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LIBRARY OF THE 
 
 UNIVERSITY OF ILLINOIS 
 
 AT URBANA-CHAMPAICN 
 
 510.84 
 
 n 0.308-315 
 cop2 
 
The person charging this material is re- 
 sponsible for its return to the library from 
 which it was withdrawn on or before the 
 Latest Date stamped below. 
 
 Theft, mutilation, and underlining of books 
 are reasons for disciplinary action and may 
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 UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN 
 
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Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/minimizationofsw309ibar 
 
vu,30<j 
 
 /n u^la\ 
 
 Report No. 309 
 
 MINIMIZATION OF SWITCHING NETWORKS 
 USING NEGATIVE FUNCTIONS 
 
 by 
 
 T. Tbaraki 
 S . Muroga 
 
 HIP \i fin 
 
 MAR 1 ? 1988 
 
 February 17, 1969 
 
MINIMIZATION OF SWITCHING NETWORKS 
 USING NEGATIVE FUNCTIONS 
 
 T. Ibaraki 
 S . Muroga 
 
 February 17, 1969 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 6l801 
 
Abstract 
 
 In this paper we discuss a minimization procedure of switching 
 networks using negative functions only, assuming that any negative function 
 can be implemented as a gate in the networks. The minimality criterion 
 used is to minimize the number of gates in the network under the conditions: 
 
 (1) The network consists of two levels, 
 
 (2) No fan-ins restriction on each gate is imposed. 
 
 This is motivated by the recent progress in the MOS integrated circuit 
 technology that a considerably complex negative function can be realized 
 by a single MOS gate. 
 
 The minimization procedure is then extended to multiple output 
 network design. Also some considerations are given to the problem of 
 minimizing the number of connections and interconnections in the network. 
 
Minimization of Switching Networks Using Negative Functions 
 1. Introduction 
 
 Recent advances in integrated circuit technology are changing 
 the entire prospect of development of logical design. It opens a number 
 of possibilities as well as complicates the logical design in many- 
 respects. 
 
 Especially, the MOS integrated circuit technology exhibits many 
 attractive and practical features. Among those is an interesting fact, 
 from which our present discussion has arisen, that each MOS gate is able 
 to represent a considerably complex negative function because of its 
 inherently high inpedance. For example, an MOS gate which represents 
 any negative function with up to 32 literals in the functional form as 
 
 g = X ll X 12* '• X lk 1 " X 21 X 22* ' '^kg V - * " " X £L X ^2 * * * X ^' 
 
 can be manufactured , as a typical implementation is shown in Fig. 1. 
 
 In Fig. 1, x. . shows a connection from an external variable, and 
 (j) and (j) are clock signal supplied from the external clock circuit. 
 
 As a consequence it is not too far from the reality to assume 
 that any negative function can be realized with one gate, as far as a 
 moderate number of input variables are considered. 
 
 With this observation, one might think of a new problem: minimization 
 of the switching network composed of negative functions only, which realizes 
 a given function (or functions). This problem is not trivial since each 
 gate in the network is able to represent any negative function and we 
 do not know in advance which of all possible negative functions each 
 gate should represent. 
 
: n-< x 2:r< 
 
 x 12 H 
 
 x -.v Hl*>iH 
 
 Ik 
 
 2k: 
 
 : a~1 
 
 K| X 22^ X i2~*l 
 
 '*} 
 
 I 
 
 "* g - X ll X 12" X 3i 1 V a ^l X 22" ,,X 2k 2 V ' 
 
 ••'" ^l'V'^k,, ' 
 
 Fig. 1 A typical MOS gate . 
 
As an example, consider a function of three variables x , x p , x : 
 
 f = x x x 2 v x 2 x v x^x. v x^x . 
 
 (1.1) 
 
 One obvious design procedure of f with negative functions is first to 
 realize three complemented variables x , x , x , using three gates 
 respectively and then to realize f by regarding f as a negative function 
 of x , x , x , x , x p , x . This is illustrated in Fig. 2, in which 
 each gate g. represents a negative function. 
 
 ± x 2 v Xp x 3 v x ± x^ ~ g 1 g 2 g 3 
 
 Fig. 2 Straightforward realization of f = x..x ^ x o x o ^ x -i x q ^ x x x 
 
 This synthesis procedure can be generalized to any function, indicating 
 that any function of n variables can be realized with at most n+1 gates 
 of negative functions within two levels. 
 
 The above network, however, can be simplified. A minimum network 
 which is obtained by the method developed in this paper is shown in Fig. 3« 
 
1" 
 
 X, 
 
 
 g=X v x 2 n/ 
 
 X 3 
 
 
 
 
 
 
 f 
 
 
 
 = a 
 
 
 x l — 
 
 
 — > 
 
 
 
 
 
 
 X 3 — 
 
 ■ 
 
 
 
 f = X 1 X 2 V X 2 X 3 V X 3 X 1 V ^1 v *2 V X 3^ 
 
 Fig. 3 Minimum realization of f = x x s/ x o x o ^ x -i x o v x -i x o x q 
 
 Here, we mean by a minimum network a network with the minimum number 
 of gates of negative functions under the conditions that; 
 
 (1) 
 
 the given function f is realized within two 
 levels (the level which consists of gates connected 
 to the output gate is called the first level and the 
 level of the output gate is the second level) , and 
 
 (2) 
 
 no fan-in restriction on each gate is imposed. 
 
 We will not minimize the number of interconnections in the network 
 except some limited cases discussed in Section "J. 
 
 Although this minimization criterion may not exactly reflect the 
 engineering requirements of the integrated circuit designers, this seems 
 to be only criterion at present, which can be theoretically treated. 
 It is to be explored whether or not a minimum network obtained under 
 the above criterion would deviate much from those obtained under different 
 minimality criteria. But we hope that the theory presented here would 
 be a basis for further development. 
 
2. Negative Functions 
 
 This section will define negative functions in terms of truth 
 table. This will he a hasis of the minimization procedure developed 
 in this paper. 
 
 Let us first consider a completely specified function. A well-known 
 definition of a negative function is that a function is negative 
 if and only if it has a disjunctive form with only complemented (i.e., 
 negative) literals. For example, x..x_ ^ x o^o v x i x -3 -*- s ne g a "tive hut 
 
 x,x v xx v x x ^ x x p x is not. Since our procedure will he based 
 
 on the monipulation of a truth table rather than Boolean expression, the 
 
 following equivalent definition, which is also well-known would 
 
 be more useful. 
 
 Definition 1 ; A completely specified function h of m variables y^, y ? , ..., y 
 
 J A.) 
 
 is negative if and only if there is no pair of input vectors y and 
 
 (j 2 ) * 
 y such that 
 
 m 
 
 and 
 
 y x >y 2 (2.1) 
 
 h(y 1 ) = 1 
 
 h(y 2 ) = 0, (2.2) 
 where y > y is defined as 
 
 Jh ] MJ 
 
 (j-,) (Jo) 
 
 y. x > Yj_ f or every i = 1, 2, 
 
 and (jJ (J P ) 
 
 y. > y. for at least one i. 
 
 * Here we exclude from our consideration the contradictory situation such that 
 
 y(Jl) = y(J2) 
 
 and h(y^^) = 1 
 
 h(y (j 2)) = o. 
 This will never occur in our procedure. 
 
