Multiple-Valued Galois Logic S/D Trees, S/D Canonical
Forms, and their Complexity
Anas Al-Rabadi, and Marek Perkowski
Dept. of Electrical and Computer Engineering
Portland State University
1800 S.W. 6th Ave.
Portland, OR 97201
U.S.A.
[alrabadi, mperkows]@ee.pdx.edu
Abstract
The idea of S/D trees for binary logic is a general concept that found its main
application in ESOP minimization and the generation of new diagrams and
canonical forms. As a special case of the most general Linearly Independent (LI)
logic, S/D trees demonstrated their power by generating forms that include a
minimum Galois-Field-Sum-of-Products (GFSOP) circuits for binary and ternary
radices. Galois Field of quaternary radix possesses some interesting properties.
Therefore, in this paper, an extension of the S/D trees to the important case of GF(4)
is presented. A formula to calculate the number of Inclusive Forms (IFs) per
variable order for an arbitrary Galois field radix and arbitrary number of variables
is derived; an issue that might be critical in finding minimum GFSOP circuits and
in image processing applications. A new fast method to count the number of IFs for
an arbitrary Galois field radix and functions of two variables is introduced; the IFn,2
Triangles. Multi-valued and binary circuit realizations of the Universal Logic
Modules (ULMs) that correspond to quaternary Shannon and Davio expansions
over GF(4) are presented. Binary and multi-valued GFSOP Filter realizations for
1
quaternary Shannon and Davio expansions over GF(4) illustrate some of the
applications.
1 Introduction
Economical and highly-testable implementations of Boolean functions [7, 13, 14,
21] based on Reed-Muller (AND/EXOR) logic play an important role in logic synthesis
and circuit design. Most of the industrial AND/EXOR circuits implement non-canonical
minimized forms of functions while circuits with desirable properties possess high
regularities that produce canonical forms of functions, i.e. expansions that are unique
representations of a Boolean function. Several large families of canonical forms: Fixed
Polarity Reed-Muller forms (FPRMs), Generalized Reed-Muller forms (GRMs),
Kronecker forms (KROs), and Pseudo-Kronecker forms (PSDKROs), referred to as the
Green/Sasao hierarchy, have been described [8, 12, 24]. Because canonical families have
higher testability and some other properties desirable for efficient synthesis, especially of
some classes of functions like symmetric and unate functions, they are widely
investigated.
Binary Green/Sasao hierarchy [3] and their extended hierarchy to the ternary case
over GF(3) [1] have found applications in many areas; finding minimum GFSOP
expressions, creation of new forms, decision diagrams, spectral decision diagrams,
regular structures [16], as well as spectral transforms with their well-known applications
in digital telecommunications, digital signal processing, and digital image processing [6].
The state-of-the-art ESOP minimizers [23, 27, 32] are either approximate and based
on heuristics or give the exact solution only for functions with a small number of
variables. The well-known formulation for finding the exact ESOP was given in [11], but
2
all known exact algorithms can deliver solutions only for certain binary functions of more
than five variables. Because GFSOP minimization is even more difficult, it is important
to investigate structural properties and the countings of their canonical subfamilies in
order to create efficient approximate algorithms (analogously to the binary case [27]).
Recently, two families of ternary canonical Reed-Muller forms, called Ternary
Inclusive Forms (TIFs) and Ternary Generalized Inclusive Forms (TGIFs) have been
proposed [1]. The second family includes a minimum ternary GFSOPs. In this paper, it
will be proposed, as a generalization of the ternary case, the general formulation of the
interesting case of the quaternary S/D trees and the countings of the multi-valued
Inclusive Forms (IFs) for an arbitrary Galois field radix and arbitrary number of
variables.
Binary logic is the simplest information carrying system. Yet, it does not provide a
deep insight into logics with more values. Ternary logic structures provide a first
glimpse, from analytical point of view, into higher radix logics, but can not provide
enough concrete information to generalize ideas into arbitrary multi-valued logic. This is
one of the reasons why quaternary systems are important; they provide a concrete
generalization, that bridges ternary logic to logics of radices higher than four. Results
form quaternary logic can utilize the binary logic and ternary logic as special cases, fivevalued logic as a first “test-bed” to detect the validity of the generalization. Therefore we
present, as an example of the multi-valued S/D trees, the special case of the quaternary
S/D (QS/D) trees and the corresponding quaternary Inclusive Forms (QIFs). One of the
major outcomes of our investigation of quaternary Reed-Muller logic in this paper, is the
discovery of a new method for a fast computation of the number of IFs over an arbitrary
3
radix of Galois field logic GF(pk) for two-variable functions; the IFn,2 Triangles. This
method will cut drastically the time needed to calculate the number of IFs over any
arbitrary order of of Galois foeld for two-variable functions, in contrast to the use of
general formulas that consume lots of time, especially when the logics are of radices
higher than four. The IFn,2 Triangle of Coefficients show a close relationship to the well
known Pascal Triangle; an issue that helps in reusing the already existing algorithms to
generate the IFn,2 Triangles.
A great interest is paid to Reed-Muller forms over quaternary Galois field GF(4) as
it represents the smallest composite field. Quaternary GF represents the smallest
nonprime field for GF(pk) where p is a prime number and k is a natural number 1.
Therefore quaternary logic can be important as it allows a single four-valued variable to
