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IEEE TRANSACTIONS ON COMPUTERS, VOL. C-27, NO. 10, OCTOBER 1978 961 Correspondence Reduction of CC-Tables Using Multiple Implication H. A. VINK, B. VAN DEN DOLDER, AND J.

1 IEEE TRANSACTIONS ON COMPUTERS, VOL. C-27, NO. 10, OCTOBER Correspondence Reduction of CC-Tables Using Multiple Implication H. A. VINK, B. VAN DEN DOLDER, AND J. AL Abstract-Gimpel has presented a minimization algorithm which realizes a minimal TANT network for any Boolean function, under a NAND gate cost criterion. Gimpel uses a covering closure (CC)table and reduces this table with the reduction techniques of Grasselli and Luccio to find a least cost subset of implicants. This correspondence presents the reduction of CC-tables using multiple implication. Closure columns are introduced which are multiply implied, instead of singly, as in GimpeL Moreover, the effect of multiple implication is discussed: a reduction of CC-tables, a simplification of some reduction rules, and the introduction of others. Inde Terms-CC-table, gate cost criterion, minimal TANT network, minimization, reduction rules. INTRODUCTION The minimization of TANT networks, three level network composed of AND-NOT (i.e., NAND gates) having only true (i.e., uncomplemented) inputs, may be illustrated by a Karnaugh map, such as Fig. 1, which shows the decimal representation of a switching function of n variables. This representation of the 1-cells, for which the function must be 1, is obtained by assigning the value 2n-i to the input variable i and treating them as a binary code. Gimpel [6] generates a sufficient set of implicants which he has called the prime permissible implicants (PPI) and determines from them the content of a covering-closure (CC-table, as shown in Fig. 2. The CC-table, an etension of the prime implicant table of Quine-McCluskey, can handle certain forms of covering problems having a nonadditive cost function. The CC-table is a rectangular matri of dots, " " entries and blanks. A row having an " " entry in a column is said to cover that column. A row having a dot in a column is said to imply that column, i.e., use of a row with a dot in the closure section requires that the column with that dot also be covered. In the CC-table, three relevant parts may be distinguished. The upper left-hand part is the cover table with a column (cover) for every fundamental product and a row (PPIR) for every PPI. The upper right-hand part is the implication table with a column (closure) for every principal tailfactor of every PPI and the same rows as the cover table. The lower right-hand part is the closure table with the same closure columns as the implication table and a row (TLR) for every tailfactor which realizes some principal tailfactor of a PPI. The closure column associated with a principal tailfactor of a PPI is implied (indicated by a dot) by the PPIR corresponding with that PPI and is covered (indicated by an " " entry) by the corresponding TLR of every tailfactor which realizes it. A set of rows Z is said to be a cover of a CC-table if every cover column is covered by some row of Z. A set of rows Z is said to be Manuscript received May, 1975; revised October 23, 1975 and April 22, H. A. Vink is with the Department of Electrical Engineering, Delft University of Technology, Delft, The Netherlands. B. van den Dolder and J. Al are with the Department of Electrical Engineering, Twente University of Technology, Enschede, The Netherlands. closed if every column implied by a row of Z is covered by some other row of Z. A solution to a CC-table is a closed cover of minimum cost. In the literature, several reduction rules for the prime implicant table (a cover table) and the CC-table are presented: [1]-[6]. Gimpel lists the reduction rules of Grasselli and Luccio [3]; these rules pertain to essential row reduction, row dominance, and column dominance. In the net section, the reduction of CC-tables using multiple implication will be presented. This reduction will be illustrated with the function shown in Fig. 1. We generate the PP implicants, determine the content of the CC-table, and obtain the reduced CC-table of Fig. 2, after the removal of the dominated tailrows. CLOSURE COLUMN REDUCTION This reduction is