As an example, the function f of variables x , x , x shown in 
 Table 1 (ignore the columns g , g_, g for a while) is not negative 
 because there exists a pair such that 
 
 1 
 
 f 
 
 x l 
 
 X 2 
 
 X 3 
 
 g l 
 
 g 2 
 
 S 3 
 
 (1) 
 
 1 
 
 
 
 
 
 
 
 1 
 
 1 
 
 1 
 
 (2) 
 
 1 
 
 
 
 
 
 1 
 
 1 
 
 1 
 
 
 (3) 
 
 1 
 
 
 
 1 
 
 
 
 1 
 
 
 1 
 
 (h) 
 
 
 
 
 
 1 
 
 1 
 
 1 
 
 
 
 (5) 
 
 l 
 
 1 
 
 
 
 
 
 
 1 
 
 1 
 
 (6) 
 
 
 
 1 
 
 
 
 1 
 
 
 1 
 
 
 (7) 
 
 
 
 1 
 
 1 
 
 
 
 
 
 1 
 
 (8) 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 
 
 Table 1 Truth table of f = x.x ? v x p x o ^ x i x -3 ^ x i x o x q 
 
 x (8) = (111) > x ik) = (Oil) 
 
 and 
 
 f (111) = 1, f (Oil) = 0. 
 This function is the same function as that discussed in Fig. 1 and Fig. 2. 
 
 The outline of our procedure which realizes a given general i^not 
 necessarily negative) function by means of a combination of negative 
 functions is now given. First we augment the truth table by several 
 columns (illustrated as columns g , gu , g in Table l) so that the given 
 function f may be considered as a negative function if those new columns 
 together with columns x , . .., x are regarded as variables of f. According 
 to our procedure, each of these new columns (g_, g~, g„ in Table l) 
 represents a negative function of variables x_, x p , ..., x . Therefore, 
 from this augmented table, we can construct a network of negative functions 
 
in which each gate in the first level corresponds to an augmented column 
 (g , gr , g in Table l) and the gate in the second level, to which the 
 outputs of gates in the first level as well as the variables x , x , 
 
 n 
 
 are fed, realizes the function f. (see Fig. k.) In general, the augmented 
 
 columns are incompletely specified, i.e., some entries of g , g^, g 
 are left blank as seen in Table 1. Generation of augmented columns will 
 be discussed later in detail. 
 
 x 
 
 l -A 
 
 X — 
 
 1 — 
 
 X 
 
 Fig. k General scheme of realization 
 with negative functions 
 
Throughout this paper let us assume that a function is generally 
 incompletely specified. 
 
 As observed in Table 1, an incompletely specified function is 
 possibly not specified in some of the output values and also in entries 
 in some input vectors. For example, the output values of g (x , Xp, x ) 
 
 in Table 1 are not specified for input vectors (100), (101) and (110). 
 Also when we consider f as a function of input vectors (x_, x p , x , 
 
 g-,* g p > go) > then some of input vectors themselves are not completely 
 
 specified. For example, the second row of g._ in Table 1 is not specified. 
 
 In this case, we interpret the output values of f to assume the values 
 
 indicated in the column f, regardless of the assignments of unspecified 
 
 entries . For example, the second row of Table 1 shows that the actual 
 
 output value of f is 1, no matter whether the input vector is ( 001110 ) 
 
 or (001111), i.e., regardless of the value in column g which is blank 
 
 in Table 1. 
 
 Now let us introduce a definition. 
 
 Definition 2 : A column c. is said negative with respect to c. , c. , ..., 
 
 i i x i 2 
 
 idj) (d 2 ) 
 
 c. if and only if for every pair of entries c. and c. of c. such 
 s 
 
 (Sj) (J 2 ) 
 
 that c. =1 and c. '' = 0, there exists a subscript I (l < & < s) which 
 
 satisfies / . \ 
 
 (j,) 
 c = 
 
 and 
 
 (j p ) 
 c, - 1 . (2.3) 
 
Consider the column f of Table 1. The column f is not negative 
 with respect to x , x and x because for the pair of the eighth and the 
 
 fourth entries, f * = 1 and f* = 0, there is no subscript I (l < i < 3) 
 such that x^ = and x\ - 1. If x , x , x , g , g^ and g are 
 
 considered as variables of f, however, then f is negative as easily 
 verified. Note that the definition of the negativeness of a column is 
 valid no matter whether related columns are completely specified or not. 
 
 In the following, first we will prove that two concepts of 
 definitions 1 and 2, the negativeness of a function and that of a column, 
 are equivalent if a completely specified case is concerned (Theorem l) . 
 Then this equivalence will be extended to the incompletely specified case 
 (Theorem 2). With those arguments, all the discussions throughout the 
 following sections will be done in terms of the negativeness of columns. 
 
 The next definition of a negative function stated as Theorem 1 
 
 [2] 
 is a variant of Aker's theorem which gave a necessary and sufficient 
 
 condition for a function being a self -dual positive function. A 
 
 comprehensive discussion is also found in [8]. 
 
 Theorem 1 : A completely specified function h of m variables y_, y p , ..., y 
 
 is negative if and only if the corresponding column h in the truth table 
 is negative with respect to columns y_, y , ..., y . 
 
Proof : Let us prove that the condition for the column h to be negative 
 
 with respect to columns y_, y p , . .., y (Definition 2) is equivalent 
 
 Ui ) 
 to the condition that there is no pair of input vectors of h, y^ 1 and 
 
 y such that y > y and h(y ) = 1, h(y ) = (Definition l). 
 
 JSj) _(o 2 ) 
 
 Now note that y > y is equivalent to non-existence* of any 
 
 (jl) (J 2 ) 
 
 subscript £ such that y, =0 and y, =1. Therefore, if for every 
 
 Jjj) _(j 2 ) 
 
 pair of entries, h(y ) = 1 and h(y ) = 0, there exists a subscript 
 
 U-l) (j 2 ) 
 
 I such that y n = and Y » = 1, it implies that there is no pair of 
 
 Jjj) Jj 2 ) JdJ Jj 2 ) Jj-,) _Jj 2 ) 
 
 vectors y and y such that y > y and h(y ) = 1, h(y ) = 0, 
 
 and vice versa. QED. 
 
 In order to extend Theorem 1 to incompletely specified functions, 
 the next definition is introduced. 
 
 Definition 3 ' A completion of h is a completely specified function 
 obtained from h by specifying all the unspecified output values of h. If 
 a resultant completely specified function is negative, it is a negative 
 completion . 
 
 Theorem 2 : An incompletely specified function h of m variables y_, y p , ..., y 
 has a negative completion if and only if the column h in a truth table is 
 negative with respect to columns y_, y p , ..., y . 
 
 Proof: The only- if part is trivial. 
 
 # 
 
 ^ y is assumed for every pair of j and j p , j ^ j p , 
 
 10 
 
Now, consider an incompletely specified column h which is negative 
 with respect to y , y_, . .., y . First, if there are blank entries in 
 columns y , y p , . .., y , expand the table by assigning and 1 to every 
 blank entry, by muliplexing each row. (i.e. if a row is (01 — l) where - 
 denotes blank entries, replace this row by (01001), (01011), (01101), 
 (Ollll).) As a result, the table has 2 rows. The entries of the column 
 h in the multiplexed rows of the new table are set to the same values 
 as the original entries. This process takes care of input vectors which 
 are incompletely specified. The column h in the new table is obviously 
 still negative. 
 
 Next consider the case in which the entries in column h is incompletely 
 specified. Denote the sets of true vectors and false vectors of h as 
 follows : 
 
 (2.fc) 
 
 A = { y (j) | h(y (d) ) = 1} 
 and B = { y* | h(y' J = 0}. 
 
