Representations of Elementary Functions Using Binary Moment Diagrams

Transcription

1 Representatons of Elementary Functons Usng Bnary Moment Dagrams Tsutomu Sasao Department of Computer Scence and Electroncs, Kyushu Insttute of Technology Izua , Japan Shnobu Nagayama Department of Computer Engneerng, Hroshma Cty Unversty Hroshma , Japan Abstract Ths paper consders representatons for elementary functons such as polynomal, trgonometrc, logarthmc, square root, and recprocal functons These real valued functons are converted nto nteger functons by usng fxed-pont representaton, and they are represented by usng bnary moment dagrams (BMDs Elementary functons are represented compactly by applyng the arthmetc transform to the functons For polynomal functons, upper bounds on the numbers of nodes n BMDs and multtermnal bnary decson dagrams (MTBDDs are derved These results show that for polynomal functons, BMDs requre fewer nodes than MTBDDs Expermental result for 6-bt precson sn(x functon shows that the BMD requres only 2% of the nodes for the MTBDD Introducton Bnary decson dagram (BDD [, ] s the most popular decson dagram, and s extensvely used n logc synthess, verfcaton, logc smulaton, etc BDDs can represent many practcal logc functons compactly, but cannot represent the multpler functon wth reasonable sze To represent such functons compactly, the arthmetc transform decson dagram (ACDD [6], the bnary moment dagram (BMD [2], the multplcatve BMD (*BMD [2], the Kronecer *BMD (K*BMD [7], and the Taylor expanson dagram (TED [3] have been proposed ACDD and BMD represent a gven nteger functon usng the arthmetc transform expanson, whle the mult-termnal BDD (MTBDD [4] represents the functon usng the Shannon expanson Decson dagrams based on the arthmetc transform represent nteger functons, such as adder and multpler, wth polynomal szes of the number of nput varables [2, 6] In ths paper, elementary functons [3] are converted nto nteger functons, and they are represented by BMDs Theoretcal and expermental results show that BMDs represent elementary functons compactly 2 Prelmnares Defnton Let B = {,}, P = {,,, p } where p 2, and R be the set of real numbers An n-nput m- output logc functon (or multple-output logc functon s a mappng: B n B m, an nteger functon s B n P, and a real functon s R R Defnton 2 The bnary fxed-pont representaton of a value r has the form d n nt d n nt 2 d d d n f rac, ( where d {,}, n nt s the number of bts for the nteger part, and n f rac s the number of bts for the fractonal part of r The representaton n ( s two s complement, and so r = 2 n nt d n nt + n nt 2 = n f rac 2 d In ths paper, when we consder only non-negatve number r, ts fxed-pont representaton excludes sgn-bt Defnton 3 Precson s the total number of bts for a bnary fxed-pont representaton Specally, n-bt precson specfes that n bts are used to represent the number; that s, n = n nt + n f rac In ths paper, an n-bt precson functon f (X means that the nput varable X s n-bt precson By fxed-pont representaton, we can convert n-bt precson real valued functons nto n-nput m-output logc functons The multple-output functons can be converted nto nteger functons by consderng m-bt bnary vectors as ntegers That s, we can convert n-bt precson real valued functons nto nteger functons: B n P, where P = {,,,2 m } In ths paper, we convert elementary functons nto nteger functons by usng n-bt fxed-pont representaton Real valued functons n ths paper are converted nto nteger functons, unless stated otherwse

