PIECEWISE LINERIZATION OF LOGIC FUNCTIONS *

Transcription

1 PIECEWISE LINERIZATION OF LOGIC FUNCTIONS * Ilya Levn, Osnat Keren, George Kolotov, Mar Karpovsy 3 Tel Avv Unversty, Ramat Avv, Tel Avv 69978, Israel, levn@computerorg Bar Ilan Unversty, Ramat Gan, Ramat Gan 59, Israel, ereno@engbuacl 3 Boston Unversty, 8 Sant Mary's Street, Boston, MA 5, marar@buedu ABSTRACT The paper deals wth a problem of lnearzaton of mult-output logc functons Specfcally, the case s dscussed, when the functons have a large number of varables and cannot be effcently lnearzed by usng nown technques For soluton of the problem a socalled pecewse lnearzaton s proposed The pecewse lnearzaton comprses decomposton of an ntal mult-output functon nto a networ of components, followed by ndependent lnearzaton of the components The decomposton s based on the theory of D -polynomals descrbed n the paper The resultng pecewse lnearzed networ s drectly mapable onto a specal type of a bnary graph called Parallel Mult Termnal BDD An effcent heurstc algorthm for the pecewse lnearzaton s provded The presented benchmars results demonstrate hgh effcency of the proposed method n comparson wth nown lnearzaton approaches The results also show that ntegratng lnearzaton technques wth the descrbed decomposton, thus obtanng the pecewse lnearzaton, are a very promsng both from practcal and from the theoretcal ponts of vew INTRODUCTION The use of the Lnear Decomposton as tool for optmzaton n logc synthess, partcularly, n Bnary Decson Dagrams (BDDs optmzaton s promsng drecton of research n the feld of logc desgn [] BDDs are a standard part of many CAD systems n logc desgn, sgnal processng, and other areas where effcent, n terms of space and tme, manpulaton of BDD representaton for a gven functon s usually estmated by the number of non-termnal nodes n the BDD The sze of a BDD s very senstve to the order of varables, rangng from the polynomal to the exponental complexty for the same functon for dfferent orders of varables Therefore, the maorty of approaches to the BDD sze reducton are related to development of effcent algorthms for reorderng of varables see for example, [], [3] Lnearly transformed BDDs are defned by allowng lnear combnatons of the varables [4] The nown lnearzaton technques use representaton of logc functons n a form of the truth table The present paper deals wth mult-output logc functons defned by ther mplcant table Moreover, we deal where the case when the mplcant table s sparse (contans a large number of don t cares, and the functon s defned by the correspondng mplcant table depends on the large number of varables In such a case the use of nown lnearzaton technques becomes neffcent or even napplcable due to the great sze of the correspondng truth table The combnaton of the lnearzaton and decomposton was used for mplementaton of planar BDDs [5] and for lnearzaton of functons defned n sum-ofproduct form [6] The present paper develops the dea of pecewse lnearzaton whch s a functonal decomposton followed by a lnearzaton procedure The paper proposes an algorthm for constructng the pecewse lnear mplementaton of an arbtrary logc functon defned by ts sum-of-products (SOP The theoretcal foundaton of our decomposton method s the algebra of D -polynomals [7] The result of the decomposton s a drect mappng of the logc functon onto a specal type of a bnary graph called Parallel Mult Termnal BDD (PMTBDD that conssts of the specfcally connected component MTBDD The decomposton approach descrbed n ths paper ams at practcal mplementatons of logc functons as specfc VLSI structures The man purposes of the proposed technque are to mnmze the sze of the resultng VLSI mplementaton by representng the ntal mult-output functon as a networ of component MTBDDs Specfcally, the above tas, named pecewse lnearzaton, can be stated as follows: Gven a mnmzed sum-of-product representaton of a multple-output logc functon, construct a networ of component MTBDDs of the mnmal sze by applyng lnearzaton of components The paper s organzed as follows Secton descrbes the general structure of PMTBDDs Secton 3 ntroduces the noton of D -polynomals Secton 4 descrbes the algorthm of decomposton, followed by the lnear transformaton of components Expermental results and the correspondng dscusson are provded n Secton 6 Conclusons are presented n Secton 5 * The wor was supported by BSF under Grant 59