applied to closure columns which are covered by the same tailrows. These closure columns can be united into one column. The resulting closure column has a dot or " " entry in every row, where an original column has a dot or " " entry. Appendi I contains the proof of this reduction. The new column is implied by several rows, and indicates that the PPI's, corresponding to these rows, have tailfactors which can be realized in the same way. The closure columns in Fig. 2, marked with the same letter, can be united. The result of applying this closure column reduction is shown in Fig. 2. From the proof of this theorem follows that multiply implied closure columns can be disunited into a series of singly implied closure columns. The impact of multiple column implication upon the prominent rules-essential row selection, row dominance, and column dominance-(the rules 1-5 of Grasselli and Luccio) is discussed in the net section. Moreover, multiple implication leads to some new reduction rules. The attention has been restricted to those reduction rules which reduce the CC-table, based on the meaning of and relation between the rows and the columns, without any knowledge about the content from which follows the specification of the table The reduction rule of Gimpel described by theorem 14 requires as additional information the function specification, but may further reduce the CC-table. The application has been simplified, however, because of the reduction of the closure section of the CC-table. ESSENTIAL CLOSURE COLUMN Because of the multiple implication, it makes sense to introduce a rule for the determination of essential closure columns: a column which is implied by every closed cover. This is the case if such an essential closure column is implied by a cover column; therefore, it can be treated as a cover column. A closure column is implied by a cover column, if the closure column has a dot in at least those rows where the cover column has an " " entry. As is proven in Appendi II, we may indicate these essential closure columns by removing the dots. We introduce for these columns the name e-closure column. Because of the multiple implication, several e-closure columns may appear. In Fig. 2, the column marked a is implied by the columns 2, 3, 8, and 9; column d is implied by /78/ $00.75 ) 1978 IEEE

2 962 IEEE TRANSACTIONS ON COMPUTERS, VOL. c-27, NO. 10, OCTOBER 1978 ~ZI) X(4) Fig 1. w 1 114, 8' 89 I (8) A function used to iliustrate some table reduction rules. P1: wy P2:wyzy yz' :wv P5*w]iz w'y' :w'y(z : w_(z tit t 1: ' :z t 4:(4B P2 P5 t I t2 t 1 co' over columns closure columns A * 0 *e X Os0 S 6@ 0 0 S. a. a b b b bb b c c dd e e X X 0 d a ( a d (c) S X X0 X a P2 P5 ti t2 P2 P5 t I K X 0 0 XXX 0 0 X XX 0 0 a XXi Fig, 2. -(f) A CC-table of the function shown in Fig (f) 1. K a d K 0 K X X 0 (d) XX 0 (d) cover columns

3 CORRESPONDENCE 963 (4) z(l) 11 1 (2) w (8) Fig. 3. CONver columns closure columns P r : w'z ' PI wyz : w'y' 0 : yz' 5 0 Ps: wz(y) yjwzj S t 1: ' t 2: (y) S t 3: (z) t 4: (w z)' t5: y' c d A function used to illustrate the reduction of implied e-closure columns. A partly reduced CC-table of the function shown in Fig 3(a4 Removing the dots in columns a and d in Fig. 2 and thus changing them into e-closure columns results in Fig. 2(c). ESSENTIAL TAILROWS An essential tailrow of a CC-table is a tailrow which is a member of every closed cover. Gimpel [6] gives a rather comple procedure for the determination of the essential tailrows. (Appendi V-Essential row selection.) If an e-closure column is only covered by one tailrow, then this row is essential. As proven in Appendi III, an essential tailrow may now be detected easily by noting that a tailrow is essential if there is an e-closure column which is only covered by this row. The tailrow t2 of Fig. 2(c) is essential because the e-closure column a is only covered by this tailrow. The tailrow is selected and can be removed, together with the covered columns. The procedure of Gimpel can, therefore, be replaced by this new rule, if it is preceded by closure column reduction and the introduction of the e-closure columns. COVER COLUMN IMPLICATION As stated by Gimpel [6], cover column dominance may reduce the CC-table further. We introduce the term "column