Multi-Valued Sub-function Encoding in Functional Decomposition Based on Information Relationships Measures

Transcription

1 Mult-Valued Sub-functon Encodng n Functonal Decomposton Based on Informaton Relatonshps Measures $UWXU&KRMQD NLDQG/H K-y(ZLDN Endhoven Unversty of Technology, Faculty of Electrcal Engneerng P.O. Box 513, EH 10.25, MB 5600 MB Endhoven, The Netherlands Abstract Functonal decomposton s becomng more and more popular, because t s more general than all other known logc synthess approaches and t seems to be the most effectve approach for LUT-based FPGAs, (CPLDs and complex CMOS-gates. The mult-level functonal decomposton can be seen as a recursve splttng of a gven functon, nto two sub-functons: the predecessor (bound-set functon and successor functon. Intally, the bound set functon s a mult-valued (symbolc functon, where a certan value (symbol s assgned to each partcular nput-cube compatblty class of the functon beng decomposed. To be mplemented wth bnary logc, the mult-valued bound-set functon must be expressed as a set of bnary functons. Ths transformaton s called the mult-valued sub-functon encodng. It can be performed by the bnary code assgnment to each partcular nput-cube compatblty class. It determnes the resultng bnary predecessor and successor sub-functons and therefore nfluences the qualty of the resultng crcut to a hgh degree. In ths paper, a new method of the mult-valued sub-functon encodng s presented. The method s based on the nformaton relatonshp measures. Expermental results from the prototype CAD-tool that mplements the method demonstrate that t s able to effcently construct extremely effectve crcuts for symmetrc functons. Results for asymmetrc functons are also very good. 1. Introducton The opportuntes created by modern mcroelectronc technology cannot be fully exploted, because of weaknesses of the tradtonal logc synthess methods appled n most today s commercal CAD tools. Partcularly n the case of (CPLDs, look-up table (LUT based FPGAs and complex CMOS-gates, the constrants are mposed not on the functon type a certan logc block can mplement, but on varous logc block s structural parameters (e.g. the number of nputs, outputs or product terms n a block and on nterconnectons between the blocks. A block s able to mplement any functon wth lmted dmensons. On the other hand, the tradtonal logc synthess methods are based on some mnmal functonally complete systems of logc gates mplementng only few very specfc functons (e.g. AND+OR+NOT. They requre a post synthess technology mappng for another mplementaton structures. If the actual synthess target strongly dffers from a gven mnmal system, any technology mappng cannot guarantee a good result, because the ntal synthess s performed wthout close relaton to the actual target. Therefore, there s presently much research n the feld of general (functonal decomposton of combnatonal crcuts and sequental machnes for VLSI synthess [2-3][6][8-11][13][15-19][21-24]. Functonal decomposton conssts of breakng down a complex system of dscrete functons or relatons nto a network of smaller and relatvely ndependent coherent sub-functons (sub-relatons, n such a way that the orgnal system s behavor s preserved, some constrants are satsfed and some obectves are optmzed. A subfuncton (sub-relaton n the network can be any sort of functon (relaton satsfyng certan specfc structural constrants. A Boolean functon wth at most k nputs s called a k- feasble functon. When all sub-functons n the network are k-feasble, the network s k-feasble and t can be drectly mapped nto LUT based FPGAs where each Confgurable Logc Block (CLB can mplement any functon up to k nputs. In practcal FPGAs, k = 4, 5 or 6. In a sngle decomposton step of the functonal decomposton, a gven functon f s splt nto two subfunctons: a predecessor (bound-set functon g and a successor functon h (see Fg. 1a. The k-feasble multlevel sub-functon network that mplements a certan functon f s constructed by repeatng the sngle decomposton steps n relaton to the orgnal and resultng sub-functons untl all sub-functons n the network become k-feasble.

