H mm IIMfUUHUHUUll: iZIMb mmMm |U Wtk * ; BubbII all lrlilt Bow? fflw&Mfuuuw ■HMHIlMlllurHKflrlH BWWWli ifflHB IliHHrailSi Hip Wffi ■ffiBSHlfiBft HH1 LIBRARY OF THE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAICN 540.^4 3.HC6Y cop' 2 ' Digitized by the Internet Archive in 2013 http://archive.org/details/computerprogramf502shin / IL^- .502. p. 2. Report No. UIUCDCS-R-72-502 wurh . COMPUTER PROGRAM FOR DESIGNING OPTIMAL NETWORKS WITH MOS GATES by Teruaki Shinozaki April 1972 COMPUTER PROGRAM FOR DESIGNING OPTIMAL NETWORKS WITH MOS GATES by TERUAKI SHINOZAKI April 2k, 1972 Department of Computer Science University of Illinois at Urbana- Champaign Urban a, Illinois 6l801 Report No. UIUCDCS-R-72-502 This work was supported in part by the National Science Foundation under Grant No. NSF GJ-503 and was submitted in partial fulfillment of the requirements for the degree of Master of Science in Computer Science, April, 1972. Ill ACKNOWLEDGEMENT The author is very grateful to his adviser Professor S. Muroga for his enthusiastic guidance and encouragement during the preparation of this thesis. The program package is based on the algorithm which was developed by Mr. T. K. Liu. The author is extremely obliged to Mr. T. K. Liu for his valuable guidance and discussions. The author thanks Dr. Y. Kambayashi for many stimulating conver- sations relevant to this paper and C. Hohulin for her careful typing. IV TABLE OF CONTENTS 1. INTRODUCTION 1 2. SYNTHESIS ALGORITHM 6 3. PROGRAM ORGANIZATION 13 k. COMPUTATIONAL RESULTS 17 5. FURTHER CONSIDERATIONS 20 5.1 INVERSION COMPLEXITY 20 5.2 MODIFIED ALGORITHM 21 5.3 COMPARISON OF TWO ALGORITHMS 27 LIST OF REFERENCES 29 APPENDIX A. f AND g. *s OBTAINED BY COMPUTER FOR ALL REPRESENTATIVE FUNCTIONS OF THREE VARIABLES 30 B. f AND g . ' s OBTAINED BY COMPUTER FOR ALL REPRESENTATIVE FUNCTIONS OF FOUR VARIABLES 35 C. EXAMPLES OF DESIGNED NETWORKS 110 1. INTRODUCTION The recent progress in integrated circuit technology has been improving the main drawback of MOS devices (i.e. slow speed), and the MOS logic circuit has become one of the important logic families for electronic digital computers. MOS logic circuits have a lot of advantages over bipolar devices such as high packing density, lower power consumption and simple production processes. In addition to these features, the MOS circuit has an interesting feature in that each MOS gate can represent a considerably complex negative function. Figure 1.1 shows conventional NOR and NAND gates and a complex gate which are implemented as MOS gates. in W B oH NOR AVB -o NAND tf *— I (AVB)CVDE o HE S-IC £4 E H T COMPLEX FIGURE 1.1 MOS GATES By using complex gates, it is usually possible to synthesize circuits with fewer MOS gates than with conventional NOR, NMD gates [1] in Figure 1.2, a binary full adder which is implemented with conven- tional negative logic gates is shown. This adder is proved to have the minimum number of gates by integer programming [2]. In Figure 1.3, a binary adder which is implemented with complex MOS gates is shown. A comparison of these adders is shown in Table 1.1. >Si > Ci FIGURE 1.2 BINARY FULL ADDER (NOR-NAND IMPLEMENTATION) B tf rf B; oHC a,H[ SHI ^HC Ci_! o 1[ i! £h Aio_| c.-iH rf " ©Si Ht FIGURE 1.3 BINARY FULL ADDER (COMPLEX GATE IMPLEMENTATION) Implementation No. of MOSFETS No. of Gates Levels Inter- Connections Power Consumption Simple gate 24 8 5 17 k Complex gate 18 k 3 Ik 2 TABLE 1.1 COMPARISON OF BINARY FULL ADDER IMPLEMENTATIONS WITH SIMPLE GATES AND COMPLEX GATES Apparently, the network implemented with complex gates is better than the other in every item. When logical networks are implemented with MOS complex gates, some of the following objectives are usually chosen by logic designers. 