2 PARALLEL MULTI TERMINAL BDDS The lnear decomposton of mult-output functons of large number of nput varables, defned by a large set of cubes of a hgh order, s a sgnfcantly hard problem due to ts complexty Decomposng of an ntal functon nto a networ of components followed by ther ndependent lnearzaton can assst n overcomng the problem of complexty for the above cases The proposed decomposton approach s based on a new concept of parallel mult-termnal bnary decson dagram (PMTBDD comprsng component MTBDDs The PMTBDD s constructed by combnng component MTBDDs usng the parallel and seres operatons: Seres connecton: replacng one termnal node of an MTBDD wth another MTBDD Parallel connecton: connectng roots of two or more MTBDDs Introducng the PMTBDD opens a way for handlng mult-output logc functons of a large number of varables, defned by ther SOP (mplcant table wth a large number of cubes of a hgh order 3 D -POLNOMIALS Ths secton ntroduces a noton of D -polynomal whch provdes a convenent mathematcal model for the manpulaton of MTBDD We show that the parallel and seres connectons of the MTBDD can be modeled as a product and substtuton of D - polynomals representng those MTBDDs 3 Representaton of Logc Functons Consder an n-nput, m-output completely specfed n m Boolean functon F : X Z, where X {,,* } and Z {, } Let F be ntally represented n mnmzed (prme and rredundant sum-of-product (SOP form, where each output Z s wrtten as a logcal sum (OR of product terms (mplcants: Z ( I( I ( denotes an ndex set of mplcants assocated wth the output Z Implcants can be shared between dfferent outputs Notce that s are functons of nput varables ( x, x,, x n We wll also refer to them as the -functons Let be the name assocated wth the output Z One can also thn of as an operator whch has to be performed when the correspondng output Z evaluates to wll denote a dummy output functon (or an empty operator whch does not produce any output Defnton D -polynomal s a polynomal defned over a set of operators D Z ( whle the coeffcents Z satsfyng the condtons: a Z (completeness, b Z Z, (orthogonalty Tang ( nto account, we have: ( ( I I( D + + where denotes th -mplcant of functon Z In ths wor we are nterested n a class of D - polynomals defned over a subset of varables whose coeffcents are mplcants of a logc functon Such D -polynomals are used to represent logc functons Coeffcents are defned explctly as the correspondng product terms of the functon, whle s defned mplctly as a complement of (to satsfy the completeness condton D -bnomal s a specal case of D -polynomal, wth exactly two dsont (orthogonal mplcants, D We dstngush between -functons belongng to dfferent D -polynomals by labelng them wth a super-scrpt ndex assocated wth the correspondng polynomal; wll ndcate that mplcant s assocated wth the D -polynomal D The same mplcant can be assocated wth dfferent polynomals, so that for arbtrary values of and Example The followng s a D -polynomal: D x + xx Here, x, xx and xx The correspondng MTBDD may be constructed n straghtforward way and can be acheved by repeatedly applyng the Shannon expanson to D Notce that the smplcty of the above transformaton s based of the followng specfc property of the D : for each applcaton of the Shannon expanson, at least one nput varable s present n all the mplcants Conceptually, a D -polynomal D can be nterpreted as follows If evaluates to, then D If all of the explct functons are equal to, then D whch means that no output s produced (or an empty operator s to be performed Let us defne a product of two D -polynomals Defnton Let D I(, and D I( The product of and D, denoted as D D s defned by: D D ( { }, l D