implication" because it gives a better description of the reduction procedure. A cover column cl implies another cover column c2, if every row which covers column cl, also covers column c2. The implied column c2 may be removed. In Fig. 2(c) we notice that column 3 implies 2 and column 9 implies 8. The result of this reduction of the CC-table is shown in Fig. 2(d). The e-closure columns may now imply each other. The reduced CC-table [Fig. 3] of the function, described by the Karnaugh map [Fig. 3], demonstrates this reduction. E-closure column c implies column d, which may be removed. The cover and closure part of a CC-table have no rows in common, so a cover column cannot imply an e-closure column; therefore, this reduction appears only among the cover columns or the e-closure columns eclusively. EX-CLOSURE CLOSURE COLUMN IMPLICATION If an e-closure column implies a closure column, then the closure column is covered if the e-closure column is covered; therefore, the closure column may be removed. In Fig. 4 the reduced CC-table of the function, described in shown. The marked closure column f of Fig. 4 is Fig. 4, is implied by the e-closure column e'and may be removed. An e-closure column implies a closure column, if every tailrow which covers the e-closure column also covers the closure column. PPI Row DOMINANCE Gimpel [6] proved that a PPI row pl is dominated by a PPI row p2 if p2 covers every column that pl covers and, if for every closure column (say c2) that p2 implies there is a closure column (say cl) implied by pl, such that c2 has " " entries in every row in which ci has " " entries. The dominated PPI row p1 may be removed together with its closure columns. Closure columns, after being united, may be multiply implied, however; therefore, only those columns are removed which are not implied by a remaining row. If a closure column does not have this property, then the dot automatically disappears with removal of the dominated row, but the column remains. The PPI row p2 of Fig. 2(d) is dominated by p3 and p5 by p4. The rows p2 and p5 are removed, together with their dot and their common closure column. Consecutive application of the following reductions results in the reduced CC-table of Fig. 2(f), which solely is composed of PPI rows without closure columns: 1) tailrow dominates tailrow. 2) tailrow is essential. 3) PPI row p7 dominates PPI row pl. 4) PPI row p8 dominates PPI row p6.

4 964 IEEE TRANSACTIONS ON COMPUTERS, VOL. c-27, NO. 10, OCTOBER 1978 ZM X(4) P1: yz P2: wy : wyz' : wy' P5: wy(z) : w(yz)' t 1: t2 (Z) t 3: (y; : (yz) (8) co) ver columns closure columns Fig. 4. A function used to illustrate the reduction of implied closure columns. A partly reduced CC-table of the function shown in Fig. 4. ef - ESSENTIAL PPI Row An essential PPI row is a row which is a member of every closed cover. The essential row is selected and may be removed from the CC-table, together with its cover columns. Its closure columns must be covered and are changed into e-closure columns: every dot of their columns is removed. The column 3 of Fig. 2(f) is only covered by p7 and 9 by p8. These rows p7 and p8 are essential and will be removed, together with their cover columns. Because they do not have a closure column anymore, no e-closure column will appear. EMPTY PPI Row An empty row is a row which does not cover a cover column; therefore, it may be removed. For this reduction, the same applies as for the dominance of PPI rows. Its closure columns may be removed, if these are not implied by a remaining row. If not, then the dot automatically disappears when the empty row is omitted. APPLICATION OF REDUCTION RULES Because of the large number of reduction rules, it is advisable to apply them in a fied order. This systematic approach also increases the efficiency in reducing the CC-table. 1) E-closure columns may be formed: a) implicit as a part of essential PPI row selection. b) after application of PPI row dominance, a comparison of the modified columns with the closure columns may give an eclosure column. c) after application of closure column reduction, a comparison of the cover columns with the modified closure columns may give an e-closure column. 