss an extenson

2 of the FPRM mnmzaton algorthm to mnmze Kronecker expressons for ncompletely speced functons. The remander of the paper s organzed as follows: Secton II ntroduces termnology and presents basc propertes. Secton III develops an exact mnmzaton algorthm. Secton IV reports expermental results. Secton V presents concluson and outlnes future work. II. Dentons and Basc Propertes In ths paper, the operators `+' and `' ndcate arthmetc and mod-2 addton, respectvely. Denton 2.1 An n-varable swtchng functon f s a mappng f : f0,1g n!f0,1g andan n-varable ntegervalued functon g s a mappng g : f0,1g n!f0,1, :::,p ; 1g, where p 2. It should be noted that swtchng functons are a subset of nteger-valued functons. Denton 2.2 An n-varable nteger-valued P functon 2 f(x 1,x 2, :::,x n ) can be wrtten as n ;1 j=0 m jx b 1 1 xb 2 2 x b n n, where m j 2f0,1, :::,p ; 1g (p 2), b 1,b 2, :::,b n 2f0,1g such that b 1 b 2 b n s the n-bt bnary number representng j, x b =x when b =0, x b = x when b =1, and = 1,2, :::,n. Then [m 0,m 1, :::,m 2 n ;1 ]sthetruth vector of f. Example 2.1 The truth vector of the three-varable swtchng functon x 1 x 2 x 3 _ x 1 s [1,0,0,0,1,1,1,1], and that of the three-varable nteger-valued functon 3x 1 + 4x 2 x 3 +2x 3 s [2,0,2,4,5,3,5,7]. Property 2.1 Let f be a swtchng functon. Then f + f = 1. Example 2.2 Let the two-varable swtchng functon f be [1,0,0,0]. Then f + f = [1,0,0,0] + [0,1,1,1] = [1,1,1,1] = 1. Property 2.2 Let f be an nteger-valued functon. Then f + f + + f = k f: {z } k operands Example 2.3 Let the two-varable nteger-valued functon f be [1,0,3,5]. Then f + f + f =3 [1,0,3,5] = [3,0,9,15]. Denton 2.3 An n-varable swtchng functon f(x 1,x 2, :::,x n ) can be wrtten as a fxed polarty Reed-Muller P2 n ;1 expresson (FPRM) a j=0 j x b 1 1 xb 2 2 xb n n, where a j 2f0,1g, b 1,b 2, :::,b n 2f0,1g such that b 1 b 2 b n s the n-bt bnary number representng j, x b = 1 when b =0, x b 2fx,x g such that for each ether x or x appear throughout the expresson when b = 1, and = 1,2, :::,n. In an FPRM, each varable can appear ether n complemented or uncomplemented form,.e., polarty ofeach varable can be chosen n two ways. Thus, for an n-varable completely speced functon there are 2 n dstnct FPRMs. Denton 2.4 Polarty vector (b 1,b 2, :::,b n ) for an FPRM of an n-varable swtchng functon f(x 1,x 2, :::,x n ) s a bnary vector wth n elements, where b = 0 ndcates varable x s used n the uncomplemented form (x ) and b = 1 ndcates varable x s used n the complemented form (x ). Example 2.4 Let the three-varable swtchng functon f be x 1 x 3 _ x 2 x 3 and (0,1,1) be a polarty vector for an FPRM of f. Snce x 1 x 3 and x 2 x 3 are dsjont, we can wrte f = x 1 x 3 x 2 x 3. By puttng x 3 =1x 3 n the expresson for f, wehave f = x 1 (1 x 3 ) x 2 x 3 = x 1 x 1 x 3 x 2 x 3,whchs the FPRM for f wth polartyvector (0,1,1). III. Mnmzaton Technques For an n-varable completely speced swtchng functon there are 2 n dstnct FPRMs, and the mnmzaton problem s to nd a polarty vector that produces an FPRM wth mnmum number of products. On the other hand, for an n-varable ncompletely speced