 There is no pair of vectors y and y such that 
 
 Jj x ) _Jd 2 ) ^(dJ Jo 2 ) 
 
 y > y and y e A, y e B, 
 
 Als) 
 
 from the negativeness of column h. Then pick up a vector y for which 
 
 the value h (y ) is not specified yet. y satisfies one of the following 
 
 three cases: 
 
 —■( r>) 
 (l) There exists a vector y such that 
 
 y (p) e A andy (p) >y (k) . (2.5) 
 
 11 
 
(2) There exists a vector y such that 
 
 y (q) eB andy (q) <y (k) . 
 
 (3) None of (l) and (2) holds. 
 If case (l) occurs, then h(y ) is specified to 1. Then the resulting 
 function h is characterized by its set of true vectors and set of false 
 vectors; 
 
 A' = A U {y^} 
 and 
 
 B' = B. 
 
 The new column h' is still negative with respect to columns y , y , . .., y , 
 because, otherwise the existence of vectors such that 
 
 y (k) >y (j) , y (k) € A', y (j ' } e b«, 
 leads to a contradiction, i.e., 
 
 y(p) > y(j) ^ y(p) g^ ^g ^ 
 
 from condition (2.5). 
 
 If (2) is the case, then h(y ) = is assigned. The proof is similarly 
 
 obtained. Finally , if (3) is the case, either or 1 is assigned to 
 
 -(k) 
 
 h(y ). Either case still keeps the column negative, even though one 
 
 more entry is specified. This is because the non-negativeness of h' 
 means the existence of 
 
 y (k) > y (i) , y (k) € A' , y^ e v 
 
 12 
 
(We assume h ' (y* ') = 1. The case of h'(y )= will be treated similarly.) 
 which is a contradiction since this y satisfies the above case (2). 
 
 The assignment to the next unspecified entry of h' can be performed 
 similarly. Repeating this process, we will eventually obtain a negative 
 completion of h. QJED. 
 
 As a result of Theorems 1 and 2, we learned that the negativeness 
 
 of a column h with respect to columns y_ , y_, .... y means that function 
 
 1 2 m 
 
 h represented by column h in the truth table is a negative function 
 (or has a negative completion) of variables y , y p , ..., y . Thus the 
 negativeness of completely specified function has been extended to the 
 case of incompletely specified function. 
 
 Since the completion procedure described in the proof of Theorem 2 
 does not always yield the unique negative completion, any negative 
 completion will be assumed in the following discussion because we are 
 concerned with only negativeness of each function. 
 
 13 
 
3. Realization of f With Negative Functions 
 
 If a given function f is not negative, more than one gate is 
 needed to realize f with negative functions. As outlined in Section 2, 
 our algorithm for the optimum network is based on the table manipulation. 
 It first augments the given truth table with columns g , gL, ..., gi 
 
 such that 
 
 (1) the column f is negative with respect to x , x , . .., x , 
 g l' ' * * ' g k* 
 
 (2) the columns g., i = 1, 2, ..., k, are negative with 
 
 respect to x_ , x^, .... x . 
 r 1' 2 n 
 
 Provided that those columns g , gp, ..., g with the above properties 
 
 are obtained, the corresponding network for f with negative functions 
 
 is shown in Fig. h. Each gate is the first level corresponds to one 
 
 of g. 's, and the gate in the second level realizes f as a function of 
 
 x l' X 2' " * " ' X n' ^l^ * * ' ' g k" 
 
 In this section, we will discuss how to obtain g , g^, ..., g 
 which satisfy the above requirements. The minimization of the number of 
 gates will be discussed in the subsequent sections. 
 
 1U 
 
(jj (j 2 > 
 
 Definition k: A pair of entries c. =1 and c. '" = in a column c. 
 * 1 l l 
 
 is unallowable with respect to c. , c. , . . . , c. if and only if there 
 
 X l X 2 X s 
 
 is no subscript i5(l<i<s) such that 
 
 (J-l) (d 2 ) 
 
 c. =0 and c. = 1. (3«l) 
 
 (J 1 ) (j' 2 ) 
 If there exists a subscript i which satisfies (3«l)> c. and c. is 
 
 said allowable due to c. . 
 
 Note that the column c. is negative with respect to columns 
 
 c. , c. , .... c. if and only if any pair of entries of c. such that 
 i- i ' i l 
 
 12 s 
 
 (jj) (J 2 ) 
 
 c. =1 and c. = is allowable, 
 l i 
 
 Now let us form the negativity table of function f of n variables 
 x , x p , ..., x . The negativity table is constructed according to the 
 following procedure: 
 
 (1) Exhaust all unallowable pairs of column f with 
 
 respect to columns x n , x^,, .... x , in the truth 
 
 12 n 
 
 table of f . 
 
 (2) For each unallowable pair of entries, f(x ) = 1 
 
 Jj 2 ) 
 and f (x ) = 0, add a new column e . whose entries 
 
 J 
 
 (j ] _) (j 2 ) 
 are e . '" = 0. e . ' = 1 and blank in all other entries. 
 
 (i.e. e. has exactly two entries which are not blank.) 
 J 
 
 e . is called a primitive additional col umn of f. 
 
 15 
 
For example, the function f shown in Table 1 has three unallowable 
 pair of entries of f, as easily seen. For these three unallowable pairs, 
 three columns e , e p , e are added to the truth table. The negativity- 
 table of f which results is shown in Table 2. 
 
 f 
 
 x l 
 
 X 2 
 
 X 3 
 
 e l 
 
 6 2 
 
 e 3 
 
 1 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 1 
 
 
 
 
 1 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 1 
 
 1 
 
 1 
 
 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 1 
 
 
 1 
 
 
 
 
 1 
 
 1 
 
 
 
 
 
 l 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 
 
 Table 2 Negativity table of f = y v x p x q '' x i x -3 v x i x p x -5 
 
 Because of the way each e . being generated, the next theorem is 
 
 J 
 obvious. 
 
 Theerem 3 - Let e , e , ..., e be all the primitive additional columns 
 
 generated for the negativity table of f of n variables x , x , ..., x . 
 
 Then f is negative with respect to x n , x_, ..., x , e n , e , ..., e. 
 
 1 2' n 1 2 t 
 
 Another property of negativity table is given next. 
 Theorem k : Each primitive additional column e . of a negativity table 
 
 J 
 
 is negative with respect to columns x n , x^, ..., x . 
 
 1' 2 n 
 
 Proof: Assume that e. is added corresponding to the unallowable pair, 
 
 J 
 Jjn) Jj 2 ) 
 
 f x ) = 1 and f(x ) = 0, to form the negativity table of f. 
 
 16 
 
(j-,) (d 2 ) (jj) 
 
 Then e. =0 and e. '" = 1, There is no I such that x„ =0 and 
 3 3 i 
 
 x„ = 1. Hence there must be a subscript r (l < r < n) such that 
 
 (jj) (o 2 ) 
 
 x =1 and x =0. 
 r r 
 
 (3.2) 
 
 Condition (3»2) shows that column e. is negative with respect to 
 
 J 
 
 columns x.. , x_, . .., x . 
 
 12 n 
 
 QFD. 
 
 Theorems 3 and k indicate that each e . can be used as a negative 
 
 J 
 
 function g. in Fig. k. Of course, this realization needs ( t + l) negative 
 
 J 
 
 functions, where t is the number of columns e ., and may not be minimum. 
 
 J 
 
 Definition 5 : A column c. is said to cover column c. if and only if for every 
 
 J 
 
 entry c. which is specified, 
 
 d 
 
 (*.) (in) 
 
 c. = c. 
 
 holds. (if an entry of c. is not specified, the above equality does not 
 
 J 
 
 necessarily hold.) 
 