2 ... a b Ψ 1...Ψ x m 1 x q x q+1 x p x p+1 x n Y 1 Y k g 1...g r Μ 1... Μ k h Θ f f Fgure 1 a Non-dsont functonal decomposton b General functonal decomposton The functonal decomposton approach has been proposed by Ashenhurst n [1] and extended by Roth and Karp n [20] and Curts n [4] (Fg. 1.a. In [8] -y(zldn presented the general decomposton scheme (Fg. 1.b and the related general decomposton theory and methodology for sequental and combnatonal crcuts [5]. All known decomposton schemes for dscrete functons and sequental machnes are some specal cases of the scheme presented n [8]. Most functonal decomposton approaches are based on recursve top-down reducton. In contrast, the bottomup constructon method based on the general decomposton theory was proposed n [23]. Based on the general decomposton theory [8], theory of nformaton relatonshps and measures [9][11][17][19], and bottom-up approach to the multple-level functonal decomposton [23], we developed a new method for the multple-level general decomposton of the multple-output ncompletely specfed Boolean functons, and we mplemented t n a prototype CAD tool. The paper ams to dscuss a new approach to the mult-valued bound-set symbolc sub-functon encodng used n our decomposton method. In partcular, we wll demonstrate how the nformaton relatonshps and relatonshp measures are appled n the process of code assgnment. 2. Representaton of nformaton n dscrete nformaton systems Let s consder a certan fnte set of elements S called symbols. Informaton about symbols (elements of S s the ablty to dstngush certan symbols from some other symbols. In Table 1 the truth table of a mult-output Boolean functon s shown. To each row of the table (product term of the functon we assgned a unque label (symbol. Informaton s represented n dscrete systems by values of some sgnals or varables. For nstance varable x 1 nduces through ts two values 0 and 1 two compatblty classes on the symbols: B 0 ={0,2,3,4} and B 1 ={1,2,3,5}. x 1 has value 0 (1 for each symbol n class B 0 (B 1 (nput don t care - means: 0 and 1. Snce symbols 4 and 5 are not placed together n any x 1 x n compatblty class nduced by x 1, x 1 s able to dstngush between these partcular two symbols (x 1 =0 for 4 and x 1 =1 for 5. x 1 s not able to dstngush symbol 0 from 2, 3, and 4 (values of x 1 for symbols 0 and 4 are the same, and for symbols 0, 2 and 3 are Table 1. 3-nput, 2-output Boolean functon S x 1x 2x 3 f 1f compatble, because don t care - s compatble wth both 0 and 1. In such a way nformaton s modeled wth set systems [2][5][8][9][10]. The set system algebra s presented n [5]. Here only the nformaton necessary to explan the encodng problem and ts soluton s recalled. A set system SS on a set S s defned as a collecton of subsets B 1, B 2,, B k of S (called blocks of SS such that: B = S B B and for. The product of set systems s defned as follows: ' ' ' ' SS 1 SS2 = { B B= B1 B2 B B1 B2 B= B1 B2 } B1 SS1 B2 SS2 ' ' B1 SS1 B2 SS2 The blocks of the product of SS 1 and SS 2 result from ntersecton of each block of SS 1 wth each block of SS 2. The blocks resultng from the ntersecton that are contaned n some other resultng blocks are then removed. Partal order operator: SS 1 s smaller than or equal to SS 2, SS 1 SS 2, f and only f each block of SS 1 s ncluded n a block of SS 2. Example 1. Let SS 1 = { 0,1,2;3,4 } and SS 2 = { 0,1,2; 2,3,4}. SS 1 SS 2 = { 0,1,2;3,4 }, SS 1 SS 2 holds, whle SS 2 SS 1 does not hold. A set system SS on S can be nterpreted as a compatblty relaton defned on S, wth the compatblty classes beng the blocks of SS. A certan set system SS gves nformaton about the elements of S wth lmted precson to the compatblty class. The set system product can be nterpreted as a product of the correspondng relatons; t represents combned nformaton about the elements of S that s provded by the relatons together. The partal orderng relaton denotes the fact that f SS 1 SS 2, then SS 1 provdes the same or the same and more nformaton about elements of S as SS 2. An elementary nformaton descrbes the ablty to dstngush a certan sngle symbol s from another sngle symbol s, where: s,s S and s s. Any set of such atomc portons of nformaton can be represented by an nformaton set IS defned on S S as follows [9][11]: IS ={{s, s } s dstngushed from s by the modeled nformaton}. Each bnary or mult-valued varable x of a dscrete functon (relaton nduces a set system π x on the functon (relaton terms. Wth each set system π x the