2 Defnton 4 Let A and B be (n n square matrces, where a a 2 a n a 2 a 22 a 2n A = a n a n2 a nn The Kronecer product of A and B s defned as the followng (n 2 n 2 matrx: a B a 2 B a n B a 2 B a 22 B a 2n B A B = a n B a n2 B a nn B 3 Arthmetc Transform Ths secton descrbes the arthmetc transform and the arthmetc spectrum For detals, see [6] Defnton 5 Let A(n be the arthmetc transform matrx defned by no [ ] A(n = A(, A( =, where the addton and the multplcaton are done n nteger For an nteger functon f gven by the functon-vector F, the arthmetc spectrum A f = [a,a,,a 2 n ] t s A f = A(nF Each a n the spectrum s called the arthmetc coeffcent Example Consder the nteger functon f (x,x 2 = x + x 2 The functon-vector s F = [,,,2] t The arthmetc spectrum s A f = A(2F = 2 = Defnton 6 Let A (n be the nverse arthmetc transform matrx defned by no [ ] A (n = A (, A ( = The matrx A (n has the same form as the Reed-Muller transform matrx R(n Thus, t s also called the nteger Reed-Muller matrx Defnton 7 In a symbolc representaton, A ( = [ x ] Thus, the nverse arthmetc transform s defned as no f = X a A f, X a = [ x ] x 2 S x x S f x 2 x 2 S x 2 2 (a MTBDD for f A A f x x 2 (b BMD for f Fgure MTBDD and BMD for f = x + x 2 Example 2 From the nverse arthmetc transform and the arthmetc spectrum obtaned n Example, the nteger functon f s represented as follows: f = X a A f = [ x 2 x x x 2 ] = x + x 2 Defnton 8 Usng A ( and A(, an nteger functon f s represented as follows: [ f = A (A(F = [ x ] [ = [ x ] ][ ] f f ] f = f f f + x ( f f, (2 where f = f (x =, f = f (x = (2 s the arthmetc transform expanson (also called A-expanson or moment decomposton The arthmetc expresson for f s obtaned by the arthmetc transform expanson The arthmetc coeffcents correspond to coeffcents of the arthmetc expresson for f 4 BMD (Bnary Moment Dagram Defnton 9 A bnary moment tree (BMT s obtaned by applyng the arthmetc transform expanson f = f + x ( f f to a gven nteger functon f recursvely A bnary moment dagram (BMD s derved from the BMT by usng the followng reducton rules: Share equvalent sub-graphs 2 If the outgong edge of a node v labeled wth x ponts to the constant zero, then delete the node v and connect the edges pontng v to the other outgong edge of v drectly Example 3 Fg (a shows the MTBDD for the nteger functon f n Example Fg (b shows the BMD for f In the fgures, the nodes labeled wth S denote the Shannon expanson, whle the nodes labeled wth A denote the arthmetc transform expanson

3 Termnal nodes n a BMD represent the arthmetc spectrum A f for a functon f, whle termnal nodes n an MTBDD represent the functon-vector F of f Thus, the number of termnal nodes n a BMD s equal to the number of dstnct arthmetc coeffcents On the other hand, n an MTBDD, t s equal to the number of dstnct functon values For X and th-order polynomal functons, we can compute the numbers of non-zero arthmetc coeffcents and nodes n BMDs from the values of precson n and polynomal order The followng lemma and theorem gve the number of non-termnal nodes n the BMD for the n-bt precson functon f (X = X, and the upper bound on the number of nodes n a BMD for an n-bt precson th-order polynomal functon Lemma For the n-bt precson functon f (X = X, the number of non-zero arthmetc coeffcents s (Proof See Appendx Lemma 2 For an n-bt precson th-order polynomal functon f (X = c + c X + c 2 X c X, the number of non-zero arthmetc coeffcents s at most (Proof See Appendx = ( n Lemma 3 For an n-bt precson th-order polynomal functon f (X = c + c X + c 2 X c X, when c > and X, the number of non-zero arthmetc coeffcents s (Proof See Appendx = ( n Lemma 4 For the n-bt precson functon f (X = X, the number of non-termnal nodes n the BMD s α(n, = (Proof See Appendx In [7], smlar problem has been consdered But t shows only an upper bound, and s not tght On the other hand, Lemma 4 gves the exact number Lemma 5 For an n-bt precson th-order polynomal functon f (X = c + c X + c 2 X c X, the number of non-termnal nodes n the BMD s at most ( n Number of nodes e Polynomal order BMD MTBDD Fgure 2 Number of nodes n BMDs and MTB- DDs for 6-bt precson th-order polynomal functons (Proof See Appendx Theorem For an n-bt precson th-order polynomal functon f (X = c + c X + c 2 X c X, the total number of non-termnal nodes and termnal nodes n the BMD s at most 2 (Proof See Appendx = Lemma 6 For an n-bt precson functon f (X, f f (X s an njecton, e, the relaton α β f (α f (β holds on any α and β, then the number of nodes n the MTBDD for f (X s 2 n+ (Proof See Appendx Corollary For an n-bt precson th-order polynomal functon f (X = c + c X + c 2 X c X, the number of nodes n the MTBDD for f (X s at most 2 n+ When c > and X, the number of nodes n the MTBDD for f (X s 2 n+ Example 4 Fg 2 compares the upper bounds on the number of nodes n BMDs and MTBDDs for 6-bt precson th-order polynomal functons Fg 3 compares the upper bounds on the number of nodes n BMDs and MTB- DDs for n-bt precson 3rd-order polynomal functons When the precson n s fxed, the upper bound on the number of nodes n BMD for th-order polynomal functon ncreases wth On the other hand, n an MTBDD, the upper bound on the nodes s 2 n+ ndependently of Thus, when s small, the upper bound on nodes n a BMD s smaller than that n an MTBDD