 For example, in Table 3, c, covers c_ but does not cover c . c 
 
 covers neither c nor c , etc. 
 
 c l 
 
 C 2 
 
 c 3 
 
 1 
 
 
 1 
 
 
 
 
 
 
 
 1 
 
 1 
 
 1 
 
 Table 3 Illustration of the covering relation 
 
 IT 
 
The motivation of introducing this concept lies in the fact that if 
 
 a column g covers a column e . of the negativity table and is negative with 
 
 J 
 
 respect to columns x , x p , ..., x , then we can replace e . by g. (i.e. 
 
 f is negative with respect to x n , x_, .... x , e n , .... e. _,g, e. _. ... e, 
 
 ^ V 2' ' n' 1' ' j-1 j+1 1 
 
 and g is negative with respect to x , x , ..., x .) 
 
 Hence, if g covers more than one e., all these e. can be replaced by 
 g reducing the number of negative functions in the network as shown in 
 Fig. k. 
 
 Again, the function f given in Tables 1 and 2 is considered. For this 
 function f, assume the column as shown in Table k. 
 
 f 
 
 x l 
 
 X 2 
 
 X 3 
 
 M 
 
 1 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 1 
 
 
 1 
 
 
 
 1 
 
 
 
 
 
 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 1 
 
 
 
 1 
 
 1 
 
 
 
 1 
 
 1 
 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 Table k 
 
 Realization of f = x..x v x o x o ^ x "\ x ? i ^ x -| x o x o 
 
 As easily checked, M is negative with respect to x_ , x p , x and moreover M 
 
 covers e , e p , e shown in Table 2. Therefore, all the three negative 
 functions e , e , e of variables x , x p , x can be replaced by a single 
 negative function M of variables x.., x , x . Fig. 2 will result by using 
 the negative completion of M (see Table 5 in the next section); 
 
 M = x 1 v x p ^ x_. 
 
 18 
 
In the next section, a systematic generation of the minimum number 
 of negative functions which together cover all of e , e , ..., e, in the 
 negativity table will be discussed. 
 
 19 
 
k. Minimum Network by Maximal Compatible Sets 
 Let us begin with a few new definitions. 
 Definition 6: The conjoint of two columns c. and c ., denoted as c„ 
 
 is defined by the following rule for each r: 
 
 M =c (r) Vc (r) 
 
 c. V c 
 
 ■P, 
 
 (r) v (r) . . 
 where c. v c. is given by 
 i J 
 
 
 1 
 
 
 
 
 1 
 
 1 
 
 9 
 
 1 
 
 
 
 1 
 
 
 
 
 
 
 1 
 
 
 
 
 The mark ? in the table shows that the operation is not defined. And blank 
 entry denotes unspecified entry. 
 
 If all entries of c, are defined, (i.e. 1, or blank but not ?) c 
 
 Xj Xj 
 
 is said definable . The conjoint of more than two columns can be defined 
 by the following inductive formula: 
 
 V c. V ... V c. = (« 
 
 V V 
 
 X l X 2 
 
 1-L i 2 
 
 c. 
 
 ) v 
 
 L k-1 
 
 L k. 
 
 Definition 7: Two columns c. and c. are compatible with respect to c. , 
 
 l j i>^ 
 
 c , . . . , c D if and only if c . V c . is definable and also negative with 
 "p s ^ J 
 
 respect to c» , c * , ..., c« . 
 
 Xj-, Xji~\ Xj 
 
 1 d- S 
 
 20 
 
We will examine the compatibility between each pair of columns, 
 e. and e . , in the negativity table. 
 
 Definition 8 : Consider a set of columns 
 
 C. ■ { c. i c. , ..., c. } . (^.l) 
 
 If and only if every two columns in C. are compatible, with respect to a set 
 
 of columns B,C. is called compatible with respect to B. A compatible set 
 
 C. covers a column c.if and only if c. V c. V ... V c. covers c . 
 i j ± 1 i 2 \ 3 
 
 C. of (U.l) is essentially c. V c. ^ ...V c. according to the 
 1 X l X 2 \ 
 
 following lemma. 
 
 Lemma 1 C. defined by (k.l) is compatible with respect to a set of columns B 
 
 if and only if c. ^ c. ^... ^c. is definable and negative with respect 
 X l x 2 \ 
 
 to B. 
 
 The proof is trivial and hence omitted. 
 
 Definition 9 : Consider all the compatible sets with respect to B. A 
 
 compatible set C. is said maximal if, and only if among these sets, there is 
 
 no other compatible set with respect to B, which includes C 
 
 Throughout this paper we will deal with only compatible sets or 
 maximal compatible sets of primitive additional columns e , e , ..., e 
 
 JL cl u 
 
 in the negativity table with respect to x_, x p , ..., x . Therefore, we will 
 refer to these simply as compatible sets or maximal compatible sets by 
 dropping the phrase "of primitive additional columns e , e , ..., e in the 
 negativity table with respect to x , x p , ..., x ." 
 
 21 
 
For example, e.., e ? and e of Table 2 are pairwise compatible. 
 Hence M = { e , e ? , e } is a compatible set. Obviously this is maximal. 
 
 The concept of maximal compatible sets is important because of the 
 next theorem. 
 
 Theorem 5 : There exists a network with the minimum number of gates for a 
 given function f in which every gate represents a negative function and those 
 in the first level represent negative completions of maximal compatible sets. 
 Proof : Consider the negativity table of f. e , e , . .., e, are all primitive 
 additional columns. Assume that the realization as shown in Fig. k is 
 minimum, i.e., k gates in the first level represent g , g_, ..., g and 
 (k+l) is the minimum number of gates which can realize f. Because the 
 last gate represents a negative function with respect to g , g , ..., gL , 
 x , x p , ..., x , any e. in the negativity table must be covered by same 
 
 -I- £— 11 (J 
 
 of g . . Now let us suppose that 
 
 g l COverS e ll' Q 12> ••" e l^ 
 gp covers e 2± , q^, ..., e 
 
 2 
 
 g k covers e^, e^, ..., e^ 
 
 k 
 
 Then since each g. is negative with respect to x , x , ..., x , and covers 
 
 e. n V e„ V ... V e. . , [e. n , e.^, ..., e.. } is a compatible set for 
 
 il i2 in. il i2 ' in. 
 
 °2. 1 
 
 i = 1, 2, ..., k (Lemma l) . Therefore for every { e, e, ..., e.. }, 
 
 J i 
 
 i = 1, 2, ..., k, it is possible to consider a maximal compatible set 
 
 which includes {e..,, e.~, ..., e.. }. 
 il' i2' ' in. 
 
 °i 
 
 22 
 
respectively. Let us call them as M_, IYL, . .., M. . Finally we can replace 
 each g. in the network by a negative completion of M. . After this replacement, 
 
 f is still negative with respect to M_, NL, . .., K, x_, x , . .., x , where 
 
 M. is a negative completion of M.. This network also gives a minimum network 
 
 since the new network uses the same number of gates as the network of 
 
 g , g^, . . . ,a which is minimum. QED. 
 
 Because of this theorem, the synthesis of a negative function 
 network is done by first generating all the maximal compatible sets and 
 then selecting the minimum number of sets from them, which together cover 
 all the primitive additional columns e , e , ..., e . The generation 
 and selection of maximal compatible sets will be discussed in the next 
 section. 
 