3 correspondng nformaton set IS(π x s assocated. In ths way, wth each (bnary or mult-valued varable x, a correspondng nformaton set IS(x s assocated. Relatonshps between varables can now be analyzed by consderng relatonshps between ther correspondng nformaton sets. Example 2. The correspondng set systems and nformaton sets for all nput and output varables of the functon from Table 1 are shown below: π f1 = { 0,1,2,3;4,5 }, π f2 = { 0,1;2,3,4,5}, π x1 = { 0,2,3,4;1,2,3,5}, π x2 = { 0,2,5;1,3,4}, π x3 = { 0,3,5;1,2,4}, IS(π f1 = { }, IS(π f2 = { }, IS(π x1 = { }, IS(π x2 = { }, IS(π x3 = { }. 3. Informaton relatonshp measures When performng analyss or desgn of nformaton systems, we often ask for relatonshps between nformaton n varous nformaton streams [9][11]. In partcular we are nterested n nformaton smlarty of two or more nformaton streams. Informaton smlarty (affnty measure ISIM: ISIM(SS 1, SS 2 = IS(SS 1 IS(SS 2. In [9][11] some relatve measures are defned, by normalzng the absolute measures, and weghted measures, by assocatng an approprate mportance weght w(s s wth each elementary nformaton. In partcular, the Weghted Informaton Smlarty Measure of two set systems π 1 and π 2 s defned as follows: WISIMπ, π = w ( s, ( 1 2 s IS( π1 IS( π2 where w(s s s a weghtng functon. Many varous weghtng functons can be defned dependent on specfc obectves. In our method, the weghtng functon s used to model the decrease of the nformaton mportance wth the number of the (bnary or mult-valued nput or ntermedate varables x of a certan functon f at whch ths nformaton s present. Buldng our heurstc weghtng functon, we made the followng assumptons: 1. w(s s =0 f nformaton s s s not requred for the computaton of a gven functon f, 2. w(s s =1 f nformaton s s s necessary for the computaton of a gven functon f, and s s s avalable only once, 3. the sum of weghts of the less mportant nformaton can not domnate the weght of the more mportant nformaton. Frst, the occurrence of an elementary nformaton s s requred for computaton of a functon f n the nformaton set of each nput varable x s checked usng the followng formula: 1: f ( s CI( f, x o( s = IS( x 0 : f ( s CI( f, x Next, the occurrence multplcty of an elementary nformaton s s n the set of nformaton sets s computed: m( s = o( s IS(x IS( x We call s s a unque nformaton n respect to IS(f and when t s provded by only one bnary or multvalued varable from,.e. f and only f m ( s = 1. To ensure that the sum of weghts of the less mportant nformaton does not domnate the weght of the more mportant nformaton, we buld the normalzaton functon n the followng way. The formula 1: f m( s = k km(( s, k = 0 : f m( s k dvdes the elementary nformaton tems nto classes of equal multplcty (k-multplcty. The normalzaton functon s defned by h (0 0, h (1 0, h( k = = = h( k 1 + ( s s IS( f km((s s,k and expresses the normalzaton coeffcents dependent on the number of elementary nformaton tems n partcular k-multplcty classes. Fnally, we formulate the weghtng functon as follows: w( s IS(f = 2 IS(f m( s 0 : f m( s 1: f m( s 2 h( m( s IS(f IS(f IS(f = 0 = 1 : otherwse Example 3. The nformaton smlarty measure (ISIM of the nput varable x 1 and output functon f 1 from Table 1 s as follows: ISIM(π x1, π g1 = {0 5, 1 4,} = 2 Occurrence multplcty of m(s s n respect to IS(f 1 and = {{IS(x } = 1,2,3} (see Table 1 s as follows: s s