4 Number of nodes e Precson n BMD MTBDD Fgure 3 Number of nodes n BMDs and MTB- DDs for n-bt precson 3rd-order polynomal functons Table Numbers of non-zero arthmetc coeffcents for randomly generated n-bt precson 4th-order polynomal functons Precson Number of arthmetc coeffcents n Upper bound Non-zero Dstnct Functon values are not rounded (e error-free, and have more bts than n When the polynomal order s fxed, the upper bound on the number of nodes n BMD for n-bt precson polynomal functon ncreases more slowly than that n MTBDD wth n Furthermore, Corollary shows that the upper bound for the MTBDD s tght when all the coeffcents c are postve Thus, when n s large, BMDs requre many fewer nodes than MTBDDs From the above observatons, we can see that for n-bt precson th-order polynomal functons, BMDs requre fewer nodes than MTBDDs when s smaller than n Usually, the polynomal order s smaller than precson n Thus, for practcal polynomal functons, BMDs are more compact than MTBDDs 5 Expermental Results 5 Number of Arthmetc Coeffcents Polynomal Functons: To verfy the tghtness of the upper bound n Lemma 2, we randomly generated n-bt precson 4th-order polynomal functons f (X = c 4 X 4 +c 3 X 3 + Table 2 Numbers of dstnct arthmetc coeffcents for 6-bt precson elementary functons Elementary Number of dstnct values Rato functons Functon Coeffcents [%] 2 x / x ln(x log 2 (x x /(x sn(x Functon values are rounded to 6-bt precson Doman of the functons s x < Rato = Coeffcents / Functon values c 2 X 2 + c X + c where X (e, we generated 5 unform random numbers for coeffcents, where each coeffcent has 6-bt precson: c 2 5 Table compares the number of non-zero arthmetc coeffcents for f wth the upper bound In Table, the columns labeled wth Non-zero and Upper bound show the numbers of non-zero arthmetc coeffcents for f (X and ther upper bounds gven by Lemma 2, respectvely The column Dstnct shows the number of dstnct arthmetc coeffcents for f (X For each precson n, we randomly generated polynomal functons For all of the generated functons, the numbers of non-zero arthmetc coeffcents are equal to the upper bounds Ths fact verfes the theoretcal result (Lemma 2 Table shows that for polynomal functons, many arthmetc coeffcents are, and many non-zero coeffcents have dentcal values as well Non-polynomal Functons: In addton to the polynomal functons, we represented the non-polynomal elementary functons shown n Table 2 Table 2 compares the numbers of dstnct functon values and dstnct arthmetc coeffcents for 6-bt precson elementary functons For each elementary functon, ts doman s x < and the functon values are rounded to 6-bt precson Table 2 shows that the elementary functons are transformed nto the compact arthmetc spectrum For x+ 77 and x +, the numbers of dstnct functon values are smaller than other functons, snce ther range s smaller than the others On the other hand, the numbers of dstnct arthmetc coeffcents are much smaller than that of functon values ndependently of the range of functons The numbers of dstnct functon values and dstnct arthmetc coeffcents correspond to the numbers of termnal nodes n the MTBDD and the BMD, respectvely Thus, BMDs requre many fewer termnal nodes than MTBDDs to represent the elementary functons 52 Number of Nodes n BMD Polynomal Functons: Table 3 compares the numbers of nodes n BMDs and n MTBDDs for randomly gener-