3 over each par of terms from D and D, ncludng the mplct terms and Here l s a logc product (AND of the correspondng - functons and s a combnaton of the respectve operators In other words: when evaluates to, both and are computed l concurrently Lemma An arbtrary D -polynomal always be represented as follows: where ( + Z, and D Z Z D can (3 Proof We wll demonstrate that the product of ( Z s equal to the D -polynomal wth the same coeffcents Frst, notce that by defnton all explct functons are parwse orthogonal, so that Z Z, for Furthermore, orthogonalty of the functons and the completeness condton that must be satsfed by each D -bnomal mply that Z,, for Hence Z Z Also, notce that the concatenaton { } means that both operators and need to be performed smultaneously Snce s a dummy operator, only has to be computed; Fnally, orthogonalty and completeness condtons of -functons mply that,, &,,, so that, Therefore, the subsequent multplcaton of the consecutve terms of expresson (3 yelds Z Z { } + Z { } + Z { } { } hence { } ( m ( m ( Z + Z ( Z m ( Z + Z + + Z mm ( Z + Z + + Zmm Z QED Theorem An arbtrary D -polynomal D can always be represented as a product of D -bnomals: D Z ( +, I ( (4 where, and After performng the multplcaton we have: D { } { } { } Proof Let D ( ( m + m ( m ( ( Based on ths, every logc functon Z of arbtrary D -polynomal D may be presented as a product of bnomals as follows: Z + ( ( ( + I I (5 Substtutng (5 nto (3 results n (4 QED An mportant concluson from ths theorem s that the product of D -polynomals can be always presented as a product of the correspondng termnal bnomals Subsequently, the termnal bnomals can be multpled to obtan hgher level D -polynomals Obvously, there are several ways to group termnal bnomals to form a D -polynomal Dfferent groupng of termnal bnomals yelds dfferent PMTBDD s, resultng n dfferent mplementatons of Boolean functons Ths fact forms the bass of our decomposton approach 3 Decomposton of D -polynomals We construct an MTBDD correspondng to a system of D -polynomals Dfferent groupngs of the termnal bnomals lead to dfferent mplementatons of the logc functon represented by ths system It defnes a way to decompose the logc functon by manpulatng the system of D -polynomals representng the functon The startng pont of our method s the specfcaton of a logc functon n the form of an mplcant table An ntal system of D -polynomals can be easly derved from the mplcant table by assocatng a sngle D -bnomal wth each product term of the table The functon can then be represented as a product of the D -bnomals Theorem A multple-output Boolean functon can always be mplemented as a product of D - polynomals Proof Represent the mplcant table of the logc functon as a product of D -bnomals and multply the orthogonal bnomals to create the consttuent D - polynomals Concluson: MTBDD correspondng to such an mplementaton can be realzed as a parallel connecton of subtrees correspondng to the ndvdual D -polynomals 4 PARALLEL DECOMPOSITION The am of the proposed parallel decomposton s creatng the PMTBDD for an arbtrary mult-output functon The proposed decomposton algorthm s based on partton of the set of product terms representng the ON-set of the functon nto a set of logc blocs It s followed by a herarchcal

4 decomposton of blocs nto a common header and a set of bloc fragments Ths s accomplshed by extractng a set of common factors (so-called prefxes from the subset of product terms of the orgnal mplcant table Defnton 4: A product term or a part of the product term, s called a prefx A set of all prefxes defnes a bloc header A subset of product terms wth a common prefx s called a bloc A set of all blocs s referred to as a bloc set Defnton 5: The set of terms ncludng the gven prefx precsely s called ts famly The remanng set of product terms, not ncluded n the bloc set, s called a remander Defnton 6: The set of all the rows of the mplcant table that do not belong to any of the famles s called the remander of the current stage A set of product terms obtaned by extractng a common prefx from all the members of the bloc wll form a bloc fragment or tal Defnton 7: The prefx s famly, after all the prefx s varables are set to don t care, s called ts tal Header s a fragment (subset of