1. Minimization of the number of MOS gates 2. Minimization of the number of MOSFETs 3. Minimization of the number of interconnections k. Minimization of logic levels Usually these objectives are not independent of each other and they often conflict with each other. In such a case we have to choose the most important objectives at the sacrifice of other objectives. For these design objectives, some algorithms have been developed. An algorithm which minimizes the number of MOS gates restricted to two level networks was first developed by Ibaraki and Muroga [3]. This method uses the truth table and the minimal network is synthesized by finding minimal cover sets. Two efficient algorithms which use dif- ferent approaches were developed by Liu [h] and Kasami et al. [5] independently. Those two algorithms are very similar and both algorithms extended the network to more than two level cases. These algorithms minimize the number of MOS gates. But the efficient algorithm which minimizes the number of MOSFETs has not been known, although this problem can be solved by integer programming for small networks [6]. In this paper, a program package which designs MOS logic networks based on Liu's algorithm* is described. And the computational results * The early version of Liu's Ph.D. thesis. 5 for functions of ten or less variables are discussed, and as a further consideration, an algorithm which was developed based on Liu's algorithm and Kasami's is described. 2. SYNTHESIS ALGORITHM An MOS gate consists of a single FET as a load and several FET's as drivers. Hence the output function of each gate is always a negative function and any negative function can be realized with an MOS gate. In this chapter, an algorithm which implements logic networks with a minimal number of negative gates is described. This algorithm is based on the early version of Liu's algorithm. Since each gate consumes almost constant power, this algorithm implements networks of lowest power consumption. The following definitions are necessary to derive the algorithm. Definition 1 Let V be the set of all n-dimensional binary vectors with each coordinate of or 1. For two vectors, A = (a_ , a^, . . ., a ) and ' 12 ' n B = (b_, b , . . . , b ) in V , A = B if and only if a. = b. for all i, 12' n n 11 ' and A < B if and only if a. < b. for all i and A 4 B. l—i ' Definition 2 Assume A is an input vector. Then a is defined as the product of complemented literals corresponding to in A and |3 is defined as the product of literals corresponding to 1 in A. (e.g. a = x x p = xx for input vector 0101 ). Definition 3 Assume A is a true vector of a Boolean function f. Then A is detached (i.e. detached from the vector (0, 0, . . . , 0)) if there is 7 a false vector of f, B such that A > B. A is said to be attached if it is not detached. Definition k Assume A.^ and A are two true vectors of f . Then A.^ is called seperated from A and denoted as A./A , if there exists a false vector of f, B, such that A. > B > A . Definition 5 Let A be a detached true vector of f. Consider every chain of vectors A_ , A , . . ., A such that A./ ' hj ... /A /A, starting with A. The number of vectors included in the maximum chain is called the degree of A. The set of all vectors with the same degree e is called a true cluster of degree e . Definition 6 For a given Boolean function f , let us define sets of input vectors m n , m , . . ., m where (a) m (i = 0, 1, . . ., d) is the true cluster of detached true vectors of degree i, (b) m_. n = {B I B & m_., B I m. ,, ., n, B is a vector smaller than at least one vector in m A . 2 (l + 1)' 2i but not smaller than any vector in m n , . _ \* }, m^. n is called a J 2 (l + 1) ' 2x+l false cluster and (c) 2d + 2 is the total number of clusters. (m_, m , .... m,^ _ ) is called the stratified structure of f . 2d + ± Definition 7 For each cluster m. (i = 0, 1, . . ., 2d + l), assign binary integer numbers L(m. ) satisfying the following conditions. (a) The least significant digit of L(m. ) must equal 1 for even i and for odd i. m is defined as the empty set. (b) L(m. ) < L(m.) for i < j. X J (c) The number of digits of L(m. ) is T = [log (d + l)] + 1 where [a] is the smallest integer not smaller than a. By representing T L(m.) = Z 2 T_k g (m.) 1 k=l k X we can construct a table shown in Figure 2.1. This table is called a stratified truth table. g l S 2 • • • om m o gl (m ) g 2 (m ) • • • g T (m Q ) m l • • • • • • • • • • m 2d + l S l (m 2d + 1 ) • S 2 (m 2d+1 ) • • ■ a • g T (m 2d + l } FIGURE 2.1 THE STRATIFIED TRUTH TABLE Definition 8 For each g (m.) = 1 in a stratified truth table, find all £ > j such that g i (m^) = 0. For each £, find an index k < i such that S k (nVj) = and