swtchng functon wth unspeced mnterms there are 2 n+ dstnct FPRMs, and the mnmzaton problem s to nd a polarty vector and an assgnment of the unspeced mnterms to 0's and 1's that produce an FPRM wth mnmum number of products. Once the polarty vector and the assgnment of the unspeced mnterms are determned, generaton of an FPRM s relatvely easy [6, 17]. Fg. 1 llustrates a method for the exact mnmzaton of FPRMs for three-varable swtchng functon. The method s based on the computaton of extended truth vector and weght vector [6, 17]. The extended truth vector [t 0,t 1, :::,t 26 ] s computed from the truth vector [m 0,m 1, :::,m 7 ] of a gven swtchng functon, and the weght vector [w 0,w 1, :::,w 7 ] s computed from the extended truth vector. In general, for an n-varable completely speced swtchng functon, extended truth vector s a bnary vector [t 0,t 1, :::,t 3 n ;1 ] wth 3 n elements, and weght vectorsannteger vector [w 0,w 1, :::,w 2 n ;1 ]wth 2 n elements. Each element of the weght vector s assocated wth a polarty vector, whch sshown at the rghtmost sde n Fg. 1. In general, for an n-varable swtchng functon f, polarty vector for w j s a bnary vector (b 1,b 2, :::,b n )suchthatb 1 b 2 b n s the n-bt bnary number representng j (j = 0,1, :::,2 n ; 1), and w j represents the number of products n the FPRM for f wth polarty vector (b 1,b 2, :::,b n ).

3 m 0 m 1 m 2 m 3 m 4 m 5 m 6 m 7 r 0 (= m 0 ) r 1 (= m 1 ) r 2 (= m 0 m 1 ) r 3 (= m 2 ) r 4 (= m 3 ) r 5 (= m 2 m 3 ) r 6 (= m 4 ) r 7 (= m 5 ) r 8 (= m 4 m 5 ) r 9 (= m 6 ) r 10 (= m 7 ) r 11 (= m 6 m 7 ) s 0 (= r 0 ) s 1 (= r 1 ) s 2 (= r 2 ) s 3 (= r 3 ) s 4 (= r 4 ) s 5 (= r 5 ) s 6 (= r 0 r 3 ) s 7 (= r 1 r 4 ) s 8 (= r 2 r 5 ) s 9 (= r 6 ) s 10 (= r 7 ) s 11 (= r 8 ) s 12 (= r 9 ) s 13 (= r 10 ) s 14 (= r 11 ) s 15 (= r 6 r 9 ) s 16 (= r 7 r 10 ) s 17 (= r 8 r 11 ) t 0 (= s 0 ) t 1 (= s 1 ) t 2 (= s 2 ) t 3 (= s 3 ) t 4 (= s 4 ) t 5 (= s 5 ) t 6 (= s 6 ) t 7 (= s 7 ) t 8 (= s 8 ) t 9 (= s 9 ) t 10 (= s 10 ) t 11 (= s 11 ) t 12 (= s 12 ) t 13 (= s 13 ) t 14 (= s 14 ) t 15 (= s 15 ) t 16 (= s 16 ) t 17 (= s 17 ) t 18 (= s 0 s 9 ) t 19 (= s 1 s 10 ) t 20 (= s 2 s 11 ) t 21 (= s 3 s 12 ) t 22 (= s 4 s 13 ) t 23 (= s 5 s 14 ) t 24 (= s 6 s 15 ) t 25 (= s 7 s 16 ) t 26 (= s 8 s 17 ) u 0 (= t 0 t 18 ) u 1 (= t 1 t 19 ) u 2 (= t 2 + t 20 ) u 3 (= t 3 t 21 ) u 4 (= t 4 t 22 ) u 5 (= t 5 + t 23 ) u 6 (= t 6 t 24 ) u 7 (= t 7 t 25 ) u 8 (= t 8 + t 26 ) u 9 (= t 9 + t 18 ) u 10 (= t 10 t 19 ) u 11 (= t 11 + t 20 ) u 12 (= t 12 t 21 ) u 13 (= t 13 t 22 ) u 14 (= t 14 + t 23 ) u 15 (= t 15 t 24 ) u 16 (= t 16 t 25 ) u 17 (= t 17 + t 26 ) v 0 (= u 0 u 6 ) v 1 (= u 1 u 7 ) v 2 (= u 2 + u 8 ) v 3 (= u 3 u 6 ) v 4 (= u 4 u 7 ) v 5 (= u 5 + u 8 ) v 6 (= u 9 + u 15 ) v 7 (= u 10 u 16 ) v 8 (= u 11 + u 17 ) v 9 (= u 12 + u 15 ) v 10 (= u 13 u 16 ) v 11 (= u 14 + u 17 ) w 0 (= v 0 + v 2 ) w 1 (= v 1 + v 2 ) w 2 (= v 3 + v 5 ) w 3 (= v 4 + v 5 ) w 4 (= v 6 + v 8 ) w 5 (= v 7 + v 8 ) w 6 (= v 9 + v 11 ) w 7 (= v 10 + v 11 ) (x 1 x 2 x 3 ) (0 0 0) (0 0 1) (0 1 0) (0 1 1) (1 0 0) (1 0 1) (1 1 0) (1 1 1) Fgure 1: Computaton of extended truth and