rows and columns of the mplcant table composed of the prefx varables The header s selected n such a way as to provde mnmzaton of the resultng PMTBDD We propose to select the header by tang nto account the followng underlyng prncple ncrease the percentage of non-don t care cells (densty of the correspondng fragment the mplcant table ensure the effcency lnearzaton of the correspondng bloc By constructon, the bloc header s a logc functon whose ON-set s a superset of the ON-sets of the logc functons assocated wth the ndvdual blocs It wll be mplemented as an MTBDD whose nternal nodes are assocated wth the prefx varables The termnal nodes of the tree represent the bloc fragments, each to be mplemented as a separate MTBDD The target of the proposed decomposton algorthm s to mnmze the total sze of the dagram The algorthm dvdes the ntal functon nto a bloc Bs and a remander R s The bloc s a sum of products of smpler sub-functons wth prefx terms The group of prefxes p the dense fragment s, chosen for the BDD mplementaton forms the header of the bloc and s, ndeed, mplemented as a MTBDD, wth the sub-functons playng the role of the termnals Each of the sub-functons and the remander functon R s can be repeatedly decomposed n the same way, untl no further decomposton s possble An mportant feature of the proposed method s ts ablty to beneft from any other optmzaton method the user may wsh to employ Sftng, K -Procedure, etc [8, ] These wll be appled to the component MTBDDs, and further reduce the total dagram s sze PMTBDD wth lnearzed blocs s denoted LPMTBDD 4 Decomposton Algorthm The followng algorthm detals the flow of a partcular teraton L Empty lst of pars {Prefx, Tal} Let R Set of prodct terms, vector of nteger outputs Let B Basc Prefx, T F ts famly, T T T F \B ts tal L [L, (B, T T ] T U R \ (T F U T R C {t: t R and t B } T R {t: t R\(C T F } C COMMON_PARTS(C Whle C >, Let S Secondary Prefx T F ts famly T T T F \S ts tal L [L, (S, T T ] T O {t: t C and t S } T NO {t: t C\T O } C T O T R T R T NO End Whle Classfy and enumerate the tals n L Construct the Bloc s BDD from L L T {l: l L, Tal(l s trval} L NT {l: l L\L T } If the remander s trval, L T [L T, Remander] Else L NT [L NT, Remander] End If For all the lst elements n L T Implement the tal End For For all the lst elements n L NT Recursvely call ths procedure End For The man part of a partcular teraton conssts of choosng the prefxes for the bloc The prefxes are chosen one by one Each tme a prefx s chosen, all the prefxes not orthogonal to t and belongng to dfferent Z -functons are moved to the remander Non-orthogonal prefxes belongng to the same functon as the prefx n the bloc are ncluded n the bloc Ths contnues untl no more sutable prefxes are present Thereafter the teraton proceeds to enumerate the tals and constructs the bloc s MTBDD The non-trval tals and remander serve as nputs to the next teratons The trval tals are mplemented mmedately as separate conuncton BDDs and do not requre addtonal teratons Example : The mlcant table used n the example s presented n Table Table : The mlcant table for Example # X X X X3 X4 F F F F3 3 4

5 BDD, LTBDD, PMTBDD and LPMTBDD for the example are presented n Fgures -4 correspondngly Fgure Straghtforward mplementaton of the MTBDD for the Example Fgure Lnearly Transformed MTBDD for the Example Fgure 3 PMTBDD correspondng to the Example Fgure 4 Lnearly Transformed PMTBDD for the Example Termnal nodes of MTBDDs n Fgures -4 are mared by decmal numbers of correspondng outputs A standard mplementaton of the exemplary MTBDD, as an ordered MTBDD, s presented n Fgure a Fgure b shows an mplementaton based on the proposed decomposton approach, n a form of PMTBDD The PMTBDD comprses two portons the bloc (left and the remander (rght The portons are assembled by the newly ntroduced parallel connecton of MTBDDs Notce that accordng to the product operaton ntroduced n Defnton, sets of termnal nodes n the PMTBDD and n the standard MTBDD are not the same It s the result of the concatenaton operaton between the orgnal termnal nodes The concatenaton s calculated as the OR functon