4 The normalzaton functon h(k n respect to IS(f 1, and = {{IS(x } = 1,2,3} s as follows k h(k The weghtng functon w(s s n respect to IS(f 1, and = {{IS(x } = 1,2,3} s s WSIM(x 1,f 1 = 2, WSIM(x 2,f 1 = 2.5, WSIM(x 3,f 1 = The encodng problem To smplfy the explanatons of ths paper the classcal form of the functonal decomposton (Fg. 1a s used. The encodng problem and ts proposed soluton method are the same n the general decomposton (Fg. 1b. Let Y = {y = 1 n} be a set of Boolean output varables of a multple-output Boolean functon f and π Y = π y be the set system nduced by these varables on the set of the functon s terms. Let π U = π x, where x U, be a set system nduced by the bound-set varables (the block s g nput varables. Let π V = π x, (x V be a set system nduced by the free-set varables (the nput varables to block h, beng not the output varables from block g, see Fg. 1a. Theorem 1. Exstence of seral decomposton [2][8][21] If there exsts a set system π g on f, such that π U π g and π g π V π Y, then the functon f has a seral functonal decomposton wth respect to U, V. In the multlevel decomposton, the selecton of the nput support U for block g, computaton of the product set system π U of the bound-set varables, constructon of the set system π g and ts encodng are repettvely appled to functons f, g and h untl a k-feasble network s constructed. The product set system of the bound-set varables π U and the block s g output set system π g of defne together the block s g mult-valued functon G:π u π g, where each partcular value B g of ths functon corresponds to a block of the set system π g. Thus, the number of values of the functon s equal to the number of blocks of π g. In ths work we focus on the bnary code assgnment to the blocks of π g. The precse explanaton on how to translate the functon G nto a set of bnary functons when the codes are already assgned s presented n [2]. For the encodng of b values of π g, mnmum l= log 2 b bnary varables are requred. Therefore, l determnes the mnmal code length and the mnmal number of the bnary functons g (=1 l representng G. The bnary code assgnment mplctly defnes a set of l two-block set systems π g ( =1 l (one for each bnary output varable. Block B 0 of the π g s the unon of the blocks of π g, that have value 0 at -th poston of the assgned code. Block B 1 s the unon of blocks that have value 1 at -th poston of the assgned code B4 Example 4. Let blocks of π { B ; B ; B ; } have the g = codes assgned as shown above the dashes. The table below shows the blocks of π g and correspondng codes. Snce the codes of blocks B 1 and B 2 have value 0 at poston 1, B 1 and B 2 wll be placed n B 0 = B 1 B 2 of π g 1. Snce the codes of blocks B 3 and B 4 have value 1 at poston 1, B 3 and B 4 wll be placed n B 1 = B 3 B 4 of π g 1. Analogously, blocks B 1 and B 4 wll be placed n B 0 = B 1 B 4 of π 2 g and blocks B 2 and B 3 wll be placed n B 1 = B 2 B 3 of π 2 g. The two resultng two-block set systems defned by ths encodng are as follows: 0 1 g B4 Code Block Poston 1 2 B B B B g B3 π = { B B ; B } π = { B B ; B }. The code length l together wth the sze of the free set V determne the number of nputs to the block h. There s a common convcton that n most cases the smaller s the number of outputs of the block g, the less complex and easer decomposable resultng network s. In the remanng cases, the decompostons wth more outputs from block g than mnmum usually do not result n much smpler networks. Therefore, usually the codes wth mnmal length are used for bnary encodng of π g, as they maxmally reduce the support of the block h, and the number of block s g functons. In the work reported n ths paper we also use the mnmal length codes. In [15] Murga et al. have notced that the problem of determnng sub-functons g and h can be consdered as an nput-output encodng problem, because functons g are outputs of the block g and nputs to the block h (Fg. 1. From the vewpont of the encodng type, one can sub-dvde the encodng algorthms nto the followng classes: 1. output-encodng: dedcated to functon g smplfcaton [6], 2. nput encodng: dedcated to functon h smplfcaton [14][15], 3. nput-output encodng: dedcated to concurrent smplfcaton of functons g and h [3][21]. Also dfferent technques of code assgnment to compatble classes were used n prevous works: 1. strct encodng (unque code assgned to each block of π g [6][13], 2. non-strct encodng (two or more codes can be assgned to a block of π g [4][24].