be fully encoded using two 2-valued variables without any don’t cares; an issue that finds
an application in the hardware realization of quaternary logic functions using binary
circuits, and in the digital processing of multi-valued input/output pictures (e.g. 4x4
blocks of a still image).
The remainder of this paper is organized as follows. The basic definitions of families
of forms belonging to the Green/Sasao hierarchy over GF(2) are given in section 2.
Quaternary S/D trees and their IFs are included in section 3. A general formula to
compute the number of IFs for an arbitrary Galois field radix and arbitrary number of
variables is presented in section 4. A new method for fast computation of the number of
IFs over an arbitrary radix of Galois field logic GF(pk) for two-variable functions is
presented in section 5. Various circuit realizations of quaternary Shannon and Davio
4
expansions over GF(4) are presented in section 6. Conclusion and future directions of our
work are presented in section 7.
2 Green/Sasao hierarchy of binary canonical forms
The Green/Sasao hierarchy of families of canonical forms and corresponding
decision diagrams is based on three generic expansions, Shannon, positive Davio, and
negative Davio expansions, which are given below respectively:
f(x1,x2,…,xn) = x1’ f0(x1,x2,…,xn)  x1 f1(x1,x2,…,xn)
f(x1,x2,…,xn) = 1 f0(x1,x2,…,xn)  x1 f2(x1,x2,…,xn)
f(x1,x2,…,xn) = 1 f0(x1,x2,…,xn)  x1’ f2(x1,x2,…,xn)
(1)
(2)
(3)
Where: f0(x1,x2,…,xn) = f(0,x2,…,xn) = f0………………...negative cofactor of variable x1
f1(x1,x2,…,xn) = f(1,x2,…,xn) = f1……………….…….…..positive cofactor of variable x1
f2(x1,x2,…,xn) = f(0,x2,…,xn)  f(1,x2,…,xn) = f0  f1
An arbitrary n-variable function f(x1,x2,…,xn) can be represented using the positive
polarity Reed-Muller (PPRM) expression:
f(x1,x2,…,xn) = a0  a1x1  a2x2  … anxn a12x1x2 a13x1x3  … a(n-1)nxn-1xn
… a12…nx1x2…xn
(4)
For each function f, the coefficients ai are determined uniquely, so PPRM is a
canonical form. If we use either only the positive literal or only thr negative literal for
each variable in equation (4), the Fixed polarity Reed-Muller (FPRM) form will be
obtained. There are 2n possible combinations of polarities and as many FPRMs for any
given logic function. By freely choosing the polarity of each literal in equation (4), we
obtain the Generalized Reed-Muller (GRM) form. In GRMs, contrary to FPRMs, the
same variable can appear in both positive and negative polarities. There are n2(n-1) literals
in equation (4), so there are
( n1)
n.2
2
polarities for an n-variable function and as many
GRMs [25]. Each of the polarities determines a unique set of coefficients, and thus each
5
GRM is a canonical representation of a function. Two other types of expansions result
from the flattening of certain binary trees that will produce Kronecker (KRO) forms and
Pseudo Kronecker (PKRO) forms for S, pD, and nD expansions. There are 3n and at most
n
1
32 different KROs and PKROs, repectively [25]. More families of canonical
expansions were introduced in [1, 3, 18, 19]. These forms are generated by flattening
certain type of trees. These forms are the Generalized Kronecker (GK) forms [18],
Pseudo Generalized Kronecker (PGK) forms [19], Inclusive Forms (IFs) [1, 3],
Generalized Inclusive Forms (GIFs) [1, 3], Free Generalized Inclusive Forms (FGIFs) [1
, 3], and Generalized Inclusive Kronecker Forms (GIKFs) [1].
3 Quaternary S/D Trees and their IFs and GIFs
Galois field (GF) is a fundamental algebraic structure in the theory of algebras. It
has proven high efficiency in various applications of logic synthesis among others,
especially in the design for test [10]. Quaternary GF logic is an important special case of
the multi-valued GF logics as it is the smallest nonprime field that allows four-valued
input/output functions to be realized in binary circuits without the need to utilize any
don’t cares (i.e. no redundant unused encoding); in contrast to the ternary case for
example. This can be achieved by using two 2-valued variables to encode a single fourvalued variable without any don’t cares; an issue that is implemented in the Universal
Logic Module (ULM) realization of four-valued input/output functions (See subsection
6.1). This method can be used for both: true realization of four-valued gates, and using
four-valued algebra only as an intermediate step in binary circuit minimization.
GF(4) addition and GF(4) multiplication are defined in Tables 1.a and 1.b,
respectively:
6
+
0
1
2
3
*
0
1
2
0
0
1
2
3
0
0
0
0
0
1
1
0
3
2
1
0
1
2
3
2
2
3
0
1
2
0
2
3
1
3
3
2
1
0
Table 1.a GF(4) addition
3
3
0
3
1
2
Table 1.b GF(4) multiplication
In general, the attractive properties of GF-based circuits, such as the high testability
of such circuits, are due to the fact that the GF operators exhibit the Cyclic Group (Latin
Square) Property. This property can be explained from the four-valued (quaternary) GF
operators as shown in Tables 1.a and 1.b, respectively. Note that in any row and column
of the addition table (Table 1.a), the elements are all different, which is cyclic, and that
the elements have a different order in each row and column. Another cyclic group can be
observed in the multiplication table. If the zero elements are removed from the
multiplication table (Table 1.b), then the remaining elements form a cyclic group. In
binary, for example, the GF(2) addition operator, EXOR, has the cyclic group property.
In general, a literal can be defined as any function of a variable. Basis functions in
the general case of multi-valued expansions are constructed using literals. Galois field
Sum-Of-Products (GFSOPs) expansions (i.e. Shannon expansions) can be performed on
variety of literals. For example, one can use, among others: K-Reduced Post literal (KRPL) to produce K-RPL-GFSOP [1], Post literal (PL) to produce PL-GFSOP,
Generalized (Post) literal (GL) to produce GL-GFSOP, or Universal literal (UL) to
produce UL-GFSOP. Since K-RPL-GFSOP is as simple as PL and more general and it is