between correspondng output vectors (see Defnton For example, termnal node 7 n the MTBDD (Fgure corresponds to two termnal nodes 3 and 6 n the PMTBDD (Fgure 3 Such cases reflect the non-dsont property, as n cubes and from Table The proposed approach, presented by Example, allows achevng sgnfcant mprovement: ndeed, the standard MTBDD has 7 non-termnal nodes (NTNs, after lnearzaton ths s reduced to NTNs (and XOR gates PMTBDD has 8 NTNs, lnearzaton further reduces t to 6 NTNs (and XOR gates 4 Choosng the Prefxes In ths secton the process of choosng the sutable bloc header of the teraton s dscussed The requrements for the frst (basc prefx and all other (secondary prefxes are gven and the crtera the choosng them are suggested The choce s performed by gradng all the potental canddates, computng a weghted average of the dfferent grades for each canddate, and then selectng the canddate wth the hghest grade The followng notatons are used: P the prefx under consderaton R the set of rows of the mplcant table For a selected prefx: T F the famly of the prefx T R the set of the terms that don t depend on any of the prefx s varables TU R\ ( TF TR the set of terms dependng on some of the prefx s varables It s called the undecded set, snce these terms are nether the prefx s famly nor ts remander T the set of terms orthogonal to the prefx T T M U the number of varables n the prefx N T F For any set A of terms: X A x : t A,t x " " In other words, ( { ( } ( A X s the set of varables n all the terms of A S ( A { x : t( x " " } In other words, ( A t A number of lterals n all the terms of A S s the

6 ( A sup, where S t s the row n the Sums S t { t A} matrx of the mplcant table correspondng tot In other words, ( A s the set of outputs correspondng to all the terms of A 4 Lst of Canddates The prefx can be ether a complete row of the Products matrx of the mplcant table, or a subset of lterals common to several terms A straghtforward procedure s proposed below for constructng the lst of the canddates from the lst of terms of the mplcant table Let x, y be varables wth values from {,, } The operator ψ ( x,y compares the two Boolean varables and returns the value of one of them f they are equal, don t care otherwse: ψ ( x,y xi( x y + " don' t care" I ( x y (5 The functon Common( T, T accepts the two terms T and T and apples ψ n a btwse manner to each of the varables n the set X ( T X ( T Fnally, the followng method summarzes the suggested procedure Constructng the Lst of Canddates INPUT: Lst of prodct terms OUTPUT: Lst of canddates Output Input Temp Output REPEAT Temp Empty set Apply Common( to every par of terms from Temp If the result s not empty, add t to Temp Output Output U Temp Temp Temp UNTIL Temp 4 Choosng the Basc Prefx The basc prefx s the foundaton of a bloc It s chosen so as to mae the bloc header the most sutable for a BDD mplementaton For ths, the basc prefx has to attract the secondary prefxes close to t and repel those far from t There are three man concerns to consder here: the nput varables, the output functons and the length of the prefx In addton, snce the secondary prefxes wll be chosen from the set T O, t s mperatve to measure the self-orthogonalty of T U The frst crteron answers the Inputs requrement: X ( TF X ( TR (6 X ( X ( TF \ X ( P X ( TR It counts the varables common to the tal and the remander correspondng to the prefx The rato has to be reduced as much as possble, n order to separate the bloc (wth ts tals from the remander Ths crteron has values n [, ] nterval, correspondng to the case when all the remander's varables are present n the tal and to the opposte The second crteron answers the Outputs requrement: ( TF ( TR (7 ( TF ( TR It counts the outputs common to the tal and the remander correspondng to the prefx The ratonale here s the same as for the Inputs requrement The thrd crteron, called Prefx Area, measures the percentage of the area covered by the prefx wthn ts famly MN P (8 S( TF The reasonng s smple: the longer the basc prefx, the longer the lst of canddates for the secondary prefxes The fnal crteron, called