5 Table 3 Numbers of nodes n BMDs and MTB- DDs for randomly generated n-bt precson 4th-order polynomal functons n MTBDD BMD Rato Bound #Nodes Bound #Nodes [%] Functon values are not rounded (e error-free Rato = (BMD nodes / MTBDD nodes Table 4 Numbers of nodes n BMDs and MTBDDs for 6-bt precson elementary functons Elementary Number of nodes Rato functons MTBDD BMD [%] 2 x / x ln(x log 2 (x x /(x sn(x Functon values are 6-bt precson Doman of the functons s x < Rato = (BMD / MTBDD ated n-bt precson 4th-order polynomal functons f (X In Table 3, the columns labeled wth MTBDD and BMD show the numbers of nodes n MTBDDs and BMDs for f (X, respectvely The sub-columns #Nodes and Bound show the number of nodes for f (X and ts upper bound, respectvely The upper bounds for BMD and MTBDD are derved by Theorem and Corollary, respectvely As shown n Table, for the polynomal functons, many arthmetc coeffcents are, and many non-zero coeffcents have dentcal values Thus, the numbers of nodes n BMDs for the polynomal functons are smaller than the upper bounds n Theorem by the reducton rule for BMDs On the other hand, n MTBDDs for the polynomal functons, the numbers of nodes are 2 n+, snce for all the generated polynomal functons, Lemma 6 holds Non-polynomal Functons: Table 4 compares the numbers of nodes n BMDs wth that n MTBDDs for 6-bt precson elementary functons As shown n Table 2, for the elementary functons, BMDs requre many fewer termnal nodes than MTBDDs Thus, the total numbers of nodes n BMDs are smaller than that of MTBDDs Especally, for an mportant elementary functon sn(x, the BMD requres only 2% of the nodes for the MTBDD 6 Concluson and Comments Ths paper consdered BMD and MTBDD representatons for elementary functons such as polynomal, trgonometrc, logarthmc, square root, and recprocal functons We derved the number of nodes n BMDs for th-order polynomal functons, and confrmed that the BMDs requre fewer nodes than the MTBDDs Especally, when the precson n s larger than the polynomal order, BMDs requre many fewer nodes than MTBDDs Expermental result usng 6-bt precson sn(x functon shows that the BMD requres only 2% of the nodes for the MTBDD Ths paper showed that the arthmetc transforms represent elementary functons compactly We conjecture that other decson dagrams based on the arthmetc transform (eg, ACDD, *BMD, K*BMD also represent elementary functons effcently In the past, many people consder the arthmetc transforms [5, 6, 8, 9,, 2, 4, 7, 8, 9] However, to the best of the authors nowledge, ths paper frst consdered representatons of elementary functons by arthmetc transform Snce BMDs represent elementary functons compactly, BMDs are promsng for verfcaton of hardware for elementary functons, and for the alternatve mplementaton of embedded RAM on FPGA for the functon tables Acnowledgments Ths research s partly supported by the Grant n Ad for Scentfc Research of the Japan Socety for the Promoton of Scence (JSPS, funds from Mnstry of Educaton, Culture, Sports, Scence, and Technology (MEXT va Ktayushu nnovatve cluster project References [] R E Bryant, Graph-based algorthms for boolean functon manpulaton, IEEE Trans on Comput, Vol C-35, No 8, pp , Aug 986 [2] R E Bryant and Y-A Chen, Verfcaton of arthmetc crcuts wth bnary moment dagrams, Desgn Automaton Conference, pp , 995 [3] M J Cesels, P Kalla, Z Zheng, and B Rouzeyre, Taylor expanson dagrams: A compact canoncal representaton wth applcatons to symbolc verfcaton, Desgn, Automaton and Test n Europe (DATE22, pp , 22 [4] E M Clare, K L McMllan, X Zhao, and M Fujta, Spectral transforms for extremely large Boolean functons, IFIP WG 5 Worshop on Applcatons of the Reed-Muller Expanson n Crcut Desgn, pp 86 9, Sept 993 [5] B J Falows, A note on the polynomal form of Boolean functons and related topcs, IEEE Trans on Comput, Vol 48, No 8, pp , Aug 999 [6] K D Hedtmann, Arthmetc spectrum appled to fault detecton for combnatonal networs, IEEE Trans on Comput, Vol 4, No 3, pp , Mar 99 [7] S Höreth and R Drechsler, Formal verfcaton of word-level specfcatons, Desgn, Automaton and Test n Europe (DATE999, pp 52 58, 999