g fc (m^) = 1. Let us denote all the indices k with k l' k 2 ? " " *' k e' " bhen & k g k . . . g is called a prohibiting 12 e product of g. (m. ). If there is no £ > j such that sA^g) = 0, then the prohibiting product is defined as = 1. Algorithm A Step 1 : Find all attached true vectors of a given function f . Denote all the attached true vectors with A , A , . . . , A . Their products of complemented literals by Definition 2 are denoted by a. , a , . . . , a . Compute Q = ql. v ql v . . . ^ a . For f , form new function fs m ^L 2 m ' by changing all the attached true vectors into false vectors. Step 2 ; Find the degree of each true vector of fs and form the strati- fied structure of fs. Step 3 : For each cluster, compute £ as follows. Assume cluster m. includes input vectors A. ,, A. , . . ., A . Then t, is obtained by 1,-L i,d x ^t^ t. ^ a. , v O. „ v i . . v a. if i is even l 1,1 1,2 l, t. k = (a i-l,l v Vl,2 - • • • - 0! i-l,t i ) (P i-l,l v P i-1,2 v . . . 8 ) if i is odd, X*~ _L • u . 7 i where a for p = ior i-1 is a product of complemented literals of an input vector A in cluster m . 8 is a product of non-comple- P,q P P*q mented literals of an input vector A in cluster m . p,q P Step k : Assign the smallest integer number to each cluster as L(m. ) satisfying the conditions of definition 7 , and construct a stratified truth table. Step 5 : For each column i of the stratified truth table, find the set of all indices j's such that g. (m ) = 1, and denote the set with J.. to i v j ' i For each j £ J. , if j - 1 e J. then eliminate the j from J., i.e., J. = J. - { j ] . After all such j's are eliminated, for each j e J. 10 form h. . = t. g J] _ g. s . . . g. e , where g^ g j2 • . . g. e is a prohibiting product of g.(m.). Form the sum of all the products h. .'s for each column, and denote it with g.. ij 1 s. = h. ._ v h. .„ v . • . v h. .. & i ljl ij2 ijk Step 6 : By using g and 0, obtained in Step 1, form Boolean function

1 = ^3 = X l V X h~ X 2 X 3 '2 = ? 1 S 1 = X 1 X 3 v X 1 X 2 X 4 g l - I I '3 ^0°l to 2 b 2°l tD l fcD 2 By Step 6 v X l X i + g l cp '3 v n = g l s 2 v x l x U g l v X 3 X ^ This network is shown in Figure 2.2. 12 x?- 9i XJ.VX4VX2X3 x 3 - x 4 - Xl- g 2 x 1 x 3 vx 1 x 2 x 4 g 1 4> f 15= Xl — X3 — x 4- gi92vx 1 x 4 g 1 VX 3 ^4 FIGURE 2.2 THE REALIZED NETWORK 13 3. PROGRAM ORGANIZATION This program package is implemented according to Algorithm A and is designed to deal with up to ten variable functions. The program package consists of one main program and Ik subroutines implemented by FORTRAN IV. The numbers of statements of the main program and each subroutine are shown in Table 3»1« The total number of statements is 6UU. Name Function No. of Statements MAIN Main program 26 BLDTT Build truth table 31 FNDAV Find attached true vectors 6o ASCEND Find next smallest vector 22 DECEND Find next largest vector 22 COUNT Count the number of l's in a vector 12 DTABLE Calculate the degrees of vectors 58 CMZETA Form t 112 COMP Find complemented function kl APDTB Add two functions 12 SIMPLE Find the irredundant form 3^ BSTTB Build Stratified Table 3^ COMFNC Form g. ' s 50 OUT Print out f and g . ' s 118 PROX Form Boolean expression of ^ and Q 38 TOTAL 6kk TABLE 3.1 THE NUMBER OF STATEMENTS OF EACH SUBROUTINE Ik Program organization is shown in Figure 3.1- Each box consists of a few subroutines and has the following functions. (T) Read a Header Card A header card indicates the number of variables of the Boolean functions which are to be processed. (g) Read Boolean Function f Build Truth Table Read cards which describe a function f , and build a truth table of the function according to the data. (3) Find Attached True Vectors Compute Q Form Function fs If the vector (0, 0, . . ., 0) is a false vector, there are no attached true vectors because all other vectors are larger than (0, 0, . . ., 0). If the vector (0, 0, . . ., 0) is a true vector, find all the false vectors of f . Eliminate all true vectors which are larger than each of the false vectors. The rest of the true vectors are attached true vectors. Make the logical product of the complements of variables which are in the vector. Q is the logical sum of all the products generated for all attached true