This characteristic is useful in detecting faults, for instance, because any single change in the inputs of the
gate will cause a change in the output (s).
7
simpler from implementation point of view than GL or UL, we will perform all the
GFSOP (i.e. Shannon) expansions in this paper utilizing K-RPL-GFSOP.
Let us define the 1-Reduced Post literal (1-RPL) as:
ix
= 1 iff x = i else ix = 0
(5)
For example 0x, 1x, 2x, 3x are the: zeroth, first, second, and third polarities of the 1Reduced Post literal, respectively. Also, let us define the quaternary shifts (inversions)
over variable x as: x, x’, x”, x’’’ which are the zeroth, first, second, and third shifts
(inversions) of the variable x respectively, and variable x can take any value of the set {0,
1, 2, 3}.
Theorem 1. Shannon expansion over GF(4) for a function with single variable is:
f = 0x f0 + 1x f1 + 2x f2 + 3x f3
(6)
Where: f0 is cofactor of f with respect to variable x of value 0
f1 is cofactor of f with respect to variable x of value 1
f2 is cofactor of f with respect to variable x of value 2
f3 is cofactor of f with respect to variable x of value 3.
Proof. From equation 5, if we substitute the values of the 1-Reduced Post literal in
equation 6, we obtain the following equations:
For x = 0  f x=0 = f0
For x = 1  f x=1 = f1
For x = 2  f x=2 = f2
For x = 3  f x=3 = f3
Which are the cofactor of variable x with value zero, cofactor of variable x with value
one, cofactor of variable x with value two, cofactor of variable x with value three,
respectively.
Q.E.D.
Example 1. Let f (x1,x2) = x1”x2 + x2’’’x1
The quaternary truth vector of the function f is: F = [0,3,1,2,2,1,3,0,3,0,2,1,1,2,0,3]T
8
Utilizing equation (6), we obtain the following quaternary Shannon expansion over GF(4)
of the function f:
f = 2  0x1 1x2 + 3 0x1 2x2 + 0x1 3x2 + 3 1x1 0x2 +  1x1 1x2 + 2 1x1 3x2+ 2x1 0x2+ 3 2x1 1x2+
2 2x1 2x2+ 2 3x1 0x2+ 3x1 2x2+ 3 3x1 3x2 .
Using the axioms of GF(4), it can be derived that the 1-Reduced Post literals defined
in equation (5) are related to the shifts of variables over GF(4) in terms of powers as
follows:
0x
= x3 + 1
= x’ + (x’)2 + (x’)3
0x = 3(x’’) + 2(x’’)2 + (x’’)3
0x = 2(x’’’) + 3(x’’’)2 + (x’’’)3
(7)
(8)
(9)
(10)
1x
= x + (x)2 + (x)3
= (x’)3 + 1
1x = 2(x’’) + 3(x’’)2 + (x’’)3
1x = 3(x’’’) + 2(x’’’)2 + (x’’’)3
(11)
(12)
(13)
(14)
2x
= 3(x) + 2(x)2 + (x)3
2x = 2(x’) + 3(x’)2 + (x’)3
2x = (x’’)3 + 1
2x = x’’’ + (x’’’)2 + (x’’’)3
(15)
(16)
(17)
(18)
3x
(19)
(20)
(21)
(22)
0x
1x
= 2(x) + 3(x)2 + (x)3
3x = 3(x’) + 2(x’)2 + (x’)3
3x = x’’ + (x’’)2 + (x’’)3
3x = (x’’’)3 + 1
Where: 0x, 1x, 2x, 3x are the: zeroth, first, second, and third polarities of the 1-Reduced
Post literal, respectively. Also, x, x’, x”, x’’’ are the zeroth, first, second, and third shifts
(inversions) of the variable x respectively, and variable x can take any value of the set {0,
1, 2, 3}.
We chose to represent the 1-Reduced Post literal in terms of shifts and powers,
among others, because of the ease of the implementation of powers of shifted variables in
hardware (See subsection 6.1). After the substitution of equations (7) through (22) in
9
equation (6), and after the rearrangement and reduction of the terms according to the
axioms of GF(4), we obtain the following equations:
f = 1 f0 + x (f1+3f2+2f3) + (x)2 (f1+2f2+3f3)+(x)3(f0+f1+f2+f3)
f = 1 f1 + (x’)(f0+2f2+3f3) + (x’)2 (f0+3f2+2f3)+(x’)3 (f0+f1+f2+f3)
f = 1 f2 + (x’’)(3f0+2f1+f3) + (x’’)2 (2f0+3f1+f3)+(x’’)3 (f0+f1+f2+f3)
f = 1 f3 + (x’’’)(f2+3f1+2f0) + (x’’’)2 (f2+2f1+3f0)+(x’’’)3 (f0+f1+f2+f3)
(23)
(24)
(25)
(26)
Equations (6) and (23) through (26) are the 1-Reduced Post literal Shannon and
Davio expansions for single variable, respectively. These equations can be rewritten in
the following matrix-based forms, respectively:
f = [0x
1x
2x
3x]
1
0

