Maitra Cascade Minimization

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1 Matra Cascade Mnmzaton Voudours Dmtros Dr. apakonstantnou George Natonal Techncal Unversty o Athens

2 Basc Dentons Comple Matra Term [] Constant 0 Boolean uncton Lteral M Matra Term a lteral G Arbtrary varable swtchng uncton ~Matra Cell Reversble Wave Cascade epresson Matra epresson [] M + Q Ga,M m M Matra Cascade Mnmzaton - Natonal Techncal Unversty o Athens

3 Basc Dentons Mnmal Eact epresson: Least number o comple terms weght Cascade realzable uncton w Restrcted Matra Cascade G mplements a complete set o 6 unctons Table Matra Cascade Mnmzaton - Natonal Techncal Unversty o Athens 3

4 Reversble Wave Cascade Cellular Archtecture Matra Comple Term X R R Rm Matra Cells Xn 0 Rn Rn Rnm XOR Collector row Generalzed n+n+ Tool gate Matra Cascade Mnmzaton - Natonal Techncal Unversty o Athens 4

5 Matra Cascade Mnmzaton - Natonal Techncal Unversty o Athens 5 Boolean Decompostons Shannon, Negatve Davo, ostve Davo decompostons New Decompostons 0 0 X X X X X X X X X X X X + + +

6 Cascade Mergng Lemma : Relaton: r, r, r Eample: r, y, y, y y, y y, y r r y holds :,,3,,3,,,,4,,4,,3,3,3,4,4,4,5,5,6,5,6,5,6,6,6 [ + 34 ] [ ] {[ + 3] [ + 3]} Matra Cascade Mnmzaton - Natonal Techncal Unversty o Athens 6

7 Matra Cell Classes Cell has non-constant nputs: Cell Class Cell Inde Cell Inde representatve Cell wth a constant nput cascaded 0 Cell Class Cell Inde representatve Cell Inde 6 Matra Cascade Mnmzaton - Natonal Techncal Unversty o Athens 7

8 Relatve Comple Terms Two comple terms RR...R n, QQ...Q wth R, Q matra cells are relatves R and Q, n belong to the same matra cell class. The ollowng comple terms are relatves:,,, n n Matra Cascade Mnmzaton - Natonal Techncal Unversty o Athens 8 n

9 Relatve Comple Terms Matra Generator Class. Matra epressons wth relatve correspondng terms. Equvalent matra epressons Represent the same swtchng uncton. Belong to the same generator class. Eample Epressons: Q , Q are equvalents snce they both represent the same swtchng uncton and urthermore terms: 34 & 4 are relatves and terms 666 & 666 are relatves. Matra Cascade Mnmzaton - Natonal Techncal Unversty o Athens 9

10 Relatve Comple Terms 3 CIL: Number o cells wth one constant nput 0. Lemma : relatve comple terms comple term., Eample s a 3455 and 4466 are relatves 3 and 4 are complements 6664 and Matra Cascade Mnmzaton - Natonal Techncal Unversty o Athens

11 Relatve Comple Terms 4 Lemma 3: I,,M,M are comple terms and: M, relatves M, M have CIL M CIL M > 0 cells o same class rom CILM+ untl the last cell Then: M and, are relatve comple terms & Eample 63 & 66 have CIL & respectvely and 43 & 44 are relatves Matra Cascade Mnmzaton - Natonal Techncal Unversty o Athens

12 Relatve Comple Terms 5 Lemma 4: I, are comple terms and: 0 < CIL < CIL cells o the same class rom CIL+ untl the last cell Then s a comple term and s relatve. Eample Matra Cascade Mnmzaton - Natonal Techncal Unversty o Athens

13 Comple Term Splttng Lemma 5 Term splttng: Q p p p Q p... q... p... p... p... q n n,q,q comple terms p, p, p, n, cells o the same class p, q, q cells o derent cell class. Eample n p p p Lemma 6:, relatve comple terms, are splt at poston :,, and, are relatves. Eample Matra Cascade Mnmzaton - Natonal Techncal Unversty o Athens

14 Matra Cascade Mnmzaton - Natonal Techncal Unversty o Athens 4 Mnmzaton Theorems Theorem : Each mnmal epresson o can be epressed as: roo XOR-sum mnmal orm o : Comparson wth Lemma.,,,,,, g z y z y y r q p q p p,...,,..., n rn r n y y Constant subuncton Equv. orms p,q3,4,3,6,4,6 and y,z subunctons p,q,r3,4,6 z y g z g y,,,, 0

15 Mnmzaton Theorems Theorem : At least one mnmal epresson o a swtchng uncton,,..., n wth less than 6 varables can be obtaned rom the mnmal epressons o ts subunctons. roo: 3 cases Theorem : Epr, y p Theorem Constant subuncton. Epr produced rom the mnmal epresson o non constant subunctons Matra Cascade Mnmzaton - Natonal Techncal Unversty o Athens 5

16 Mnmzaton Theorems 3 Epr, y, z, y,z subunctons. wwy+wz p, q Epr produced by the mnmal epressons o subunctons. Epr p, y q, z r, g 0,, y g, z g, y wy wz wg w 5 wy w 3 wy wz wg w w common term w wy + wz + wg m 4,5 wy m w0 + w * wy + wg + wz m + w0 + w m. roo ollows that o prevous case. w0 + w m + 0 y g, y z mnmal eprs o these subunctons or ths partcular case, At least one mnmal epr o 0 and wth one common term y to be merged. z Matra Cascade Mnmzaton - Natonal Techncal Unversty o Athens 6