5 Moreover, dfferent crtera for evaluaton of the encodng qualty were proposed: 1. maxmzaton of the number of the output don t cares n functon g [3][21], 2. support mnmzaton of functon g [6][24]. 3. mnmzaton of the number of compatblty classes n functon h [7]. We propose below a new nput-output strct boundset functon encodng method wth a new qualty crteron based on nformaton relatonshp measures. Informaton that s necessary for computng values of a certan output of a consdered mult-output functon f s dstrbuted on a number of ts nputs x. The nputs also delver some nformaton that s not needed for the output, and nformaton on the nputs s represented otherwse than on the consdered output. Each block of the multlevel functonal decomposton representng a certan subfuncton g or M (Fg. 1 can be therefore consdered as a block where an ntermedate nformaton transformaton s performed whch nvolves an approprate combnaton, abstracton and re-structurng of the nput nformaton. Ths transformaton s defned by constructon of an approprate set system π g from a set system π U nduced by the set of selected bound-set varables U and encodng of the set system π g. The set system π g should carry a part of nformaton delvered by the set system π U, whch n combnaton wth nformaton delvered by the set system π V nduced by the varables from the free-set, s essental to compute the requred output nformaton. Snce the number of block s g outputs should be n practcal cases much smaller than the number of ts nputs, π g must have much less blocks than π U. π g s created by mergng the blocks of π U and a lot of nformaton s removed durng ths mergng. To preserve the behavor of the functon beng decomposed, the transformaton must preserve the unque nformaton,.e. the nformaton that s represented n only one nformaton set. The transformaton process should also try to preserve the almost unque nformaton (.e. delvered by only few nput or ntermedate varables, because ths prepares better condtons for the followng decomposton steps, by makng the almost unque nformaton avalable at more ntermedate varables. It should abstract from the nformaton that s not requred for computaton of the functon beng decomposed and nformaton that s delvered by all or many nput or ntermedate varables. The encodng procedure substtutes the block s g mult-valued functon by a set of bnary functons. It s possble to buld such a set of bnary functons that the number of unque or almost unque nformaton tems n ths set s lower than n the orgnal mult-valued functon. In result, the functon h wll tend to be easer decomposable (the nput encodng approach. Ths s acheved by repeatng the unque or almost unque nformaton provded by the block s g nput varables on two or more block s g bnary output varables. Snce n ths way the nformaton wll be present at more nput varables of h, t wll be made non-unque. Although for smplfcaton of the blocks g that are not k-feasble more aspects are mportant. If the block s g bnary output varables of block g provde less unque nformaton tems then they provde more common nformaton. Thus, t wll be easer to buld some common sub-functons for the block s g bnary functons. Therefore, we formulated the bound-set functon encodng problem as follows: Fnd such mnmal length assgnment of bnary codes to blocks of the set system π g that the number of unque or almost unque elementary nformaton tems delvered by the block s g bnary output varables s mnmal. 