0

0
f = [1 x x2 x3]
1
0

0

1
f = [1 x’ (x’)2 (x’)3]
0
1

1

1
f = [1 x” ( x”)2 ( x”)3]
0
3

2

1
0
2
2
3
f = [1 x’’’ (x’’’) ( x’’’) ] 
3

1
0 0 0  f 0 
1 0 0  f 1 
0 1 0  f 2 
 
0 0 1  f 3 
0 0 0  f 0 
1 3 2  f 1 
1 2 3  f 2 
 
1 1 1  f 3
1 0 0  f 0 
0 2 3  f 1 
0 3 2  f 2 
 
1 1 1  f 3
0 1 0  f 0 
2 0 1  f 1 
3 0 1  f 2 
 
1 1 1  f 3 
0 0 1  f 0 
3 1 0  f 1 
2 1 0  f 2 
 
1 1 1  f 3 
(27)
(28)
(29)
(30)
(31)
Observe, that equations (27) through (31) are expansions for single variable. Yet,
these canonical expressions can be easily generated for arbitrary number of variables (N)
10
using the Kronecker product, analogous to the binary case [28]. This can be expressed
formally as in the following forms for Shannon (S), and Davio (D0, D1, D2, and D3)
expressions, respectively:
f

N
 [ 0xi
1x
i
2x
i
3x ][S][
F
i
]
(32)
i 1
f

N
[ 1
xi xi 2 xi3 ][D0][ F ]
(33)
xi’ (xi’)2 (xi’)3 ][D1][ F ]
(34)
xi” (xi”)2 (xi”)3 ][D2][ F ]
(35)
xi’’’ (xi’’’)2 (xi’’’)3 ][D3][ F ]
(36)
i 1
f