vectors. Form function fs by changing all the attached true vectors into false vectors. (J) Build Degree table Set the degree of the vector (l, 1, . . . , l) to 0. For a vector A = (a , a , . . . f a ) find all the true vectors B = (b , b , . . ., b ) such that B > A and b. > a. for only one i , and denote them as li ' 15 @ READ A HEADER CARD @ READ BOOLEAN FUNCTION f BUILD TRUTH TABLE NO MORE DATA Amk (5) FIND ATTACHED TRUE VECTORS COMPUTE & FORM FUNCTION f c BUILD DEGREE TABLE 5) FORM Z's 6) BUILD STRATIFIED TABLE I (?) COMPUTE f AND g j's OUTPUT FIGURE 3.1 PROGRAM ORGANIZATION 16 B_ , B , . . . , B. . Find all the false vectors C = (C_ , C , . . . , C ) 12k. J. 2 n such that C > A and c. > a. for only one i , and denote them as Q, , f . . . , C* . Let the degrees of B_, B , . . . , R be d., d , . . . , d and the degrees of C , C , . . ., C^ be e , e , . . ., e- respectively. If the vector A is a true vector, then its degree is the maximum of d. , d , . . . , d . Otherwise its degree is the maximum of d + 1, d + 1, . . . , d + 1, e , e , . . ., e .. By this procedure, we can construct the stratified structure of function f at the same time. (£) Compute £ ' s For each cluster of vectors, compute £ . © Build Stratified Table Assign smallest binary integer number to each cluster as L(m.) satisfying the conditions of definition 7 in chapter 2. Construct a stratified truth table. © Form g.'s Form g. 's according to Algorithm A. ® Output Print f and g.'s in the form of Boolean expressions. 17 k. COMPUTATIONAL RESULTS This programming package was applied to all representative functions of three and four variables and some functions of more than four variables. For negative gate network design, all functions can be classified into equivalence classes with respect to permutation of variables. According to this classification there are 80 representative functions for three variable functions and 398^ representative functions for four variable functions [7l« The numbers of gates which are required for three variable representative functions and four variable representative functions are shown in Table k.l. 71% of three variable functions require two gates and 60% of four variable functions require three gates. The average computation time per function is shown in Figure k.l. The average computation time is the average of ten functions which are randomly chosen for each number of variables. Figure k.l shows that the computation time exponentially increases according to the number of variables. In the case of ten variable functions, about 95% of the computation time is required to form the Boolean expres- sions of t ' s. The computation time will be improved by using the modified algorithm described in chapter 5» We can expect to decrease the computation time to half that of Algorithm A. The computational results of all three and four variable functions are listed in appendices. Examples of networks which are designed based on the computer printouts are also shown in appendices. 18 gate 1 gate 2 gates 3 gates Total No. of functions 3 8 57 12 80 THREE VARIABLE FUNCTIONS gate 1 gate 2 gates 3 gates Total No. of functions 3 28 1577 2376 398^ FOUR VARIABLE FUNCTIONS TABLE k.l THE NUMBER OF FUNCTIONS VS. THE NUMBER OF REQUIRED GATES . In these examples the Boolean expression of each function given in the computer printouts is factored with parentheses whenever it is found to decrease the number of literals. Then based on this Boolean expression, each gate is synthesized by series or parallel connection of MOSFETs corresponding to the logical product or sum, respectively. Since the minimization of the number of MOSFET's is not considered, the number of MOSFET's of these networks may not be minimum. This problem has not been solved yet. 19 100 r uj 10 .. o < 0. 1 " 0.1 -- o u 0.01 + + + + 2 3 4 5 6 7 8 NUMBER OF VARIABLES 10 FIGURE k.l COMPUTATION TIME VS. NUMBER OF VARIABLES 20 5. FURTHER CONSIDERATIONS The algorithm -which was developed by Kasami et al. is similar to Liu's early algorithm. In this chapter, an algorithm is developed by combining Liu's algorithm and Kasami 's algorithm in order to improve the processing speed. This modified algorithm is efficient to find the irredundant disjunctive form of each gate function. In this chapter we discuss the modified algorithm and compare it with Algorithm A in chapter 2. 