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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
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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
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