N
[ 1
i 1
f

N
[ 1
i 1
f

N
[ 1
i 1
As analogous to the binary case [3] and the ternary case [1], the expansions can mix
Shannon (S) for certain variables and Davio (D0, D1, D2, and D3) for other variables. This
will lead to, analogously to the ternary case [1] for instance, to the Kronecker quaternary
decision tree (KQDT). Moreover, the mixed expansions can be extended to include,
among others, Pseudo Kronecker quaternary decision tree (PKQDT).
The basic S, D0, D1, D2, and D3 quaternary expansions (i.e. flattened forms) over
GF(4) can be represented in quaternary DTs (QDTs) and the corresponding varieties of
reduced quaternary DDs (RQDDs) (i.e. according to the corresponding reduction rules
that are used). For one variable (i.e. one level of the DT), Figure 1 represents the
expansion nodes for S, D0, D1, D2, and D3, respectively.
11
S
0x
1x
2x
3x
Equation (6)
(a)
D0
1
D1
x2
x
Equation (23)
(b)
x3
1
D2
(x’) (x’)2 (x’)3
Equation (24)
1
(x’’)
(x’’)2 (x’’)3
D3
1 (x’’’) (x’’’)2 (x’’’)3
Equation (25)
Equation (26)
(d)
(e)
(c)
Figure 1. Quaternary S (a), D0 (b), D1 (c), D2(d), and D3 (e) DTs, respectively
In correspondence to the binary S/D trees [3], and ternary S/D trees [1], the concept
of the Quaternary S/D Trees can be introduced. Analogously to the work in [1, 3], the
quaternary S/D trees are generated through the definition of the Generalized Davio
expansion over GF(4) as follows:
D
1
x
(x )2
( x )3
Figure 2. Generalized quaternary Davio (GQD) expansion
The notation here is that ( x ) corresponds to the four possible shifts of the variable x
as follows:
x  {x, x’, x”, x’’’} over GF(4)
(37)
Utilizing the definition of quaternary Shannon (Figure 1.a) and quaternary
Generalized Davio (Figure 2), and analogously to the work done in [1, 3], one can obtain
12
the quaternary Shannon/Davio trees (QS/DT) for two variables. The number of these S/D
trees per variable order is: 2(4+1) = 32. The number of QIFs per S/D tree will be later
derived in two different ways: the first method is by using the general formula for
arbitrary number of variables over GF(4) developed in this section. The second method is
using the very general formula developed in section 4. Example 2 illustrates some of the
quaternary S/D trees and some of the quaternary trees they produce. The numbers on the
top of S/D trees in Figures 3.a and 4.a are the numbers of the total QIFs (i.e. total number
of quaternary trees) that can be generated (Full derivation of these numbers will be
provided later on in this section and in section 4).
Example 2. Let us produce some of the quaternary trees for the following quaternary S/D
trees. Utilizing the notation in (37), we obtain, for the S/D trees in Figures 3.a and 4.a, the
following S/D trees in Figures (3.b and 3.c) and Figures (4.b and 4.c), respectively:
S
N = 4,096
0
1
a
D
2
a
D
a
S
b (b)2 (b)3 1 b’ (b”)2 (b’’’)3 0b 1b 2b
1
3
a
S
3b
0b
1b
2b
3b
(a)
S
0
D
1
a
S
2
a
D
3
a
0
a
S
1 b (b’)2 (b’)3 1 b” (b’)2 (b)3 0b 1b
(b)
S
2b 3b
0b
1b
2b 3b
D
a
1
2
a
D
a
3
a
S
1 b (b)2 (b”)3 1 b’ (b)2 (b)3
0b 1b
(c)
S
2b 3b
0b
1b
Figure 3. A quaternary S/D tree for two variables of order {a, b} with three Shannon
nodes and two Generalzied Davio nodes (a), and some of the quaternary trees that it
generates (b and c).
13
2b 3b
N = 262,144
0
1
b
D
2
b
D
a (a)2 (a)3
1
S
1 a
3
b
D
S
(a)2 (a)3 0a 1a
b
2a
3a
1 a (a)2 (a)3
(a)
S
0
b
D
1
1
2
b
D
a (a)2 (a)3
S
b
S
1 a’ (a”)2 (a’’’)3 0a
1a 2a 3a
3
0
b
D
1 a”(a’’’)2 (a’)3 1
b
D
a (a)2 (a’)3
1
2
b
D
3
b
D
S
1 a (a”)2 (a’’)3 0a
(b)
b
1a 2a 3a
1 a” (a)2 (a’)3
(c)
Figure 4. A quaternary S/D tree for two variables of order {b, a} with two Shannon
nodes and three Generalzied Davio nodes (a), and some of the quaternary trees that it
generates (b and c).
From the quaternary S/D DTs shown in Figures 3 and 4, by taking any S/D tree.
Multiplying the two-level cofactors (which are in the QDT leafs) each by the
corresponding path in that QDT, and next summing all the resulting cubes (terms;
products) over GF(4), one obtains the flattened form for the function f, as a certain
GFSOP expression (expansion). For each QDT in Figures 3.a and 4.a, there are as many
forms obtained for the function f as the number of all possible permutations of the
polarities of the variables in the second level branches of each QDT.
Theorem 2. For GF(4) and N variables, the total number of QIFs per variable order is
equal to:
14
# QIFs =  =
N 1
(4) (4)
N 2
(4)
N 3
(4)
0
( N 1)
4
   ... {[
k 1 0
k 2 0
0
3.(4)
k 30
1
3.(4)
k1
(4
( N 1)
(4
kN  0
) (4
 k 1)!k 1!
( N 1)
3.(4)
k2
( N 2)
4
!
) ...(4
kN
)
( N 2)
(4
0
!
...
 k 2)!k 2!
4 ! ][
(4  k )!k !
0
N
N
(38)
]}
Proof. A very general proof that will include the quaternary Galois field as a special case
will be provided in section 4.
Q.E.D.
Properties and extended Green/Sasao hierarchy for quaternary S/D trees and their
corresponding forms can be developed similar to the work in [1, 3]. The extension of the
concept of S/D trees to higher radices of Galois fields (i.e. higher than four) is a
systematic and direct process that follows the same methodology developed for the
ternary case [1], and the quaternary case that has been presented in this section. The
following example demonstrates the countings of QIFs using Theorem 2.
Example 3. For number of variables equal to two (N=2), equation (38) reduces to:
1
0
( 4)
(4)
k 1 0
kN  0
= 
=
 {[
|
k 10, k 2 0
(1)
4
(1)
(4
+
(0)
4
!
 k 1)! k 1!
|
k 11, k 2  0
+
(0)
0
!
(4