5.1 INVERSION COMPLEXITY Let V be the set of all n-dimensional binary vectors with each n coordinate of or 1. Each vector in V is represented as a vertex n of an n-dimensional cube C so that two vectors A and B are adjacent n if and only if A differs from B at exactly one component. Each edge — > is directed from A to B if A > B and is denoted by AB. Let f .. , f , . . . , f be completely specified Boolean functions of n vari- ables. When each vertex A of C is labeled with a binary number m C, (A; f_, . . ., f ) = E f. (A) 2 m_1 , C is referred to as 1 m . , i ' n l =1 C (f , . . ., f ). A directed edge AB is called an inverse edge of n ± m — 3 — C n (f 1 , . . ., fj if and only if t (A ; f , . . . , fj > I (B ; f^ . . A f ). The number of inverse edges included in a directed path P of c n ( f l> ' • ') f ) is called the number of inversions of the path P . And the maximum number of inversions over all directed paths of 21 C (f, , . . ., f ) is called the number of inversions of C (f, , . . . , f ) n v i' > m n— 1^ -* — m^- and denoted by R(f., . . . , f ). J v 1' ' m A Boolean function f is negative if and only if £(A ; f) < i(B; f) —* for every edge AB of C . When a logic network is realized with negative gates only, the minimum number of negative gates N(f ) is given by N(f ) = [log 2 { R(f ) + 1 } ] + 1 where [a] is the smallest integer not smaller than a. 5.2 MODIFIED ALGORITHM A generalized feed-forward network which consists of r negative gates is shown in Figure 5.1. Any networks with r negative gates can be obtained by deleting some of the interconnections from Figure 5.1. In Figure 5-1 x _, x , . . . , x denote input variables, g. denotes the i-th gate from the left, g. (x , . . . , x , u , . . . , u ) is the function which gate g. realizes, and u. (x , . . ., x ) is the output function of gate g. in terms of x , . . ., x . The following lemma and algorithm are shown by Kasami et al. Lemma 1: For a network of the form in Figure 5«1» C (u , . . ., ' n 1 u r n , f) has no inverse edge. Conversely, if C (u_, . . ., u _, f) -1 ' n 1 r-1 has no inverse edge then u_ f . . . , u 1 , f can be realized by a network of the form in Figure 5«1« This lemma immediately leads to the following algorithm which gives a minimum number of negative gates and a set of u, , . . • , u , , f for a given single output function f. 22 ■1 T Xl 92 >--- u 2 "l u r , 2 - Xl X n Qr-1 u 2 ui u r-l Xl 9r > FIGURE 5-1 GENERALIZED FEED-FORWARD NETWORK WITH r NEGATIVE GATES 23 Algorithm 1 . Step 1 ; Assign the value of f (x) to L(l), where L(A) denotes a label to be attached to a vertex A of C . I denotes the vector (1, . . ., 1) € V n . Step 2 : When L(A) is assigned to every vertex A of weight W of C (weight is the number of 1 elements in a vector), assign the smallest binary integer as L(B) to vertex B of weight W- 1, such that (a) The least significant bit of L(B) is f(B). (b) L(B) > L(A) for every vertex A whose weight is W. Step 3 : Repeat step 2 until L(0) is assigned, where is the vector (o, o, . . ., o) € v n . Generally there are other networks with a minimum number of negative gates than the network obtained by Algorithm 1, because as long as L(0) is less than 2 and the two conditions in Step 2 are satisfied, a greater integer (instead of the smallest integer that Algorithm 1 describes) can be assigned to L(B) without increasing the number of negative gates. This alternative affects the number of interconnections and also the structure of each gate. Appropriate binary integer assignment may make the complexity of each gate more equally distributed, and also may reduce the total number of MOSFETs. In this paper this problem will not be discussed further. The Boolean expressions of g.'s are obtained by the following algorithm. This algorithm is developed based on Liu's early algorithm and Kasami et al. 