17 Mnmzaton Theorems 4 Theorem 3: Let Q be a mnmal epresson o, produced by the mnmal epressons, o, respectvely. An equvalent to Q epresson Q o can be obtaned rom two other mnmal epressons, o, whch are equvalents to,. Matra Cascade Mnmzaton - Natonal Techncal Unversty o Athens 7

18 Matra Cascade Mnmzaton - Natonal Techncal Unversty o Athens 8 Mnmzaton Theorems 5 roo: I w w 0,, then the proo s trval. I w w + w then the proo s trval. I w w + w. Common term:,,, relatves. It holds: r q Q..., ,... k k k k k Q Q r q r q r q M M r r q q

19 Mnmzaton Theorems 6 urthermore: M and M,M wll have cells o the M M same cell class rom poston MAXCILM,CILM+ untl the last cell Lemma. Wthout loss o generalty: CILM CILM. CILM < CILM M s relatve to M Lemma M... and r r Q Q Q k k k 3 3 k k... r r 3 M merges wth at most terms k... k CILM CILM M wll have cells o the same class wth M & M rom poston > CILM + untl the last cell, whle the rest o them wll represent a, lteral. The proo ollows that o prevous case Lemma 3. r M k... r k r k r k... r k Matra Cascade Mnmzaton - Natonal Techncal Unversty o Athens 9

20 Heurstc Algorthm outlne Input : a swtchng uncton n mnterm ormulaton Decomposton recursvely usng ETDDs Shannon and Davo epansons. Composton recursvely: roduce epressons or rom the mnmal epressons o ts subunctons. I comple terms wth the same generator are ound between two such epressons, : Merge them to produce a by-product comple term. Try to merge that by-product wth the rest o the terms n,. I ths by-product s merged, then the weght o the uncton s reduced by. At the end keep those epressons wth the least number o comple terms. Matra Cascade Mnmzaton - Natonal Techncal Unversty o Athens 0

21 Eample Matra Cascade Mnmzaton - Natonal Techncal Unversty o Athens

22 Epermental results Lee s Algorthm: G.Lee, R.Drechsler: ETDD-Based Synthess o term-based GAs or ncompletely speced boolean unctons, AS-DAC 998 Mnct: N. Song, M. erkowsk: Mnmzaton o eclusve sums o multvalued comple terms or logc cell arrays, ISMVL 98, p 3 Matra Cascade Mnmzaton - Natonal Techncal Unversty o Athens

23 Conclusons and uture work Contrbuton New boolean decompostons are presented. An algorthm has been descrbed or producng mnmal reversble wave cascades or swtchng unctons up to 5 varables and near mnmal or unctons wth more varables, wthout the need to calculate equvalent epressons. uture work Mult-output swtchng unctons. Mnmal soluton or unctons o up to 6 varables. Matra Cascade Mnmzaton - Natonal Techncal Unversty o Athens 3

24 Bblography K.K. Matra: Cascaded swtchng networks o two-nput leble cells, IRE Trans. Electron. Comput., pp, 36-43, 96. R. C. Mnnck: Cutpont cellular logc, IEEE Trans. Electron. Comput., vol. EC-3. pp , Dec,964. J. T. Butler: Restrcted Cellular Networks, IEEE Trans. Computers 5: 39-4, 976. A. Mshchenko, M. erkowsk: Logc Synthess o Reversble Wave Cascades, Internatonal Workshop on Logc And Synthess 00, New Orleans, Lousana, June 4-7, 00. A. Sarab, N. Song, M. Chrzanowska-Jeske, M. erkowsk: A comprehensve approach to logc synthess and physcal desgn or two-dmensonal logc arrays, DAC 994, I. Schaeer, M. erkowsk, H. Wu: Multlevel logc synthess or cellular GAs based on orthogonal epansons, roc, II WG 0.5 Workshop on Applcatons o the Reed-Muller Epanson n Crcut Desgn, Hanburg, Germany, pp. 4-5, Sept Matra Cascade Mnmzaton - Natonal Techncal Unversty o Athens 4

25 Bblography G. Lee: Logc synthess or celullar archtecture GA usng BDD, AS-DAC 97, pp Jan 997. G. Lee, R. Drechsler: ETDD-based Synthess o term-based GAs or ncompletely speced boolean unctons, AS-DAC Lndgren, R. Drechsler, B. Becker: Look-up table GA synthess rom mnmzed mult-valued pseudo kronecker epressons, ISMVL 98. G. apakonstantnou,. Grtzal: Modulo- epressons o swtchng unctons, Electronc Letters, G. apakonstantnou: Synthess o cutpong cellular arrays wth eclusve-or collector row, Electronc Letters, J. reskll: Lecture notes n quantum computng, G. apakonstantnou: Cascade Transormaton, IEEE Transactons on computers, Jan 976. Matra Cascade Mnmzaton - Natonal Techncal Unversty o Athens 5

26 Bblography 3 N. Song, M. erkowsk: Mnmzaton o eclusve sums o mult-valued comple terms or logc cell arrays, ISMVL 98, p 3. N. Song, M. erkowsk: A new approach to and/or/eor actorzaton or regular arrays, roc. 998 Euromcro, pp , Vasteras, Sweden, August 5-7, 998. C. Bennet: Logcal Reversblty o Computaton, IBM Journal o Research and Development, 7, 973, pp D. Voudours, S. Stergou, G. apakonstantnou: Mnmzaton o reversble wave cascades, IEICE, under revson. S. Stergou, D. Voudours, G. apakonstantnou: Eact and Heurstc MVESO Mnmzaton Algorthms, IEICE Trans. on undamentals, vol.e87-a,no,jan Matra Cascade Mnmzaton - Natonal Techncal Unversty o Athens 6

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