Orthogonalty answers the addtonal requrement Sb ( TO T (9 Sb ( TU It counts the number of lterals n the terms orthogonal to the prefx relatve to the number of lterals n all the canddates 43 Choosng the Secondary Prefx In selectng the basc prefx t s mportant to establsh a sold foundaton for the bloc For secondary prefxes the goal s dfferent, and the requrements change accordngly The prefxes already chosen nto the bloc have to be taen nto account In the followng equatons, the superscrpt ndces and + stand for current stuaton and after addng the consdered prefx, respectvely The frst crteron, called Addtonal Inputs, counts the number of varables common to the tal and to the remander of the consdered prefx, but only those not htherto present n the bloc + + X ( TF X ( TR X ( TF X ( TR ( β X X ( T \ X ( P X T ( ( F The second crteron, called Addtonal Outputs, counts the number of output functons common to the tal and to the remander of the consdered prefx Here, le n the prevous crteron, only the newly added outputs are consdered + + ( TF ( TR ( TF ( TR β ( + + ( TF ( TR The thrd crteron, called Addtonal Area, measures the addtonal area brought to the bloc and to the remander by selectng the consdered prefx + + S ( TF S ( TF S ( TR S ( TR β S ( + + S ( TF S ( TR Ths equaton can be rewrtten as follows: S ( T F S ( T R S ( TR S ( TF (3 βs S ( TF S ( TR S ( TR S ( TF Each of the two fractons s lmted to the nterval [, ], but the total value of β S s n the nterval [, ] In R

7 ths t dffers from all the other crtera and runs somewhat the elegance of the total, but does not bear any serous mpact on the results 44 Combnng the Crtera The four crtera and the three crtera for the basc prefx are combned n the followng weghted average ax X + a + ap P + at T (4 β b β + b β + b β X X S S When choosng the basc prefx, the canddate wth the hghest s taen Lewse, when choosng the secondary prefx, the canddate wth the hghest β s taen The coeffcents of the crtera have to be chosen so as to get the optmal result 5 EXPERIMENTS The experments demonstrate that the proposed decomposton, when successful, greatly reduces the sze of the BDD Its success strongly depends on the densty of the mplcant table Therefore, ts effectveness can be predcted qute relably by mang some prelmnary study of the mplcant table functons' representaton The goals of the conducted experments are as follows: Comparng the effectveness of the complete lnearzaton (performed wth K -Procedure and that of the Parallel decomposton Investgatng the relatve mportance of the coeffcents and formulatng the gudelnes concernng the weghts Durng the experments the mplcant table representatons of the standard combnatoral-crcut benchmars (LGSNTH93 were used The experments were lmted to the followng subset of benchmars: Number of terms the mplcant table s lmted by Ths lmtaton may seem somewhat severe, and, ndeed, the Parallel decomposton can be performed n certan cases for mplcant tables of over 5 terms In general, however, snce the search for common parts grows polynomally n the number of nput terms, t s suggested that ths value s ept below Number of nput varables lmted by 3 Ths s a necessary condton for runnng the Exhaustve K - Procedure n ts tabular form Number of output functons lmted by 3 Ths s the weaest of all requrements and s dctated by the treatment of the outputs as ntegers Both the Parallel decomposton and the K -Procedure based lnearzaton of resultng blocs were checed on standard combnatoral-crcut benchmars (LGSNTH93 For each benchmar, the szes of the followng dagrams were recorded and compared: orgnal BDD; LTBDD; PMTBDD; and LPMTBDD The results are shown n the Tables and 3 The frst lsts the benchmars for whch the Parallel decomposton s more effectve than the lnearzaton of the ntal mplcant table wthout decomposton The second shows the falures The columns n the tables are as follows For each benchmar, the number of the nput varables and the mplcant table densty are followed by the four columns of the results The two addtonal columns show the mprovement/degradaton ratos: from BDD to PMTBDD, wth and wthout the