5. Encodng algorthm The constructon process of the set system π g s based on mergng some blocks of the set system π U. Please note that even f partcular nformaton s provded by several nput varables, t may become a unque nformaton f t s provded only by the bound-set varables and the multvalued varable correspondng to π g s the only varable provdng ths nformaton. As t was shown n Example 4, the bound-set functon encodng mplctly defnes a set of two-block set systems. These set systems are nduced by the bnary code assgnment to each block of π g. In general, mergng the blocks of a set system reduces the amount of nformaton provded by ths set system. Let us defne the cost of mergng of any two blocks of π g as the sum of weghts of ncompatblty pars removed by the mergng: bmc( B1, B2 = w( s, where w(s s s the s B1 s B2 ( s IS( πu weghtng functon defned n Secton 3. The block mergng cost functon bmc descrbes how many and how mportant (unque, almost unque, etc. tems of elementary nformaton wll be lost f we merge blocks B 1 and B 2 together. Hammng dstance hd of two bnary vectors s the number of the correspondng postons at whch these vectors dffer. For example for c 1 =00111 and c 2 =10110, hd(c 1, c 2 =2. In order to decrease the number of unque and almost unque elementary nformaton tems n the set of nformaton sets IS(π g nduced by the bnary encoded varables correspondng to π g, the Hammng dstance between the codes assgned to the pars of blocks wth the hgh mergng cost bmc should be hgh. If the values of two codes assgned to two blocks of π g dffer at a certan poston, then these two blocks are placed n dfferent blocks of π g and the nformaton nduced by these blocks of π g s avalable n π g also. When codes dffer at several postons, the nformaton s avalable at each bnary

6 Table 2 Boolean functon 1 of 4 S x 1x 2x 3x 4 y varable correspondng to the code poston at whch they dffer. The more dfferent postons n codes assgned to two blocks of π g, (hgher Hammng dstance the more bnary varables π g provde nformaton nduced by these two blocks. In ths way we ntroduce multplcaton of the unque and almost unque nformaton n the set of π g. The above stated encodng problem s an optmzaton problem that can be solved n many dfferent ways. We propose a fast greedy encodng algorthm executed nsde a beam search. The beam search selects beam most promsng search drectons for the encodng algorthm, and the encodng algorthm constructs encodngs n these drectons n a greedy way. Frst, we look for the ntal beam pars of blocks of π g havng the maxmum mergng cost bmc. Parameter beam s set by the user and lmts the search space. These pars are the startng ponts for the greedy encodng algorthm that s terated beam tmes. The encodng results from the greedy algorthm are evaluated, and the soluton wth the lowest cost s selected. The greedy algorthm assgns the codes wth maxmum Hammng dstance to the ntal par of blocks. The assgned codes are removed from the avalable codes, because they must not be used n next steps. Then, the algorthm looks for the next par (B 1, B 2 of blocks wth the maxmum mergng cost untl all blocks are encoded. If B 1 (B 2 s already encoded the avalable code wth the maxmum Hammng dstance to code of B 1 (B 2 s selected and assgned to B 2 (B 1. Example 5. Table 2 presents a 4-nput symmetrc Boolean functon. The bound-set U={x 1, x 4 } s selected and the block s g output set system π g = {1,2,4,6; 5,6,7,8,9,10; 0,3,5,6,7,8,10} from ts product set system π U s constructed. Three unque code assgnments exst. Others are some permutatons of π g and/or nversons one of π g (e.g. the assgnment B 1 11, B 2 00 and B 3 01 s equvalent to assgnment B 1 00, B 2 11 and B 3 01 (column 1 after nverson of each bt and permutaton of π 1 g and π 2 g. π 1 2 g π g π 1 2 g π g π 1 2 g π g B B B The block mergng costs for these blocks are as follows: B B Bmc(B,B B 1 B 2 The cost of each assgnment n terms of a number of the unque elementary nformaton tems s shown n Table 3. The assgnment from column 3 s selected, because ts cost s mnmal. Table 3 Costs of the code assgnment n Example 5 π 1 2 g π g π 1 2 g π g π 1 2 g π g B B B cost Expermental results To better analyze the nfluence of the symbolc functon encodng we decreased the nfluence of the bound-set selecton procedure on the results of experments, by usng symmetrc Boolean functons as test cases. Functons of ths class are nvarant under any permutaton of prmary nput varables, and therefore, the bound set selecton from