 Let us consider the previous example given in Table 1. Table 2 is 
 the negativity table of f . We have seen that M = { e , e , e } is the 
 
 maximal compatible set and no other maximal compatible set exists. M 
 
 is the one shown in Table k. Now, we can realize f with two gates, one 
 
 is for M and the other is the last gate which gives f . A negative completion 
 
 of M, M , is shown in Table 5« 
 
 Table 5 
 Completion of M 
 
 f 
 
 x l 
 
 X 2 
 
 x 3 
 
 * 
 
 M 
 
 1 
 
 
 
 
 
 
 
 * 
 
 1 
 
 1 
 
 
 
 
 
 1 
 
 1 
 
 1 
 
 
 
 1 
 
 
 
 1 
 
 
 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 * 
 
 1 
 
 
 
 1 
 
 
 
 1 
 
 1 
 
 
 
 1 
 
 1 
 
 
 
 1 
 
 ' 1 
 
 1 
 
 1 
 
 1 
 
 
 
 23 
 
The entries of M with stars were newly specified, following the procedure 
 given in the proof of Theorem 2. (All entries were determined uniquely 
 
 at 
 
 by cases of (l) and (2) of the proof of Theorem 2.) M is written as 
 
 M = x. v x v x . 
 
 Note that f is now incompletely specified if it is regarded as a function 
 of x,, x , x„ and M . A negative completion easily obtained by inspection 
 
 is 
 
 f = xx v xx v xx v M . 
 
 These completions of M and f give the network shown in Fig. 2. The 
 procedure described so far explains how the minimum network of Fig. 2 
 was synthesized. 
 
 2k 
 
5. Generation and Selection of Maximal Compatible Sets 
 
 Let us start this section with a new example whose negativity 
 table is shown in Table 6. 
 
 f 
 
 x l 
 
 X 2 
 
 x 3 
 
 e l 
 
 6 2 
 
 e 3 
 
 % 
 
 e 5 
 
 e 6 
 
 e 7 
 
 1 
 
 e 8 
 
 
 
 
 
 
 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 1 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 1 
 
 l 
 
 
 
 
 
 
 
 1 
 
 1 
 
 
 
 
 
 
 
 1 
 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 1 
 
 
 
 
 
 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Table 6 
 Negativity table of f = x.x ? v x i x -5 v x i x ? x -3 
 
 From primitive additional columns e , e p , . .., e, specified in a 
 negativity table, we will construct the compatibility matrix of f. The 
 compatibility matrix is a triangular matrix in which columns correspond 
 to e_, e , ..., e, and rows correspond to e , e , ..., e , as shown in 
 Table 7. A cross is entered if e. and e. which correspond to the row 
 and the column of the entry are not compatible. Otherwise it is left 
 as blank. The compatibility matrix of f derived from Table 6 is given 
 in Table 7 • 
 
 25 
 
e 2 
 
 
 
 
 
 
 
 e 3 
 
 
 
 
 
 
 % 
 
 
 
 
 
 
 e 5 
 
 
 
 
 
 
 % 
 
 
 
 
 
 
 
 e 7 
 
 X 
 
 
 
 
 
 
 
 e 8 
 
 X 
 
 X 
 
 
 
 
 
 
 
 e l 
 
 e 2 
 
 v 
 
 % 
 
 e 5 
 
 e 6 
 
 e 7 
 
 Table 7 
 Compatibility matrix of 
 
 X 1 X 2 y X 1 X 3 V X 1 X 2 X 3 
 
 The compatibility matrix tells us that all pairs of primitive 
 additional columns e. and e. are compatible except three pairs (e ,e ), 
 
 (e^eg) and (e 2 ,eg). 
 
 With the aid of compatibility matrix, we can generate all the 
 maximal compatible sets in the following way. Let us consider primitive 
 
 additional columns e , e , 
 
 > V 
 
 (1) 
 
 (2) 
 
 Obtain all the maximal compatible sets of e only. 
 Obiviously there is only one maximal compatible set 
 
 [e^. 
 
 Given all the maximal compatible sets M , M , ..., 
 
 of e n , e~, .... e, (l < k < t-1 ), introduce e. _ 
 12' k — — k+1 
 
 \ 
 
 and form the sets according to the following rule for 
 each i, i = 1, 2, ..., b. 
 
 26 
 
(a) If the set M.U {e _} is compatible, i.e., 
 e n is compatible with every element of 
 M., then form 
 
 (b) Otherwise let M. "be the set of all the 
 
 elements of M. which are compatible with 
 
 e, ., « Then form two new sets: 
 k+1 
 
 11 
 
 M. = M. , 
 l l ' 
 
 M i" ■ "i u {e k + i' • 
 
 (3) Delete all the sets which are included in some other 
 
 sets generated by (2) above. Note that only M. could 
 
 be included in other set. The remaining sets are all the 
 maximal compatible sets of e , e , . .., e . 
 
 {h) Increase k by one. If the new k is equal to t, the procedure 
 
 is completed, otherwise go to (2) again. 
 
 In the above recursive procedure, it is not obvious whether all 
 
 the maximal compatible sets of e_, e , ..., e are generated after 
 
 completing (3). This will be affirmatively proved in Theorem 6. For 
 
 illustration let us obtain all the maximal compatible sets of e_ , e^,, .... e 
 
 12 o 
 
 whose compatibility relations are given in Table 7« The generation 
 
 procedure is illustrated in Fig. 5» 
 
 27 
 
Fig. 5 shows that only one maximal compatible set is generated 
 for each k not more than 6, since any two of e_,e , ...,e,- are compatible 
 as seen from Table 7« When e„ is added, two maximal compatible sets 
 result 
 
 k = 1 2 3 k 5 6 
 
 {e 1 } -* {e^e^ -* {e^e^e ) -* {e^e^e^e^} -* {e^e^e^e^e^} -* {e^e^e ,e^,e ,eg} 
 
 [e l> e 2> e 3>%> e 5> e 6 ] 
 
 i Q 2 ,e y e k' e 5' e 6' e 7^ 
 
 te l' e 2' e 3'VV e 6 } 
 {e 2 ,e 3' e ^ e 5 ,e 6 ,e 7 } 
 
 {e 3' e VV e 6'V e 8 } 
 
 7 k = 8 
 
 Fig. 5 Generation of maximal compatible sets of f given by 
 Table 6. 
 
 since procedure 2(b) is applied because of non-compatibility of e with e . 
 
 Finally for k = 8, three maximal compatible sets are obtained. 
 
 {e ,e,,e ,e/,eJ is deleted because this is included in {e ,e,,e ,e^,e ,eo)» 
 
 Theorem 6 : Assume that M_, M_, •••> ^L. are a H ^ ne maximal compatible 
 sets of e., e p , ..., e,» All the sets obtained as a consequence of the 
 above steps (2) and (3) are maximal compatible sets of e ,e , ..., e . 
 
 -L c. .Kt-L 
 
 Moreover there is no other maximal compatible set of e , e , ...,e , -, • 
 
 28 
 
Proof : To prove that the above procedure generate all the maximal sets, 
 let us assume that M is a maximal compatible set of e , e p ,...,e and 
 
 is not generated by steps (2) and (3) in the above procedure. M must 
 contain e _ because otherwise M is equal to one of M , M , . .., M, . 
 
 Let M' be M from which e is deleted. Then M' is included in some of 
 M ,Mp,...,M, because any two elements in M' are compatible. Let M. jz>M' 
 
 for some i of i = 1, 2, ...b. Then M' must be included in or equal to 
 
 t ' ' ' 
 M. or M. of step (2). This is a contradiction with the initial assumption 
 
 that M is maximal and is not generated by steps (2) and (3). Next it is 
 
 obvious that after step (3) no compatible set which is not maximal remains. QED. 
 
 Definition 10 : A minimal compatible cover of e , e , ...,e for f is a 
 set of the minimum number of maximal compatible sets such that every e. 
 belongs to at least one of maximal compatible sets chosen. 
 