's. 2h Algorithm 2. Step 1: Divide all the vectors into groups so that all vectors in each group have the same label. Let us call these groups m , m , m ..... m , where m. has a smaller label than m. _ (j = 0, 1, . . . , 2' ' k Z 3 + 1 k- 1). Step 2: For each vector, make the logical product of only the variables which are 1 in the vector. If there is no such variable, the logical product is defined to be 1. For each group, make the logical sum of the products corresponding to vectors which belong to the group. Let us call these logical sums £ Q for m , £_ for m , £ for m , etc. Step 3 : Choose m.'s from m , m , . . . , til whose label L(m. ) has u - 0, where u. „ is the i + 1 st digit of the label from the i + 1 i + l left. And for eacn selected m., make the logical product of £. and J J g.'s which have u. = 1. i i Step k: Make the logical sum of all the products obtained by Step 3« Let us denote this sum of products with g. . Step 5 : Repeat Step 3 and Step k for all i(i=0, 1, . . ., r-l), where r is the number of negative gates N(f). g. is the Boolean function of the i th gate from the left, g represents the function f of the last gate. The difference between this algorithm and Kasami's is dividing the vectors into groups which have the same labels. This grouping technique makes the calculation of g.'s faster especially when the number of variables increases. 25 Example f = Z (0, k, 6, 8, 10, 11, 12, 13, Ik, 15) First assign binary integer numbers to all vertexes as labels by Algorithm 1. All labels must satisfy the two conditions in Step 2. This assignment is shown in Figure 5*2. Functions g.'s are obtained from the labels by Algorithm 2. VECTORS -►1111*0014 LABELS 1100 1000 0000 101 • TRUE VECTOR O FALSE VECTOR FIGURE 5.2 BINARY INTEGER NUMBER ASSIGNMENT 26 By Step 1, group the input vectors. = {1111, 1110, 1101, 1011, 1100, 1010} m. label 001 = (0111, 0011, 0101, 1001, 0001} m. label 010 m = {1000, 0100, 0110} label Oil m 3 = {0010} label 100 m = {0000} label 101 By Step 2 5 ^ - x i x 2 v x l x 3 h =X k ^2 = X l " X 2 ^*3 By Steps 3 and ^ j x =^0 ^ ? 1 v ^2 = X 1 X 2 ^ ^3 ^ >: x_x„ ss x,, v X_ n/ X 2 = x„ \/ x^ s/ x. 'l ' = X„X^ V X n X v X g v 1 A 3 v ^3 & 1 °1 = x l x 2 v X 1 X 3 " g l f = ^g 2 - ^ = ^2 v X 3 S 1 27 From this result, we can obtain the logic network with MOS gates shown in Figure 5.3. tf ft X ciH x 4-|[ *^l FIGURE 5.3 MOS NETWORK DESIGNED BY ALGORITHM 1 AND 2 ' 5.3 COMPARISON OF TWO ALGORITHMS The modified algorithm is better than Algorithm A in chapter 2 in the following points. (a) In Algorithm A, attached true vectors have to be found and removed from function f. Since in the modified algorithm, it is not required to find attached true vectors, this procedure can be completely omitted. 28 (b) In Algorithm A, two equations must be calculated in order to obtain the Boolean expressions of 5 ' ■■ * f we need ^redundant dis- junctive forms for the expressions ofg.'s, the calculation is 1 complicated and it requires considerable computation time. In the modified algorithm, only one equation is necessary, and an irredundant disjunctive form can be obtained by simple calculation. (c) In the modified algorithm, the Boolean expression of g. is more easily found than Algorithm A because in Algorithm A, the procedure to find prohibiting products is complicated. (Generally, if we assign the smallest binary integer to each group of vectors as a label L(m.), we can simplify the procedure to find prohibiting products.) The network obtained by the modified algorithm is usually differ- ent from the network by Algorithm A, because the Boolean expression obtained by the modified algorithm is a sum of product form, and the other one is a product of sum form. In addition to this difference, there are some other differences caused by the following reasons. (a) In Algorithm A, attached true vectors are removed from the given function f, and g.'s are formed for the new function fs. (b) In Algorithm A, true vectors are used to find the functions g 's, and all don't care vectors are set to 0. But in the modified algorithm, false vectors are used, and all don't care vectors are set to 1. Hence even if we compare the two results in the same form (e.g. sum of product form) by converting one of them to the other form, they are usually different. 