2) Closure columns may be united: a) after the formation of the CC-table, if the closure columns are compared with each other. b) after application of tailrow dominance, if the modified closure columns and the closure columns are compared with each other. 3) Detection of essential and empty rows may easily, and therefore often, be performed. 4) To discover PPI row dominance is time consuming because every pair of PPI rows must be compared. Applying this rule in the following order may result in a decrease of time. a) determine those pairs of PPI rows, each consisting of a potential dominant and dominated PPI row, by comparison of the covered fundamental products. b) if a "dominant" row has no closure columns, then this pair denotes a dominated row. c) if from a remaining pair a "dominated" row has no closure columns, then this pair does not denote a dominated row. d) compare the tailrows of the remaining rows. e) remove the dominated rows. FURTHER INVESTIGATION In report [7], a description of the minimization method of Gimpel is given in a modified and etended way. Gimpel proved that a quasi-simple and isolated PPI is dominant. This report gives some new rules concerning isolated and "quasi-isolated" PPI. A computer program for the minimization of TANT networks has been written for the IBM360/50 at Twente University of Technology. The program is part of a Computer Assisted Instruction (CAI) project and is written in APL. The program accepts the

5 CORRESPONDENCE 965 4I 1. k cover implication P table table C t co imp closure table Fig. 5. The subdivision of the CC-table. decimal equivalent of the fundamental products of the specified function, the prime implicants and the prime permissible implicants in their symbolic product form, and gives a Karnaugh map and a CC-table as shown in the Figs. 1 and 2, respectively. We have the following eperience with this CAI project, based on APL: 1) This CAI project accepts a specification of a function by a user and calculates the result with the algorithm the user wants to practice, which means the interest of the eerciser noticing. 2) The reduction of the CC-table may be performed in a highly conversational way. 3) In a very short time (several weeks) a first release of a CAI system has been obtained. CONCLUSION This correspondence presents closure columns which are multiply implied. The introduction of multiple implication causes the number of closure columns to be strongly reduced; moreover, some closure columns may change into an e-closure column. These new columns may reduce the CC-table further. Multiple implication causes most of the prominent rules to be somewhat modified, sometimes in an etension of the scope of the reduction rule or in a simplification of the application. Beside these modifications of the prominent reduction rules, some new ones have been introduced. APPENDIX I The minimization of a CC-table can be epressed in the following way: p: a binary vector of the selected PPI. t: a binary vector of the selected tailrows. c: the cost vector of the PPI and the tailrows. co: the binary cover table; a "1" is interpreted as an " " entry and a "0" as a blank. imp: the binary implication table; a "1" is interpreted as a dot and a "O" as a blank. ci: the binary closure table; a " 1" is interpreted as an " "' entry and a "0" as a blank (see Fig. 5). Minimize suibject to n m Eci.p + EC. i=l i=l i+n i n E coij - >, 1 i=l t ( ) mn ij. ti) >1 (= i ij.p,2 J=t12... Because of the multiple implication, (1.3) changes into (1.4): at most, I closure columns may be united, so the left epression will be multiplied with 1: (1.3) i=1,2, ~~~~(1.4) Theorem: Closure columns which are covered by the same tailrow, can be united into one column. The resulting closure column has a dot or" " entry in every row, where an original column has a dot or " " entry. Proof: Suppose that the last o closure columns are covered by the same tailrows. We have to prove that the symbolic notation of the CC-table without multiply implied closure columns (2.1) is equal to the symbolic notation of the CC-table with the multiply implied closure columns (2.2). k n co 11 A A V impi ) C V (VClij J= Ci= D= A t I ~ ~ ~ ~ ~ ~~ /\ ((\AMj, A i) V( /Cl, A tj)) imija. iml1 J=I-0+1 =\:/im j= I-0o1 cli =Clij I-o<i sl \Vimm A)V( \</Clij At ji-o=+il \n/( m ( V(Vimp1j ) A) \V /cij At' t j= 1,2.- k (1-2) i.a) i=1 j=i-o+1 i=1 A (2.1) (2.2) (2.3)