 Now note that to find a minimal compatible cover is equivalent to 
 designing a minimal network of negative functions. However, finding 
 a minimal compatible cover is nothing but a set-covering problem which 
 has been discussed in the literature in conjunction with a variety of 
 problems. 
 
 Let M_,]VL,...,M be all the maximal compatible sets. The covering 
 
 table is a table, the i th row of which corresponds to e. and the j th 
 
 column of which corresponds to M.. The (i, j) th entry of the covering 
 
 J 
 
 table is 1 if e. is covered by M., and blank otherwise. The set-covering 
 
 J 
 
 problem is to find the minimum number of columns so that every row has an 
 entry 1 in at least one of the selected columns. A number of techniques to 
 solve the set-covering problem through the manipulation of the covering 
 table are discussed in [7], for example. 
 
 29 
 
Another interesting approach is the integer programming approach. 
 The formulation of a set covering problem "by an integer programming is 
 given in [3][*+L f° r example. Recent computational result shows that 
 some problems which correspond to the cases (30 rows, 50 columns) and 
 (30 rows, 90 columns) are solved in 0.15 seconds and 1.60 seconds 
 respectively on the IBM 3^0/751 computer. 
 
 The covering table for the above example (Table 6 and Fig. 5) 
 is now shown as an example. 
 Let 
 
 M x = {e^ e 2 , e , e^, e , e^} 
 
 V^ = {e 2 , e , e k , e , e g , e } 
 
 M 3 = {e 3 ' %> e 5 > %> V S 8 } 
 
 Then the covering table is: 
 
 
 M l 
 
 Mg 
 
 M 3 
 
 e l 
 
 1 
 
 
 
 6 2 
 
 1 
 
 1 
 
 
 e 3 
 
 1 
 
 1 
 
 1 
 
 % 
 
 1 
 
 1 
 
 1 
 
 e 5 
 
 1 
 
 1 
 
 1 
 
 e 6 
 
 1 
 
 1 
 
 1 
 
 e 7 
 
 
 1 
 
 1 
 
 e 8 
 
 
 
 1 
 
 Table 8 
 
 Covering table of the 
 example given by Table 6 
 and Fig. 5* 
 
 * The correspondence is not exact in the sense that the matrices are 
 generated randonly and may not represent switching functions. 
 
 30 
 
Obviously IVL and IVL constitute the minimal compatible cover of 
 e ,e p ,...,en which is unique in this particular example. 
 
 When IVL and M are considered as functions of variables X-^x^x , 
 
 *■ -*■ 
 their negative completions M , M are given in Table S. 
 
 f 
 
 x l 
 
 X 2 
 
 x 3 
 
 -* 
 
 "l 
 
 -* 
 M 3 
 
 
 
 
 
 
 
 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 1 
 
 
 
 1* 
 
 
 
 
 
 1 
 
 
 
 1 
 
 1 
 
 
 
 
 
 1 
 
 1 
 
 0* 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 
 
 1* 
 
 
 
 1 
 
 
 
 1 
 
 0* 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 Table 9 
 Completion of VL and M 
 
 -x- * 
 
 M and M are written as 
 
 M l = x l x 3 
 
 M = x ± v x^ . 
 
 The minimum network of f using M and M as gates in the first level is 
 given in Fig. 6. 
 
 31 
 
_* _■*,__ 
 
 f = M 3 v M^ (x^ ^ 5^x 3 ) 
 
 Fig. 6 A minimum network of f given by Table 6. 
 
 Up to this point, an entire procedure to obtain a minimum network 
 with a single output has been presented. Next two sections will deal 
 with a case in which multiple outputs are required and some topics 
 related to the minimization of the number of connections and interconnections. 
 
 32 
 
6. Minimum Multiple Output Networks 
 
 One of the advantages of our approach is that it can he easily 
 extended to the synthesis of a mutiple output network. When we are 
 to realize more than one non-negative function , only necessary modification 
 is that primitive additional columns e. are generated in the negativity 
 table for each given function. The rest of procedure, i.e., the generation 
 of maximal compatible sets, the selection of a minimal compatible cover 
 and the completion of each selected maximal compatible set, is exactly 
 the same as that discussed so far. The procedure will also result in a 
 network with the minimum number of gates. 
 
 Let us obtain a minimum network of two functions: 
 
 f l " X 1 X 2 v X 1 X 3 v X 1 X 2 X 3 
 
 f 2 = X] ,x 2 - Xl x 3 - x 2 x 3 . 
 
 (1 
 (2 
 (3 
 
 (k 
 
 (5 
 (6 
 (7 
 (8 
 
 f l 
 
 f £ 
 
 x l 
 
 X 2 
 
 X 3 
 
 e l 
 
 6 2 
 
 e 3 
 
 % 
 
 e 5 
 
 e 6 
 
 e 7 
 
 e 8 
 
 e 9 
 
 *• 
 
 "l 
 
 -* 
 M 3 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1* 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 l 
 
 1 
 
 
 
 1 
 
 1 
 
 
 
 
 
 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 l 
 
 
 0* 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1* 
 
 
 
 1 
 
 1 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 1 
 
 2 
 
 1 
 
 1 
 
 l 
 
 1 
 
 l 
 
 1 
 
 
 
 
 
 
 
 
 2 
 
 2 
 
 
 • 
 
 2 
 
 
 2 
 
 
 
 
 
 
 Table 10 Negativity table of f and f 
 
 * When some of functions to be realized are negative, the synthesis would be 
 simpler. In other words each function can be realized by a single gate. 
 Furthermore, the outputs of gates for these negative functions can be used as 
 inputs to other gates which are to realize non- negative functions. The 
 modification of algorithm required is straightforward, however, and will 
 not be given here. 
 
 33 
 
The entries given underneath primitive additional columns e , e , ...,e 
 
 L d y 
 
 in Table 10 show from which of f and f the primitive additional column 
 
 is generated. For example, the column e.. is generated from both f and f p . 
 
 The compatibility matrix is shown in Table 11. The generation of all the 
 maximal compatible sets is also shown in Fig. r J. 
 
 e 2 
 
 
 
 
 
 
 
 
 
 e 3 
 
 
 
 
 % 
 
 
 
 
 
 e 5 
 
 
 
 
 
 
 e 6 
 
 
 
 
 
 
 
 e 7 
 
 
 
 
 
 
 
 
 e 8 
 
 X 
 
 
 
 
 
 
 
 
 e 9 
 
 X 
 
 X 
 
 X 
 
 
 
 
 
 
 
 e l 
 
 6 2 
 
 e 3 
 
 e u 
 
 e 5 
 
 % 
 
 e 7 
 
 e 8 
 
 Table 11 Compatibility matrix of f and fp 
 
 3^ 
 
{e^ -* (e^eg) - ... -* [e^e^eye^e ,e^,e ) 
 
 ¥ (e l ,e 2 ,e 3'W e 6 ,e 7 ] 
 
 9 ] 
 
 (e 1 ,e 2 ,e 3 ,e u ,e 5 ,e 6 ,e 7 ] - _^ {eTT^zS^J 
 
 {e 2 ,e 3 ,e^,e 5 ,e 6 ,e 7 ,eg} 
 
 M 2 ={e 2 ,e 3'V e 5 ,e 6 ,e 7 ,e 8 } 
 
 M 3 ={€ VV e 6'V e 8' e 9 ] 
 
 Fig. 7« Generation of maximal compatible sets of f and f 
 
 Minimal compatible covers can be determined in the same way as the 
 single output network problem. The covering table consists of 9 rows 
 corresponding to e ,e , ..., e and 3 columns corresponding to M_,M 
 
 and M_. The minimal compatible cover for this problem is tM^M^J . 
 