29 LIST OF REFERENCES [1] Spencer, R. F., "MOS Complex Gates in Digital Systems Design," IEEE Computer Group News , Vol. 2, No. 11, September I969, pp. i+7-55. [2] Liu, T. K. , L. E. Shiau and S. Muroga, "Optimal One-bit Full Adder with Different Types of Gates," to be published in IEEE Transactions on Computers . [3] Ibaraki, T. and S. Muroga, "Synthesis of Networks with a Minimum Number of Negative Gates," IEEE Transactions on Computers , Vol. C-20, No. 1, January 1971, pp. ^9-58. [k] Liu, T. K. , "Synthesis of Logic Networks with MOS Complex Cells," Doctoral thesis (in preparation), Department of Computer Science, University of Illinois at Urbana- Champaign, Urbana, Illinois. [5] Nakamura, K. , N. Tokura and T. Kasami, "Minimal Decomposition of a Logical Function into Monotone Functions," The Institute of Electronics and Communication Engineers of Japan , Vol. 5^4— C, No. 1, January 1971, pp. 18-25. [6] Liu, T. K. , T. Nakagawa and S. Muroga, "Synthesis Networks of MOS Cells by Integer Programming, " Memorandum, Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana, Illinois (unpublished paper) . [7] Culliney, J., "On the Synthesis by Integer Programming of Optimal NOR Gate Networks for Four Variable Switching Functions," Report No. 480, Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana, Illinois. 30 APPENDIX A f AND g . ' s OBTAINED BY COMPUTER FOR ALL REPRESENTATIVE FUNCTIONS OF THREE VARIABLES In Table h, the three digit numbers in the far left column of the list represent the functions with octal numbers which are formed from the output column in the truth table. The binary digits in the output column (downward from the top) are divided into three groups and each group is expressed by an octal digit as shown in Figure A. f = x^ v Xl x 2 v x 2 x 3 v Xl x 2 x 3 = Z (0, 2, 3, 5, 6) 12 3 12 3 Binary 1 Octal 2 123 123 123 1 6 X 1 X 2 X 3 X 1 X 2 X 3 X 1 X 2 X 3 1 1 6 FIGURE A OCTAL REPRESENTATION OF A THREE VARIABLE FUNCTION In the representation of the function of each gate, the following notations are used. XI GI F Complement 31 For example,, Gl - n (XI + X2X3) F = -i (Xl) -, Gl means g = x ^ x x f = Xl gl 32 TABLE A f AND g. *s OBTAINED BY COMPUTER FOR ALL REPRESENTATIVE FUNCTIONS OF THREE VARIABLES 3h X CM x x r-fc ^ r»i "f> o x x X r + ^J f\J ■ X X m — — ' -*■ x r r •-Hi «M« x r + 4- ^ —i — 1 r + c '—1 o --J r 1 '.*■; -iq h ci . — * a h 4 in ro ro vjr >j- LH Lf\ s4» •jj -4 -u 'A ro rt rn ro r° rO ro r<4 ro •q r- ■-> h ji ^ .[) -if -v r- r- f-» r- p** rO rO r^ ro rrj rO fO 35 APPENDIX B f AND g . ' s OBTAINED BY COMPUTER FOR ALL REPRESENTATIVE FUNCTIONS OF FOUR VARIABLES In Table B, four digit hexadecimal numbers are used to represent the functions. Each digit represents four binary digits of truth table in the same way as three variable function. 36 TABLE B f AND g.'S OBTAINED BY COMPUTER FOR ALL REPRESENTATIVE FUNCTIONS OF FOUR VARIABLES 1 LS r 13 r US r M ** LS X m P ■> X ^ L3 M m r MM] 1*1 X X LS MM X + M X ^ r M X + -» ♦ IN mm: ♦ •J- X X LS MM X M * ♦ M r M m X X IS X X p -M ♦ M 1*1 H r M ♦ + CJ> X X O MM o * r X 1*1 ♦ C9 r * J- . . r o r IN X * r (*! X 4> J- X M LS r MM X X X L3 L3 MM L3 r — « 1*1 MM r»l MM f r IN 1*1 r J- -M _( O X r X MM, r X Mm' X X 4- '3 LS r (3 L9 M X * r M X 4- ^ ^ » X + _ f IN 13 r r r ' 4- M •— * PM IN IN m MMi -»■ m P 1-5 -* O 4- O 13 13 X X X X X X ♦ MM MM r X rri i*l X r M il r r r <*i X o r r (*1 4- r*i * M MM r IN X M LS r L3 r 13 P M X + fl X M MM X X X r MM MM o X X X L3 r + L3 X —A m + M M -— r -^ M M -t r ♦ 4. MM r nj 4- ■M 1*1 r _■ * -^ o X X X ♦ 'J L3 X X X MM P il '.3 X L3 MM MM X J! LS r r r X ♦ * -J- _^ IN o r r II LU r ♦ -J- 4. MM r m M X 4. • X * 4- PA P X PA — . X r X X (3 — H X x ♦ X r r L3 13 X L3 7 MM X X X MM X — in MM — — r — * in LU M^ ♦ M f*l rn r r * MM] r * 4 IN MM AJ PA — m X MM -» r * r r o X X U3 X X ♦ — . 