6 966 IEEE TRANSACTIONS ON COMPUTERS, VOL. c-27, NO. 10, OCTOBER 1978 Fig. 6. Fig. 7. t21 1E1 (P1V P2)A(P1 vtl)a(p2'vtl) _ (2.9) P1 P2jX. (PI vp2) A((PI'AP2)vtb) tl _ (2.10) A CC-table with closure columns which can be united. A CC-table with a multiply implied closure column. P2 Il* P V I1vP2) 3 Vtl (2.11)P' p3 a (Pl V)A^ (P1V2 A t I P2 (P1 v) A(PlPvP2) Atl t 1 (1t^3)t) 2al (2.12) A CC-table with a closure column which is implied by a cover column. A CC-table with an e-closure column. j n (,i I m j j) YV j (=V+, V imp AP,)) Vciij At1 1=1 j=i-o+1 j 1=1 Closure column jl will be implied by every closed cover, so there (2.5) must be a cover column which implies closure column jl. Q.E.D. (IK(\2inp.AP ) j'=-o+1 1= = =Iv''V 1CI.A (j(vimpj A )) vj clvjat - (2.7) 1=1-0+1 i=1 '1li I A\((VimP.j4)v(Vcl'At. (2.8) =I-o+. l'=i ii Eample: See Fig. 6 and. Q.E.D. APPENDIX II This appendi contains the proof that the dots of an e-closure column can be removed. An e-closure column is a column which is implied by a cover column. Theorem: The dots of an e-closure column in a CC-table can be removed. Proof: Suppose that cover column jl implies closure column j2. Column jl must be covered, so for j = jl epression A of (2.1) will be high ('1") for a certain il. Epression C will also be high for j = j2 because column jl implies j2. There must be a certain i2 for which epression D is high. Q.E.D. Column j2 must be covered and the dots, symbolized by epression C, may be removed for j = j2. Eample: See Fig. 7 and. APPENDIX III This appendi contains the proof that an essential tailrow is the only row which covers an e-closure column. An essential tailrow is a tailrow which is a member of every closed cover. Theorem: An essential tailrow is the only row which covers an e-closure column. Proof: Suppose that tailrow il is essential and a closure column jl which only is covered by row il. In (2.1) we can see that epression C will be high forj = jl because tailrow il is essential. ACKNOWLEDGMENT (.6) The authors wish to thank Prof. G. A. Blaauw for his many helpful suggestions and his discussions of the paper /78/ $00.75 ( 1978 IEEE REFERENCES [1] J. F. Gimpel, "A reduction technique for prime implicant tables," IEEE Trans. Electron. Comput., vol. EC-14, pp , Aug [2] S. U. Robinson, III and R. W. House, "Gimpel's reduction technique etended to the covering problem with costs," IEEE Trans. Electron. Comput., vol. EC-16, pp , Aug [3] A. Grasselli and F. Luccio, "A method for minimizing the number of internal states in incompletely specified sequential networks," IEEE Trans. Electron. Comput., vol. EC-14, pp , June [4] R. W. House and D. W. Stevens, "A new rule for reducing CC tables," IEEE Trans. Comput., vol. C-19, pp , Nov [5] H. A. Curtis, "The further reduction of CC-tables," IEEE Trans. Comput., vol. C-20, pp , Apr [6] J. F. Gimpel, "The minimization of TANT networks," IEEE Trans. Electron. Comput., vol. EC-16, pp , Feb [7] H. A. Vink, "The analysis and synthesis of TANT networks," Twente University of Technology, EF-library, The Netherlands, to be published. On Computing the Discrete Cosine Transform B. D. TSENG AND W. C. MILLER Abstract-Haralick has shown that the discrete cosine transform of N points can be computed more rapidly by taking two N-point fast Fourier transforms (FFT's) than by taking one 2N-point FFT as Ahmed had proposed. In this correspondence, we show that if Haralick had made use of the fact that the FFT's of real sequences can be computed more rapidly than general FFT's, the result would have been reversed. A modified algorithm is also presented. Inde Terms-Comple FFT, comple operations, discrete cosine transform, real FFT. Manuscript received November 26, The authors are with the Department of Electrical Engineering, University of Windsor, Windsor, Ont., Canada.

Minimal TANT Networks of Functions with DON'T CARE'S -and Some Complemented Input Variables

Minimal TANT Networks of Functions with DON'T CARE'S -and Some Complemented Input Variables CORRESPONDENCE 1073 l 0 1 2 0 0 0 1 1 0 1 1 x2 xi\ 0 1 2 0 0 1 1 1 1 1 1 2 1 1 2 2 1 1 2 (a) (b) (c) Fig. 5. MT*(3) elements for binary (a) AND*, (b) OR*, and (c) NOT* gates. TABLE I Decimal Equivalent