 Negative completions of M and M are also shown in Table 10. M_ and 
 
 M_ are written as 
 
 * 
 
 "l~3 
 
 M. 
 
 * 
 M = x x v x 2 . 
 
 ■i and M 3 
 
 A minimum network of f , and f„ obtained from those M_ and M 
 
 is given in Fig. 8. f p does not require 
 
 35 
 
X 
 
 X 
 
 X 
 
 
 x 2 
 
 x. 
 
 
 
 
 
 
 J 
 
 
 
 
 M* - »- Y_ 
 
 > 
 
 1 
 
 "l 
 
 "J. "3 
 
 
 
 
 
 
 
 
 
 3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 
 
 3 
 
 M ! = 
 
 X;L -x 2 
 
 
 X l 
 
 ■ ■ ^ J 
 
 
 
 
 2 
 
 
 
 3 
 
 
 „ f 2 = M* (£gV x ) 
 
 f x = M* sy M* (X-jXg N/ XgX ) 
 
 Fig. 8 Minimum network of f and f 
 
 M because all the primitive additional columns e ,e p ,e ,e.,e^ generated 
 from f _ can be covered by a single maximal compatible set M . (A more 
 rigorous discussion will be found in the next section. ) 
 
 36 
 
(e^ -* [e lf e 2 ) -* ... ■* {e^egje^e^e^e^e ) ■* 
 
 [e lf e 2 ,e f e^,e ,e^,e^) 
 
 M^U^ V e 3' %' V %> e 7^ 
 
 ( e 2 , e^, e^, e,_, eg, e^, eg} 
 
 M 2 ={e 2 ,e 3 ,e u ,e 5 ,e g ,e T ,e 8 ) 
 
 M 3 ={e V e 5 ,e 6 ,e 7 ,e 8 ,e 9 } 
 
 Fig. 7« Generation of maximal compatible sets of f and f 
 
 Minimal compatible covers can be determined in the same way as the 
 single output network problem. The covering table consists of 9 rows 
 corresponding to e , e p , ..., e and 3 columns corresponding to 1YL,M 
 
 and M_. The minimal compatible cover for this problem is IM-^M^J • 
 
 Negative completions of M_ and M are also shown in Table 10. M.^ and 
 
 1VL are written as 
 
 * 
 
 V3 
 
 M. 
 
 M = x x v x 2 . 
 
 '1 and M 3 
 
 A minimum network of f„ and f„ obtained from those VL and M 
 
 is given in Fig. 8. f p does not require 
 
 35 
 
X 
 
 
 
 x 2 
 
 X 3 — 
 
 
 
 
 
 M"* — V"- Y_ 
 
 11 > 
 
 1 
 
 "l 
 
 -JL "3 
 
 
 
 
 
 
 
 
 
 3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 
 
 3 
 
 M 3 = 
 
 x 1 ^ ic 2 
 
 
 x l 
 
 x 2 
 
 .> J 
 
 
 
 
 
 
 3 
 
 
 > f 2 = % ( *2 V S } 
 
 ^ = M* sy M* (x x x 2 v a^x ) 
 
 Fig. 8 Minimu.ii network of f and f 
 
 M "because all the primitive additional columns e ,e p ,e ,ei , e^ generated 
 from f p can be covered by a single maximal compatible set M . (A more 
 rigorous discussion will be found in the next section. ) 
 
 36 
 
7. Some Considerations on the Reduction of Interconnections . 
 
 Although we can minimize the number of gates in a network by 
 following the method presented so far, no consideration was given on the 
 minimization of the number of interconnections. 
 
 The overall minimization of the number of interconnections (between 
 the first and second levels) and connections of inputs to the first and 
 second levels (from variables x , x , . .., x ) seems difficult at present 
 unless we exhaust all the possible networks and then pick up the best one 
 among them. Here we will discuss two cases in which the number of 
 interconnections and connections can be minimized in a restricted sense. 
 One is minimizing the number of interconnections between the first and 
 second levels, when designing a multiple output network. The other is the 
 
 minimization of the number of connections from variables x.. , x«, .... x 
 
 1' 2' ' n 
 
 fed to the first level. 
 
 Let us consider the former problem. 
 
 The minimality criterion is to find a network with the minimum 
 number of interconnections between the first and second levels among those 
 with the minimum number of gates. 
 
 Note that we are not concerned about the number of connections 
 
 fed from the external variables x n , x n , ..., x . 
 
 12' ' n 
 
 The problem is trivial if the network has a single output, because 
 it is then the same as minimizing the total number of gates in the network. 
 
 The following property is useful for our problem. 
 Theorem 7 : Under the minimality criterion as stated above, there is a 
 
 minimum network in which each gate in the first level represents a negative 
 completion of maximal compatible set. 
 
 37 
 
The proof is similar to that of Theorem 5. This theorem enable 
 us to limit our consideration only to maximal compatible sets. 
 
 Before discussing the whole scheme of minimization, assume that a 
 
 minimal compatible cover for functions f , , f p , ... f which are to be 
 
 realized is already determined by the method described in Section 6. The 
 
 minimal compatible cover consists of maximal compatible sets, M-.,Mp, ..., M . 
 
 Then in the negativity table of f , , f~, ... f , it is shown which of 
 
 primitive additional columns e. are necessary for each function f . (see 
 
 the bottom row of Table 10). Every primitive additional column must 
 
 be covered by some of maximal compatible sets chosen above. However, some 
 
 function f . may not require all the primitive additional columns for f . to 
 J J 
 
 be negative and therefore some of maximal compatible sets may be omitted 
 
 for f.. 
 
 Assume that e , e , ..., e , j = 1, 2, ..., q, are necessary 
 
 J 
 
 in order for f . to be negative. The interconnections from the first 
 
 J 
 
 level to the output gate for each f . can be minimized by selecting the minimum 
 number of maximal compatible sets out of the minimal compatible cover 
 {1VL, 1VL, ..., M } chosen for f . , f p , ..., f , such that each e . ., 1 < £ < k., 
 is covered by some of selected maximal compatible sets. Apply this process 
 for j = 1, 2, . . . , q. This is again a set-covering problem and can be 
 solved similarly. 
 
 If the minimal compatible cover is unique, the network obtained 
 above is the minimum network in the present sense. Otherwise, we have to 
 apply the above procedure to every minimal compatible cover and then 
 find the best one among those obtained. This procedure is not so inefficient 
 as it appears. 
 
 38 
 
Consider the example given by Table 10. This example has the 
 unique minimal compatible cover (M , M }. First consider f . It requires 
 primitive additional columns e , e , e., e , e^, e , e^, e Q for f to 
 
 be negative as shown in Table 10. This means that f needs both M and 
 M„ to cover those 8 primitive additional columns. On the contrary, f 
 
 needs only e , e , e , e, , e^ in order to be negative, and these e. are 
 
 all included in one maximal compatible set M_ = {e_, e , e„, e, , e,_, e^, e„} 
 
 l 1 d $ 4 p o ( 
 
 * - - 
 
 In fact, the realization of Fig. 8 shows that f p uses only M = x x„ 
 
 although f uses both NL = x x and M = x_ ^ x . 
 
 Next the same problem is formulated as an integer programming 
 problem. Although the formulation seems neater than that given above, 
 probably it will take more computation time, judging from the present 
 status of integer programming. 
 