 k 2)! k 2!
(4
3.(4)
k1
1
3.(4)
) (4
k2
)
]}
 |k 2,k 0 +  |k 3,k 0 +  |k 4,k 0 +  |k 0,k 1 +
1
2
1
2
1
2
1
2
 |k 1,k 1 +  |k 2,k 1 +  |k 3,k 1 +  |k 4,k 1
1
2
=  00 +
1
10 +
2
1
2
1
2
 20 +  30 +  40 +  01 +  11 +  21 +  31 +  41
= 1 + 256 + 24,576 + 1,048,576 + 16,777,216 + 16,777,216 + 4,294,967,296 +
412,316,860,416 + 1.75921860444* 1013 + 2.81477976711*1014
= 2.99483809211*1014 .
15
Utilizing MVL map representation, we can easily prove that there are 416 =
4,294,967,296 quaternary functions of two variables, and 2.99483809211*1014 quaternary
Inclusive Forms generated by the S/D trees. Thus, on the average every function of two
variables can be realized in approximately 69,729 ways. This high number of realizations
means that most functions of two variables are realized with less than five expansions,
and all functions with at most five expansions.
4 General formula to compute the number of IFs for an arbitrary number
of variables and arbitrary Galois field radix
Although the S/D trees and Inclusive Forms were developed in section 3 for
quaternary Galois logic, the same concept can be directly and systematically extended to
the case of nth radix of Galois fields and N variables (i.e. N DT levels). The following
Theorem provides the total number of IFs per variable order for N variables (i.e. N
decision tree levels) and nth radix of any arbitrary algebraic field, including GF(pk) where
p is a prime number and k is a natural number  1. This generality of the following
theorem comes from the fact that algebraic structures specify the type of operations (e.g.
addition and multiplication operations) in the functional expansions but do not specify the
countings which are an intrinsic property of the tree structure and independent of the
algebraic operations that are performed. Therefore the following theorem is valid, among
others, for Galois fields of arbitrary radix (pk) where p is a prime number and k is a
natural number  1 (e.g. 3, 4, 5, ,7, 8, 9, 11, 13, …).
Theorem 3. The total number of Inclusive Forms for N variables and nth radix Galois
field logic is equal to:
16
# QIFs =  n,N =
N 1
( n) ( n)
N 2
( n)
N 3
( n)
0
   ... {[
k 1 0
k 2 0
k 30
(n 1).( n)
(n
0
k1
kN  0
( N 1)
n
( N 1)
(n
(n 1).(n)
1
) (n
( N 2)
n
!
( N 2)
 k 1)!k 1!
k2
(n
( N 1)
(n 1).(n)
) ...(n
kN
)
!
 k 2)!k 2!
0
...
n ! ][
(n  k )!k !
0
]}
N
N
(39)
Proof. The following is the derivation of the general equation (39) to calculate the
number of IFs per variable order (For clarity refer to Figures 1, 2, 3, and 4):
The total number of nodes for any nth radix Galois field (GF(n)) tree with N levels (i.e. N
variables) equals:
N 1
 (n)
k
(40)
k 0
For any S-type (i.e. Shannon type) node there is only one type of nodes as the
branches of the Shannon node have the possibility of single value each. Yet, for D-type
(i.e. Davio type) node there are N possible types of nodes (where N is the number of
variables, which is equal to the number of levels). The highest possible number of forms
for the D-type node exists when the Davio node exists in the first (highest) level, and the
lowest possible number of forms for the D-type node is when the Davio node exists in the
Nth-level (lowest level).
Therefore, for certain number (M) of S-type nodes the following formula describes
the number of the D-type nodes for N variables:
#S=M#D=
N 1
 (n)
k
- M
(41)
k 0
It can be shown that for GF(n) (i.e. N-ary decision tree with N-levels (i.e. N
variables), the general formulas that count the number of D-type nodes, and the number
17
of all possible forms for the D-type node in the kth level (where k is less than or equal the
total number of levels N) are, respectively:
# Dk =
( k 1)
(n)
(42)
( N K )
|Dk| =
(n 1).(n)
(n)
(43)
Where: # Dk is the number of D-type nodes in the kth level.
|Dk| is the number of all possible forms (per node) for the D-type node in the kth level.
Let us define the S/D tree category to be: the S/D trees that have in common the
same number of S-type nodes and the same number of D-type nodes within the same
variable order. Let us define the following entities for nth radix Galois field and N
variables (i.e. N decision tree levels):
 n,N = number of variable orders
 n,N = number of S/D tree categories per variable order
 n,N = number of S/D trees per category
 n,N = number of IFs per variable order
(44)
(45)
(46)
(47)
From equations (42) and (43), and using some elementary counting rules, one can
derive by mathematical induction the following general formulas for N being the number
of variables, and n being the field radix:
 n,N = N!
 n,N =
(48)
N 1
 (n)
k 0
N 1
 n,N =
k
+1
[ n ]!
k
k 0
N 1
[ n  k ]! k!
k
(49)
N 1
, Where k =0, 1, 2, 3, …,
 (n)
k
(50)
k 0
k 0
18
 n,N =
N 1
( n) ( n)
N 2
( n)
N 3
( n)
0
( N 1)
n
   ... {[
k 1 0
k 2 0
k 30
(n 1).( n)
0
(n
( N 1)
(n
kN  0
(n 1).(n)
1
k1
) (n
( N 2)
n
!
 k 1)!k 1!
( N 2)
(n
( N 1)
(n 1).(n)
k2
) ...(n
kN
)
0
!
...
 k 2)!k 2!
n ! ][
(n  k )!k !
0
N
N
(51)
]}
Q.E.D.
Example 7. Let us produce the number of QIFs over GF(4) for two variables (i.e. N=2
and n = 4), then one obtains:
 4,2 =
2 1
(4) (4)
22
( 2 1)
4
 