lnearzaton The ratos are gven n per cents Both tables are sorted by the ascendng the mplcant table densty Table Benchmars results, LPMTBDD < LTBDD Ttle X Denst y BDD LTBDD PBDD LTPBDD PBDD / BDD LTPBDD / LTBDD ALU B DK DK CON ALU DUKE ALU MISEX3C WIM F5M DK APLA INC MEAN RATIO Table 3 Benchmars results, LPMTBDD > LTBDD Ttle X Densty BDD LTBDD PBDD LTPBDD PBDD / BDD LTPBDD / LTBDD ADD RADD CLIP Z ROOT SQR SQN MLP SAO DIST BW RD MEAN RATIO The analyss of these results shows that the densty of the mplcant table s, ndeed, a relable ndcator of the success of the functon s Parallelzaton The successful cases ( PMTBDD < MTBDD and/or LTPBDD < LTBDD are mostly n the low-densty area (Densty up to 45% and the unsuccessful ones are mostly n the hgh-densty area (Densty at least 6% The mddle or gray area functons (Densty wthn 4-6% are dvded more or less evenly between the successes and the falures Moreover, there are several examples where the hgh-densty functons are successfully decomposed, and no examples where the method faled to wor on lowdensty functons The Parallel decomposton, on the other hand, reles upon extractng dense fragments from the gven mplcant table, and treatng the sparse remanders and tals separately Therefore, sparse mplcant table can be easly dealt wth by splttng them nto a networ of concurrently worng BDDs Wth dense mplcant tables, choosng sutable blocs s dffcult, and arbtrary choces lead to neffectve mplementatons Ths s summarzed n Table 4

8 Table 4 Analyss of the results Method PLA BDD Parallelzaton Densty Low Hgh Lots of don t cares Few don t cares Lots of replcatons Number of replcatons small BDD not compact BDD compact Several component Dense PLA dffcult to determne fragments Easy choce of separate fragments component dagrams Unsutable for PBDD Sutable for PBDD 6 CONCLUSIONS The paper descrbes a new method of a so-called pecewse lnearzaton of logc functons, whch comprses a decomposng the ntal mplcant table of a logcal functon nto a networ of component Mult Termnal BDDs, and b separate lnearzaton of these BDDs Two connectng operatons are used n the decomposton networ, e, seral and parallel connectons of the component BDDs A new mathematcal bass of the parallel connecton of the BDDs s formed by a newly ntroduced algebra of D - polynomals Ths new algebra, as well as a number of theoretcal statements proven n the paper are used as a theoretcal bacground of the proposed decomposton algorthm The decomposton algorthm s presented n detals Benchmar results demonstrate effcency of the proposed approach n comparson wth straghtforward mplementatons of MTBDDs (ncludng or not ncludng lnearzaton solutons wth lnearzaton and wthout lnearzaton The proposed pecewse lnearzaton opens a way for usng the lnearzaton technque to mplement logc functons havng a great number of varables 7 REFERENCES [] MGKarpovsy, RStancovc, J Aastola, "Reducton of Szes of Decson Dagrams by Autocorrelaton Functons", IEEE Trans on Computers, May, 3, pp59-67 [] M Futa, Kumoto, R K Brayton, BDD mnmzaton by truth table permutaton, Proc Int Symp on Crcuts and Systems, ISCAS 96, May -5, 996, Vol 4, [3] Rudell, R, Dynamc varable orderng for ordered bnary decson dagrams, Proc IEEE Conf Computer Aded Desgn, Santa Clara, CA, 993, 4-47 [4] W Gunther, R Drechsler, Lnear transformatons and exact mnmzaton of BDDs, Proc 8th Great lae Symp on VLSI, February 9-, 998, [5] Levn I, Stanovc R, Karpovsy M, Astola J, (5 "Constructon of Planar BDDs by Usng Lnearzaton and Decomposton" Proceedngs of Fourteenth Internatonal Worshop on Logc and Synthess, Lae Arrowhead, Calforna, pp 3-39 [6] Keren O, Levn I, "Lnearzaton of the Logc Functons Defned n SOP Form" Procs of the Wor n Progress Sesson, DSD 5, Porto (Portugal, September 5, E Grosspetsch, K Klocner (eds, SEA-Publcatons: SEA- SR-9, July 5 [7] Levn, I, Levt, V (998 "Controlware for Learnng wth Moble Robots Computer Scence Educaton", 8(3, 8-96 [8] Menel, C Somenz, F Theobald, T "Lnear Sftng of Decson Dagrams and ts Applcaton n Synthess" IEEE Transacton on Computer Aded Desgn of Integrated Crcuts and Systems, Vol 9; part 5, pages 5-533