 Let us assume that M , IVL, ..., M are all the maximal compatible 
 
 sets for f , f p , ..., f . Each f . requires primitive additional columns 
 
 e. n , e „, ..., e .. in order to be negative. Let us introduce zero-one 
 jl' jd Jk^. 
 
 variables € . ., 3 = 1> 2, . . . , q, i = 1, 2, . . . , p € assumes 1 if 
 
 M. is selected and actually used to cover some of e._, e ._, ..., e ., 
 
 J 
 
 for f ., and otherwise. Then prepare Table 12, by filling 1 in the entry 
 
 J 
 
 a, „,, . if e .is included in M., otherwise, and a, „ .. = if k J= j. 
 
 kiki ki i &£, ji ' 
 
 Let us denote the matrix of these entries with A. Then consider the set 
 
 q 
 
 of 2k. inequalities with p»q variables £.. 
 
 d=i 3 J1 
 
 A € > 1 . (7.1) 
 
 39 
 

 
 
 
 
 1 
 
 
 for f 
 
 
 for fg 
 
 
 for f 
 
 
 € 11 € 12 •*• 
 
 e lp 
 
 € 21 € 22 '•* G 2p 
 
 ■ • • 
 
 € ql e q2 • " € qp 
 
 e ll 
 
 
 
 
 
 
 6 12 
 
 • 
 
 A^ 
 
 
 
 
 • * • 
 
 
 
 X 
 
 
 
 
 
 
 e 21 
 
 
 
 
 
 
 e 
 22 
 
 
 
 
 A< 2 > 
 
 • • • 
 
 
 
 6 2k 2 
 
 
 
 
 
 
 • 
 
 ! 
 
 
 • 
 
 • 
 
 • 
 
 • 
 
 • 
 
 
 • 
 
 • 
 
 • 
 
 • 
 
 • 
 
 
 • 
 
 • 
 
 • 
 
 V 
 
 
 
 
 
 
 V 
 
 
 
 
 
 
 • • • 
 
 A («) 
 
 • 
 
 
 
 
 
 
 I >1 
 
 'qk 
 
 &. 
 
 Table 12. Illustration of integer program for 
 minimization of interconnections. 
 
 This set of inequalities (7-1) requires that each e^ be covered by 
 
 at least one of M. which are to be selected. Note that I e.. represent 
 
 i=l 
 
 Ji 
 
 the number of interconnections from the first level to the last gate for f 
 
 J 
 
 1+0 
 
Let us introduce new zero-one variables X., i = 1, 2, . .., p, such 
 that 
 
 g. 
 
 s 6 -H < q. • V i = 1, 2, ..., p. (7,2) 
 
 j=l J 
 
 Then if one of £.., j = 1, 2, ..., q, is 1, i.e., M. is selected to cover 
 
 some e . g , then A., must assume 1. Otherwise (7*2) is not restrictive. 
 
 With these preparation, let us solve a integer linear programming 
 (7.1) and (7*2) with the minimization of 
 
 U Z \. + Z Z £.., 
 
 i=l i i=l j=l J1 
 
 where U is a sufficiently large positive number such that U > p q. It 
 
 primarily minimizes Z X. which represents the number of maximal compatible 
 
 sets actually used, and secondarily minimizes Z Z €.. which is the total 
 
 number of interconnections between the first level and the second level. 
 Obviously, this is exactly the same minimality criterion as we have 
 discussed before. At present no computational experience has been obtained 
 for this integer program. 
 
 Next let us discuss another minimization problem. For this problem, 
 assume that we have already determined a minimal compatible cover for a 
 given function (or functions). Each maximal compatible set will define a 
 negative function of variables x , x , ..., x which is then implemented 
 as a gate in the first level. 
 
 in 
 
Here we minimize the number of variables actually used for these 
 gates, i.e., the total number of connections fed to the first level, 
 assuming that maximal compatible sets for those gates in the first level 
 are given. Contrary to the previous minimization problem, this assumption 
 is necessary because compatible sets which are not maximal may reduce 
 the number of connections further. However, the development of algorithm 
 which eventually yields the network of the minimum number of interconnections 
 and connections seems difficult at present. In this sense, the present 
 procedure gives only an approximate solution to the problem. 
 
 Now let us consider a maximal compatible set in the given minimal 
 compatible cover (the negative completion are not obtained yet) and the 
 corresponding incompletely specified negative function g of variables 
 
 x l' X 2* * * * ' X n* 
 
 Each pair of entries g =1 and g = are allowable because 
 g is assumed to be negative. Denote all such pairs of entries as P , 
 
 P^, ..., P . Each pair P. is allowable due to x. n , x >r ,, ..., or x., 
 2' ' u ^ l il' i2' ' lk. 
 
 l 
 
 (see Definition k) . Then the determination of the minimum number of 
 variables for g is to select the minimum number of variables such that 
 every P., i = 1, 2, ..., u, is allowable due to these selected variables. 
 This is also a set-covering problem and therefore can be solved, though 
 the detail is not given. This set-covering problem should be solved for 
 every maximal compatible set corresponding to a gate in the 
 first level. 
 
 k2 
 
As an example, consider the maximal compatible set M„ derived 
 
 •* 
 
 from M_ of Table 9> by deleting the entries with stars. Table 11 shows 
 
 the covering table of this problem. Each row represents a pair of entries 
 P. and each column represents a variable x.. (For the above example, 
 
 each pair is specified by a pair of row numbers of the column M_. ) The 
 
 (i, j)th entry of the table is 1 if P. is allowable due to x., and blank 
 
 otherwise. 
 
 Table 13 . Covering table for the determination of 
 the minimum number of inputs to M_ 
 
 derived from M of Table 9« 
 
 P. 
 
 1 
 
 x l 
 
 X 2 
 
 X 3 
 
 (1.7) 
 
 l 
 
 
 
 (1.8) 
 
 1 
 
 
 1 
 
 (3-7) 
 
 1 
 
 
 1 
 
 (3.8) 
 
 1 
 
 
 
 (k.l) 
 
 1 
 
 
 
 (h.Q) 
 
 1 
 
 
 
 (6.7) 
 
 
 1 
 
 
 (6.8) 
 
 
 1 
 
 
 Table 13 shows that we need only x.. and x to realize M as a negative 
 function of x , x , x . In fact, function M = x v- x in Fig. 6 needs 
 only x, and x . 
 
 ^3 
 
Determination of inputs to the output gate can also "be, done 
 similarly. In this case, however, we know the negative functions which 
 are fed from the first level. Therefore, in determining external variables 
 which are connected to the output gate, we need not consider pairs of 
 entries which are already allowable due to those. The subsequent manipulation 
 should be applied on this shrinked covering table. 
 
 kk 
 
8. Conclusion 
 
 The minimization of the number of gates in the network in which 
 each gate can represent only negative function was converted into a set- 
 covering problem. Since the set-covering problem is rather easy to solve, 
 this approach appears helpful in obtaining such networks. Although some 
 attempts are made, the minimization algorithm of the total number of 
 interconnections and connections is still an open question. 
 
 It would be a direction of the future research to develop algorithms 
 based on the other minimality criteria, such as the number of interconnections 
 and connections? the total cost of gates which is evaluated differently 
 from the number of gates, and also the extension of the approach in this 
 paper to more general cases such as feed-forward networks. 
 
 ^5 
 
REFERENCES 
 
 (1) S. B. Akers, "A truth table method for the synthesis of 
 combinational logic, " IRE Transaction on Electronic Computers, 
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 (2) S. B. Akers, "Synthesis of combinational logic using three-input 
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 October 1962. 
 
 (3) M. L. Balinski, "Integer programming: methods, uses, computation," 
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 (5) E. N. Gilbert, "Lattice theoretic properties of frontal switching 
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 1*6 
 

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