k 1 0
=
( 2 1)
(4
k 2 0
( 22)
4
!
( 2 2)
 k 1)!k 1! (4
(4 1).(4)
0
!

 k 2)!k 2!
(4
k1
(4 1).(4)
1
) (4
k2
)
 10|4,2 +  20|4,2 +  30|4,2 +  40|4,2 +  01|4,2 +  11|4,2 +  21|4,2 +
00|4,2 +
 31|4,2 +  41|4,2
= 1 + 256 + 24,576 + 1,048,576 + 16,777,216 + 16,777,216 + 4,294,967,296 +
412,316,860,416 + 1.75921860444* 1013 + 2.81477976711*1014
= 2.99483809211*1014.
Which is the same result that was obtained in Example 3.
Corollary. From [3] and equation (51), the following mathematical corollary can be
obtained to count the number of Inclusive forms for N variables and second radix:
n 1
 (1 2
2
n  k 1
)
k 0
N 1
(2) (2)
N 2
2
(2)
k

N 3
(2)
0
   ... {[
k 1 0
k 2 0
k 30
(2 1).(2)
0
(2
k1
kN  0
( N 1)
2
( N 1)
(2
(2 1).(2)
) (2
1
k2
( N 2)
2
!
 k 1)!k 1!
( N 2)
(2
( N 1)
(2 1).(2)
) ...(2
kN
)
!
 k 2)!k 2!
]}
0
...
2 ! ][
(2  k )!k !
0
N
N
(52)
19
In general, expansions of functions can be produced over basis functions of a single
variable, two variables, or any number of variables. The interesting case of expansions of
functions utilizing pairs of variables can be produced using the general procedure for
expansions over Linearly Independent (LI) logic [20]. This can be achieved by the
recursive expansions of a multi-variable function over bases of two variables. The
advantage of such expansions is the regular usage of universal blocks with two control
variables that generalize multiplexers (data selectors) with two control variables (i.e.
variables of control functions) and four data inputs (data functions) [20]. The following
section introduces a fast method to calculate the number of IFs (that are special cases of
Linearly Independent (LI) logic) for an arbitrary Galois field logic for functions with two
variables.
5 A fast method to calculate the number of IFs over an arbitrary radix of
Galois field GF(pk) for functions of two variables
Counting the number of IFs is important in many applications, especially in
providing upper numerical boundaries for efficient search of a minimum GFSOP.
Calculating the numbers of Inclusive Forms (IFs) can be very time consuming due to the
time required to perform the mathematical operations in the general equation (51). This is
why a fast method to generate the number of IFs is needed. Because functions with two
variables find an important application in the ULMs for pairs of variables [20], the
following subsection provides a fast method to calculate the number of IFs over an
arbitrary radix of Galois field GF(pk) for two variable functions (i.e. N = 2).
5.1 IFn,2 Triangles
Functions with two variables are attractive in logic synthesis since many functional
decomposition methods exist that produce two control inputs for primitive cells in a
20
standard library of standard cells (such as in a multiplexer with two address lines). It is
shown in [20] how to produce Universal Logic Modules (ULMs) for pairs of control
variables that generalize Shannon and Davio expansion modules. Theorem 4 will
introduce a fast method to calculate the number of Inclusive Forms for functions with two
input variables over an arbitrary radix of Galois field.
Theorem 4. The following IFn,2 Triangles provide a fast method to calculate the number
of IFs over an arbitrary nth radix of Galois field (GF(pk)) for two variable functions
(N=2).
1
1
1 5
1 6 15
7 21 35
1
4
10
20
35
1
3
6
10
15
21
2
3
4
5
6
7
1
1
1
1
1
1
1
1
1
1
1
1
2
3
4
5
6
7
1
3
6
10
15
21
1
4 1
10 5 1
20 15 6 1
35 35 21 7
1
(a)
40
50
54
30
43
58
20
32
46
21