weght vector for three-varable swtchng functon. For an n-varable swtchng functon wth unspeced mnterms d 1,d 2, :::,d, extended truth vector s a vector of swtchng functons t (d 1,d 2, :::,d )( = 0,1, :::,3 n ;1), and weght vector s a vector of nteger-valued functons w j (d 1,d 2, :::,d )(j = 0,1, :::,2 n ; 1). For three-varable case, all the t 's and w j 's can be obtaned from Fg. 1. Extenson to the functons wth more varables s straghtforward. Expressons for several t 's and w 0 for threevarable functon [m 0,m 1, :::,m 7 ] are shown n the followng: w 0 = t 0 + t 2 + t 6 + t 8 + t 18 + t 20 + t 24 + t 26, (3.1) where t 0 = m 0, t 2 = m 0 m 1, t 6 = m 0 m 2, t 8 = m 0 m 1 m 2 m 3, t 18 = m 0 m 4, t 20 = m 0 m 1 m 4 m 5, t 24 = m 0 m 2 m 4 m 6, t 26 = m 0 m 1 m 2 m 3 m 4 m 5 m 6 m 7 : (3.2) Denton 3.1 Let the mnmum value of the -varable nteger-valued functon w(d 1,d 2, :::,d ), denoted by w mn,bemn 02 ;1 m,where[m 0,m 1, :::,m 2 ;1 ]represents the truth vector for w. Let [w 0,w 1, :::,w 2 n ;1 ] be the weght vector for an n-varable ncompletely speced swtchng functon f(x 1,x 2, :::,x n ), and wj mn be the mnmum value for w j (d 1,d 2, :::,d )whered 1,d 2, :::,d represent unspeced mnterms of f. Let 0 k 2 n ;1anda 1,a 2, :::,a 2f0,1g such that w k (a 1,a 2, :::,a ) = mn 0j2 n ;1 wj mn. Let c 1 c 2 c n be the n-bt bnary number representng k. Then, (a 1,a 2, :::,a ) represents an assgnment of (d 1,d 2, :::,d ) and (c 1,c 2, :::,c n ) represents a polarty vector that produce a mnmum FPRM for f. Example 3.1 Consder a three-varable swtchng functon f(x 1,x 2,x 3 ) whose truth vector [m 0,m 1,m 2,m 3,m 4, m 5,m 6,m 7 ]=[d 0,0,d 2,d 3,1,1,1,0], where d 0, d 2,andd 3 are unspeced mnterms. By puttng the value of m ( = 0,1, :::,7) n (3.2), we have t 0 = d 0, t 2 = d 0, t 6 = d 0 d 2, t 8 = d 0 d 2 d 3, t 18 =1 d 0, t 20 = d 0, t 24 = d 0 d 2, t 26 =1 d 0 d 2 d 3 : (3.3) By usng Property 2.1 to (3.3), we have t 18 + t 20 = 1 and t 8 + t 26 = 1. Also, by usng Property 2.2 to (3.3), we have t 6 + t 24 =2(d 0 d 2 )andt 0 + t 2 =2d 0. Thus, from (3.1) and (3.3), we obtan w 0 =2+2d 0 +2(d 0 d 2 ): (3.4)

4 Equaton (3.4) shows that w 0 cannot be less than 2 and t s ndependent ofd 3. By nspecton, we have w 0 =2, when d 0 = d 2 =0 however w 0 = 6, when d 0 = 1 and d 2 =0. Thus, the mnmum value for w 0 s 2. Smlarly, we can obtan mnmum value for w, when = 1,2, :::,7. To manpulate nteger-valued functon we use multtermnal bnary decson dagram (MTBDD) [5]. An MTBDD, whch s a natural extenson of bnary decson dagram (BDD) [2], s a drected acyclc graph wth multple termnal nodes each ofwhch has an nteger value. Arthmetc operatons, such as addton and multplcaton, between nteger-valued functons can be ecently performed by usng MTBDDs. It should be noted that swtchng functons are a subset of nteger-valued functons and an MTBDD for a swtchng functon s a BDD. We use MTBDD data structure to perform Boolean operatons between swtchng functons. A straghtforward method to buld MTBDDs for weght vector requres excessve computaton tme and memory resources, because they represent all possble FPRMs for the gven