the prmary nput set s a trval problem for symmetrc functons. Although our decomposton method and tool are general and do not use any a pror nformaton on the functon type, t turned out that they are extremely effectve for symmetrc functons [10]. In Fg. 2a through 2c varous decompostons (2- feasble networks are presented for the 1 of 4 functon. They dffer n encodng of the mult-valued output functons of the decomposton blocks. From all possble encodng combnatons only one produces the optmal crcut from Fg. 2a. Ths s the combnaton constructed by our decomposton method based on the nformaton relatonshp measures. All other combnatons produce worse crcuts: wth more logc blocks, more levels and more nterconnectons. From these results we can conclude that the bnary representaton (encodng of the multple-valued functons of the decomposton blocks has a bg nfluence on the qualty of the resultng multlevel network. Usng our prototype tool that mplements our bottomup multple-level decomposton method, we performed a number of experments wth more than 30 varous symmetrc functons. For all checked functons our method effcently constructed the extremely effectve mult-level networks. In Fg. 3 through 5 some of the mult-level networks found by our decomposton tool are presented for a number of symmetrc functons. The realzaton constructed by our tool for functon 9sym (Fg. 3 s better than found by the specfc method for symmetrc functons presented n [12]. Please note that the networks shown n Fg. 3 can be mapped nto 5 CLBs of

7 XC3000 and XC4000 FPGAs, whle ths reported n Table 4 requres 6 CLBs of these types. In Table 4, our bottom-up decomposton tool based on the nformaton relatonshp measures (column lsynth s v v v v v v v v v v v v v v v v v v v 13 v 10 v 12 v v 9 v 13 v 18 v 9 v 13 v v v v v v v 10 v 12 v v 9 v 13 v 18 v 9 v 12 v 15 v 10 v 13 v 17 v 15 v 18 v 20 v 15 v 18 v 21 v v 29 v v 28 v v v v v 15 v 17 v v o Fgure 3 Realzatons of the symmetrc functon 3,4,5,6 of 9 (9sym from [25] a v 15 v 18 v 20 b v 17 v 21 v v 20 v 13 v 23 v 20 Fgure 2 Dfferent realzatons of the symmetrc Boolean functon, 1 of 4 c Table 4 Comparson of 5-LUT counts compared to the top-down decomposer IMODEC [24], by applyng both approaches to a set of MCNC benchmark functons [25]. IMODEC s known as one of the best functonal decomposers. Our tool constructs always better or equally good decompostons as IMODEC for symmetrc functons. However, for most asymmetrc functons t was also able to fnd better or equal qualty decompostons. 7. Concluson In ths paper a new formulaton of the mult-valued sub-functon encodng problem has been proposed and a novel approach to soluton of ths problem. Ths approach consderably dffers from other approaches used n multple-level functonal decomposton. It s based on nformaton relatonshp measures and uses a novel cost functon for the qualty evaluaton. We appled ths approach n our prototype decomposton tool based on the theores of general functonal decomposton and nformaton relatonshp measures. The expermental results demonstrate that our tool s able to effcently buld extremely effectve networks for mult-level mplementatons of symmetrc Boolean functons. They also show that the encodng of the multvalued bound-set functons has a bg nfluence on the qualty of the fnal network n the terms of number of LUTs and number of levels n the network. Results for asymmetrc Boolean functons are also very good. They are comparable to the results from IMODEC [24] that s known as one of the best functonal decomposers. 8. References # #o IMODEC lsynth tme LUTs LUTs/levels mm:ss 5xp /2 0:32 9sym /3 1:57 alu /5 6:31 b /3 2:10 clp /3 17:58 count /3 0:24 duke /5 14:26 f51m /3 1:17 msex /2 0:05 msex /3 0:16 rd /2 0:36 rd /2 2:31 sao /3 0:52 z4ml /2 0:17 [1] R.L. Ashenhurst: The decomposton of swtchng functons, Proceedngs Internatonal Symposum on the Theory of Swtchng Functons, p , Aprl 1959.

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