6 [8] S L Hurst, D M Mller, and J C Muzo, Spectral Technques n Dgtal Logc, Academc Press, Brstol, 985 [9] J Jan, Arthmetc transform of Boolean functons, Chapter 6 n [5] [] S K Kumar and M A Breuer, Probablstc aspects of Boolean swtchng functons va a new transform, Journal of the ACM, 28(3, pp 52 52, 98 [] C Menel and T Theobald, Algorthms and Data Structures n VLSI Desgn: OBDD Foundatons and Applcatons, Sprnger, 998 [2] V-D Malyugn, Paralleled Calculatons by Means of Arthmetc Polynomals, Physcal and Mathematcal Publshng Company, Russan Academy of Sc, Moscow, Russa, 997 [3] J-M Muller, Elementary Functon: Algorthms and Implementaton, Brhauser Boston, Inc, Secaucus, NJ, 997 [4] S G Papaoannou and W A Barrett, The real transform of a Boolean functon and ts applcaton, Computer and Electronc Engneerng, Vol 2, pp , Pergamon Press, 975 [5] T Sasao and M Fujta (eds, Representatons of Dscrete Functons, Kluwer Academc Publshers, 996 [6] R Stanovc, T Sasao, and C Moraga, Spectral transform decson dagrams, Chapter 3 n [5] [7] R Stanovc and T Sasao, A dscusson on the hstory of research n arthmetc and Reed-Muller expressons, IEEE Trans on CAD, Vol 2, No 9, pp 77 79, Sept 2 [8] R Stanovc and J Astola, Spectral Interpretaton of Decson Dagrams, Sprnger Verlag, New Yor, 23 [9] M A Thornton, R Drechsler, and D M Mller, Spectral Technques n VLSI CAD, Sprnger, 2 Appendx Proof for Lemma From the property of the arthmetc transform, the number of non-zero arthmetc coeffcents s equal to the number of terms n the expresson that s obtaned by expandng and rearrangng the followng: X = ( 2 n x n + 2 n 2 x n x + 2 x, where x 2 = x ( =,,2,,n because x s a Boolean varable In the expresson expanded and rearranged, the number of terms wth a sngle lteral (e, terms of x s ( n And, the number of terms wth two lterals (e, terms of x x j ( < j s ( n 2 Smlarly, the number of terms wth lterals s ( ( n Therefore, the total number of terms s n Example A Consder the 4-bt precson functon f (X = X 2 From X = 8x 3 + 4x 2 + 2x + x, we have X 2 = ( 8x 3 + 4x 2 + 2x + x 2 = (64x x x2 + x2 + 2( 32x 2x 3 6x x 3 8x x 3 + 8x x 2 + 4x x 2 + 2x x = 64x 3 + 6x 2 + 4x + x 64x 2 x 3 32x x 3 6x x 3 + 6x x 2 + 8x x 2 + 4x x Note that ths expresson has terms The number of ( non-zero arthmetc coeffcents obtaned by Lemma s 4 ( = =, and t s dentcal to the number of terms n the above expresson Proof for Lemma 2 From the proof for Lemma, t s clear that the sets of terms obtaned by expandng and rearrangng X ( =,2,, are proper subsets of the set of terms for X f coeffcents of the terms are gnored When ( c, the arthmetc coeffcent for the constant term n must be consdered Therefore, Lemma 2 holds Proof for Lemma 3 From the proof for Lemma, when X, all the non-zero arthmetc coeffcents for X ( =,2,, are postve Snce c >, the number of nonzero arthmetc coeffcents for f (X s equal to ts upper bound Proof for Lemma 4 In a BMT for X, let the varable order be x,x,,x n, from the root node to termnal nodes From the proof for Lemma, t s clear that the arthmetc expresson for X conssts of terms wth at most lterals The arthmetc expresson for X can be represented by a tree structure representng all the terms n the expresson Let β(n, be the number of nodes n the tree Then, β(n, satsfes the followng relaton: β(n, = + β(n, + β(n,, (A where s the number of root node, and β(n, and β(n, are the numbers of nodes n the left sub-treee and the rght sub-tree, respectvely We show that α(n, satsfes the relaton (A + α(n, + α(n, = + + = + = [( ( ] n n = + = Therefore, we have the lemma = α(n, Proof for Lemma 5 From the proof for Lemma 2, t s clear that the arthmetc expresson for f (X conssts of terms wth at most lterals The arthmetc expresson for f (X can be represented by tree structure representng the all terms n the expresson From the property of the BMT, a BMT always requres a constant term ndependently of the value of the constant term Therefore, BMD for f (X has at most ( n non-termnal nodes Proof for Theorem From Lemma 2, the number of termnal nodes n BMD for f (X s at most ( n = From Lemma 5, the number of non-termnal nodes n BMD for f (X s at most ( n Therefore, we have the theorem Proof for Lemma 6 When the number of dstnct values of X s 2 n, the number of dstnct values of f (X s also 2 n Then, the MTBDD for f (X s the complete bnary tree, and the number of nodes n the tree s 2 n+