ncompletely speced functon. However, we are only nterested n an FPRM wth the fewest products. Suppose we have an FPRM for the gven functon wth t threshold + 1 products, then t s sucent to search for an FPRM wth t threshold or fewer products. If suchan FPRM does not exst then the FPRM wth t threshold +1 products s the mnmum FPRM. Thus, to restrct the search space wthout sacrcng the mnmalty of the soluton, we use threshold value, t threshold, durng constructon of MTBDDs. The threshold value can be obtaned by usng any smplcaton program for FPRMs. Based on the above dscussons, we develop the followng algorthm for exact mnmzaton of FPRM for ncompletely speced n-varable swtchng functon f: Algorthm 3.1 (Exact mnmzaton) 1. Get the user suppled threshold value, t threshold. 2. Prepare extended truth vector [t 0,t 1, :::,t 3 n ;1 ] for f. (Fg. 1 shows the computaton method for extended truth vector for three-varable functons. Extenson to the functons wth more varables s straghtforward. Each element of the extended truth vector s a swtchng functon represented as an MTBDD.) 3. Let [w 0,w 1, :::,w 2 n ;1 ] be the weghtvector for f. For =0to2 n ; 1, do the followng: (a) Gather elements from the extended truth vector such that w = X t2t t, (3.5) where T ft 0,t 1, :::,t 3 n ;1 g. (Fg. 1 shows how to gather elements correspondng to w from the 1 procedure Construct MTBDD(a,S,D all ) f 2 D reman, D ths : set of support varables 3 S save, S ths : subset of S 4 c save, c ths :nteger (counter) 5 w a 6 untl S 6= do f 7 D reman D all ; support(w ) 8 c save 0 9 for each d reman 2 D reman do f 10 D ths fd reman g[support(w ) 11 c ths 0 S ths 12 for each b j u j 2 S do f 13 f support(u j ) D ths then f 14 c ths c ths + b j 15 S ths S ths [fb j u j g g g 18 f c ths >c save then f 19 c save c ths 20 S save S ths g g 23 f S save = then 24 S save an element ofs 25 for each b j u j 2 S save do 26 w w + b j u j /*MTBDDoperaton*/ g S S ; S save 29 return w /* MTBDD */ 30g Fgure 2: Pseudocode Construct MTBDD. extended truth vector for three-varable functon. Extenson to the functons wth more varables s straghtforward. It s obvous from Fg. 1 that the number of elements n T s 2 n.) (b) Apply Propertes 2.1 and 2.2 to (3.5). Thus, we have w = a + X 0jL;1 b j u j, where a 0, b j 1, u j 2 T, and L 2 n. (c) Construct an MTBDD for w. Durng constructon () f any termnal value of an ntermedate MTBDD s greater than t threshold,set that termnal value to 1 ()fanntermedate MTBDD represent constant 1, stop the constructon and assgn MTBDD for w to 1. (d) Obtan mnmum value w mn from the MTBDD for w. (Ths corresponds to ndng a path to a termnal node of the MTBDD that produces a mnmum value for w.) (e) If w mn <= t threshold : () save the polarty vector and an assgnment of the unspec-