4. L9 ii X X X r X x r x + 1*1 X rn X * U5 * * ^ r X C3 p * r X * p LL * X 'J O L9 X ii 11 + r o X II X MM r *• II (J X — -J- -J- r -^ r r + LU 4- -^ r IN r -v LL -- J- MM 4- J- 1*1 -M ^M LL r MM t ■t r X _ x _ r _ _ ■J- X IN 5 'J r 13 r ♦ M r lS r X L9 r -r » X X M _, L3 r ii LU -3 r _M r X X + 7 ta T 19 o o X X X -~ O — ♦ r X x 7 X O 13 + p ls r m r MM' r r r -V — • ♦ (*> m — 1 r -»• rj —4 MM m mM r ~-> -^ r PL r ls * U * N r (J X MM X 13 X o o ii * MM r X 4- X MM 19 13 L3 4- ♦ r PL 13 ^ r M ii LU ii o ii ♦ r ■J- X M X 4- X M X r L3 7 r r LL 'N L9 J- X 4. r LS * _ IN X '3 r r ii r AJ LS M - LS pm r ■ MM* -* LL -M r — - ♦ + r3 ■M —4 ♦ r LL PA (3 ii r ii MM MM X X 13 J x (9 19 X C3 L3 X MM H L3 ii X X J ii X II r Li- ii X LU J ^ mm X -J- X — + ^ ii + r r _ . r 1*1 X r -■ r U3 4- m x LL J il r LL J- X x x + r * x MM "■ —• LU X -4 ll IN r m j- M in r^ J" ->• — < LU -J- j- ^1 r Ml 4 IN r LU m rvj + ■t PA PL —■ -- + — J X X X X X X X X L9 X MM ^ . — 11 — X MM X MM MM O X X MM X X X MM x X L3 LS -t -M r — ' -p — MM M rfl + rj M m (N r m MM vt r — IN ♦ 4> 4> MM LU m CI —I Ml 4- J- -M r 4. -f * * * 1-4 J" J- PA * M> r — . x — i o X II — ■< r r r r ii r x r X MM i r x X X X X MM X MM X x IX x X X X MM MM — * PA (•1 m r PL LL -»■ — J- LU r<^ MM l»l 1*1 M m LL — - + M ♦ iN J- IN -f + J fN P*l rn m AJ PA m 4- 4 LS ii X X x X X r X n n ii — ii ii X — ii r X X X X ii <3 X X X X X MM X X X X X X X X X X X X X| r ll -* 4- in ~j -m pn ro LU LU r\; j- r\i IN MM rj ^j —4 *-* — * — < ^i r J- ~> ■* —< m -- m J- 1*1 m —■ M ■Ml ^J M,J- IN -4 * M M A3 N d X x — . X MM x ii X (*i x L9 l"> r X d ii x X X X :3 — X X X X X X X x X X X X X X XX X X X X X X| r 4" 4> 4- -M 1*1 LL AJ CNjI M n o X l LS o II K X X' o X o X XI X X X — . L3 X X X X — ii ii ii LS LS LS LL r : ,_ ll ♦ + s x * + + MM * m X * fN X ♦ x X m X ■T X X m X + X IN X M IN -* M ^ ll * m * -r Mrf + J- p*t r^ ^ r<-> — 1 m —* |*| ♦ •N * J- (Si MM ■*■ * * 4- — < + 4. Mi + — " + 4- + 1*1 m r^ m ^J ro _i + 1*1 + IN IN f\J ^J — « ■N 4- — < 4- ♦ * 4- 4- 4- 4- + 4- ♦ + * * ♦ 4- ♦ * ♦ ■> + A", -A PA 1 ♦ PA| X X — x X X X X X X X X X X X X X XI X X X X X X XI X PA -- m * t ^J rsj MM r\l * M * rsl M 4- 4- 4- 4. 4- * + ■T + 1-1 1*1 (*! f^ r-l 1-1 m r*l m 1*1 m <*! 1*1 1*1 •-A f*l (*! 1*1 m PA AJ A0 Ml PA M X X X X X X X X X X X X X X X X X X X X x X X X X X X X X X X X x X X X X XXX X X x X XXX IM ♦ n t AJ +■ + 4. •* pa PA -t + 4- + f> ro + ♦ + + wf + 4- + * rr\ 4- 4- 4- -r J- J- **! * ♦ ♦ ♦ 4- 4- ■f ♦ + + 4> * + ♦ 4- + ♦ +1 ♦ * 4- 4- + ♦. * * X x X X X X X X X X X 1 X X X X X X X X X X X 1 — -J — • PA -* J- -r ■J- pa AJ -4 (N 1 -f IN ■t r\i — - ~3- J- J- -* m J- (*> ■J- 4- IN fl r*i m rr\ HI f*| M -M J- -J" st -f 4" J- ■J- J- J" J- J- >T J- * ts 3 j J U ■3 '•> L9 19 o 'J -j. 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X X x X X X X X x X X X X X X X X X X X X X X X X X X X x X x X X X x X X X X rn _. —. — 1 —Imp 'NJ ^ '■J -n >NJ rvji — i — 1 — i —i — — - — " ~-» —4 —* — i mw mM ppp —4 — i — p —4 pn — 1 —4 — 1 fM rn* mm M _i PM —4 MJ pp4 fM — < Ml —i IM -■ fM xxx X X X X 1 X X X X X X 13 X X X < X X X X X x r X r x X X X X X X X X X X X X X X X X X X X X X X X X — < * + + + fM — IfM — fM w — * —i * r 4- ♦ + i. * 4 ♦ ♦ + + * mm + * PM mm PM + PM ♦ —• mm pp p> PM ♦ — ' 4- PM + ~m + — ♦ PM PV X x p|x r X r 13 Xi «- — II ii r ppp -* -~ r r r r r r r r 1 r r r r r ■» c, .J. *r\ ml — ** — 1 r + -r ^J -f *i J- m J- UL m LL fM pn fM * IM J- -f M* M-' fM t J- J" j- X X X X x II ' X ii x II ' X — . X X x X X x X x X X X X X II x X X X ■x ii ii n X ii X il ii ii X ii X II x II X II X i X II X ■41 r\l -n ■M ISI + X (3 X ■J J> X (3 X L3 - u!x :9 X 13 X L3 X n.— r\j — — J- * •J- 1 •r r\j| r\i r-, + * + + ♦ * + p> + + + + ♦ * + p> * ♦ x r x r r|x X X 1 X X X x. 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J 19 — « M X X -4 ^—4 ^|— U. X XX xlx + + + + I + t*^ r*^ ' i*i f*i | (*i x xlx XX — #■ J- X X X- X XX *\ l#l 4 X X ♦ I* ■* X m j- ^ -o x xlx X r r r * ii n :9 'J p P r pi r r ii i + n j ao o, o -O B o -o o o -> o p pip pip p II + 1 II II I II It -9 jj o u u .* LTI J3 ( l^ CC « i-a <, «o ! 49 n P P| P P P r II II | II II : ♦ II II II II O 0!U ri r + " < <,«i < i o|lu ll'ci 49j •O <49 o olo o|o o i fl p III II IT' < CD li CO 1 CO CO CO o o o o 09 iO O p! p pi p P ii i ii ii I ii ii —i** —:-< — ■J|l3 O O O ^ UJ LLlO —4 /n! co col uj uj O -O -C C5 >o o'oon a r p p p| p p ii ♦ n ii i ii ii 3 !H JOU N (*l ^ N- CO JJ UJ UJ JJ LU O -O -0 O -O 3 O O O O P P, P P P P II II II II II II OO.J JOO II II + II - _l - tl 9 O *t CO JJ u, o UJ UJ UJ LU , UJ U- J5 -O ^ 13 — « ♦ r 4^ mi r f p. -J) 13 a 441 o * — ' f r ^^ IN #-« r p. X o •• (J 4* X 13 p. o — < (SI .^ f IN m* r rsi 4 r rsj r ti X a X *■ zi X X p» 5 r •^ r ♦ IN -4 ♦ m 4 p. (SJ —- •— t X ^ X X — . 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