5 ed mnterms correspondng to w mn () t threshold w mn ; If any polarty vector s saved n step 3(e) then obtan an FPRM by usng the most recently saved polarty vector and assgnment of the unspeced mnterms, otherwse report \No soluton exsts wth t threshold or fewer products." To buld an MTBDD for w at step 3(c) we must do arthmetc addton of a set of MTBDDs. The MTBDDs can be arranged n numerous ways to perform addton. The arrangement nuences the computaton tme and the sze of the ntermedate MTBDDs durng addton. Anave arrangement requres excessve memory resources and long computaton tme. To buld MTBDD for w we use procedure Construct MTBDD(a,S,D all ), the pseudocode of whch s shown n Fg. 2, where S represents fb 0 u 0,b 1 u 1, :::,b L;1 u L;1 g, D all represents fd 1,d 2, :::,d g, and support(w ) represents the set of varables on whch w depends. The procedure rst chose an MTBDD that depends on the fewest varables and then arranged other MTBDDs to slowly ncrease the number of varables n the ntermedate MTBDDs. The mnmzaton method presented n ths secton can be adapted to obtan exact mnmum Kronecker expressons [1, 6, 15] for ncompletely speced swtchng functons. In ths case we must compute extended weght vector [1, 6] nstead of weght vector. IV. Expermental Results We mplemented Algorthm 3.1 n C by usng CUDD package [19] and conducted experments by usng a set of swtchng functons. The detal expermental results are shown n Table 1. The factors on whch the computaton tme manly depends are the threshold value, the number of varables n the functon, and the number of unspeced mnterms. The current mplementaton works favorably for many functons wth eght orfewer varables and wth any number of unspeced mnterms. However, for functons wth nne or more varables t often requres excessve CPU tme and memory resources when the number of unspeced mnterms s more than 30. A comparson wth the other algorthms s dcult, because none of them explctly reported the benchmark functons. V. Conclusons and Comments Exact mnmzaton of FPRMs for ncompletely speced swtchng functons s a computatonally hard problem, Table 1: Expermental results. f(n,t,d,s) Prod y Thre z Peak x Tme { f(6,15,30,25) f(6,12,40,50) f(7,35,50,5) f(7,20,80,5) f(7,20,90,5) f(8,25,200,50) f(8,100,80,10) f(8,35,180,10) f(8,60,160,5) f(8,80,100,50) f(9,250,50,5) f(9,15,480,80) f(10,500,40,25) f(12,2000,30,25) f(14,8000,30,50) f(n,t,d,s): Functon (dened n Appendx). y Prod: Number of products n mnmum FPRM. z Thre: Threshold value. x Peak: Peak number of lve MTBDD nodes. { Tme: CPU seconds on SUN ULTRA 10. because the search space ncreases exponentally wth the ncrease n the unspeced mnterms. Although Algorthm 3.1 requres only one MTBDD at a tme, the present mplementaton of t uses a xed varable order for all the MTBDDs. Currently,we are workng to nd a better varable order for ndvdual MTBDDs. Ths strategy would be useful to solve larger problems wth less memory resources. Acknowledgement Ths work was supported n part by a Postdoctoral Fellowshp of the Japan Socety for the Promoton of Scence and n part by a Grant-n-Ad for the Scentc Research of the Mnstry of Educaton, Scence, Culture, and Sports of Japan. The authors thank anonymous referees for gvng constructve suggestons. References [1] G. Boul, M. Davo, and J.-P. Deschamps, \Mnmzaton of rng-sum expansons of Boolean functons," Phlps Res. Repts., Vol. 28, pp. 17{36, 1973.

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