34
49
22 22
36 36
412 412
23
38
415
24
310
418
312
421 424
512
516
520 520
524
528
532 536
N0(N-1)N1(N-1)N2(N-1) N3(N-1)... N(N-1)(N-1)NN(N-1)NN(N-1) N(N+1)(N-1) N(N+2)(N-1)
…
540
N2N(N-1)
(b)
Figure 5. IFn,2 Triangles; the Triangle of Coeffecients (a) and the Triangle of Values
(b) for a fast calculation of the number of Inclusive Forms for an arbitrary radix Galois
field and functions of two input variables (N=2).
21
Proof. The proof of Theorem 4 follows directly from the mathematical induction of the
number of IFs over an arbitrary radix of Galois field GF(pk) for two variable functions.
This can be deduced directly from equation (51); if the IFn,2 Triangles are valid for n = q
then they will be also valid for n = q + 1, where n = pk (p is a prime number and k is a
natural number  1).
Q.E.D.
It can be observed that the IFn,2 Triangle of Coefficients possesses a close
similarity to the well known Pascal Triangle. This occurs as follows: if one omits the
first two rows of the Pascal triangle and duplicates each row into another horizontally
adjacent row, the IFn,2 Triangle of Coefficients will be obtained. This fact helps in
creating computer algorithms that generates the IFn,2 Triangle of Coefficients since many
efficient and optimized algorithms already exist to generate the Pascal triangle [9].
The following example illustrates the concept of counting the number of IFs over an
arbitrary radix of Galois field GF(pk) for two variable functions through the IFn,2
Triangles that were demonstrated in Figure 5.
Example 6. Utilizing the IFn,2 Triangles from Figure 5, we can calculate the number of
Inclusive Forms for GF(2), GF(3), and GF(4) for two variables, respectively, as follows:
 2,2 = 1 20+2 21+1 22+1 22+2 23+1 24 = 1 + 4 + 4 + 4 + 16 + 16 = 45
 3,2 = 1 30+3 32+3 34+1 36+1 36+3 38+3 310+1 312 = 730,000
 4,2 = 1 40+4 4 3+6 46+4 49+1 412+1 412+4 415+6 418+4 421+1 424
= 2.99483809211*1014
(53)
(54)
(55)
One can observe that the results from equations (53), (54), and (55) are the same
countings that were obtained in [3], [1], and in sections 3 and 4 of this paper,
respectively.
22
5.2 Properties of IFn,2 Triangles
The following are some properties of the IFn,2 Triangles presented in Figure 5.
1. The number of positions (elements) in each row of the triangles in Figures 5.a and
5.b are even starting from six.
2. The sum of elements in each row in Figure 5.a equals to the number of S/D trees
per variable order.
3. The triangle in Figure 5.a possesses even symmetry around an imaginary vertical
axis in the middle of the triangle.
4. The minimum number of columns required to generate the whole triangle in
Figure 5.a is equal to three (due to even symmetry): one wing, one column
neighbor to the middle column, and one middle column.
5. The triangle in Figure 5.a can be generated by the process of “Shift Diagonally
and Add Diagonally” (SDAAD); shift the left wing diagonally from west to
southeast direction and add two numbers diagonally from east to southwest
direction, and shift the right wing diagonally from east to southwest direction and
add two numbers diagonally from west to southeast direction.
6. The difference in powers in the triangle in Figure 5.b per row element is: (N-1).
7. The first number in each row of the triangle in Figure 5.b is: N0 and the last
number per row is: N 2N(N-1), where N is the number of variables.
8. The middle two numbers in each row of the triangle in Figure 5.b are always
equal to: NN(N-1) , where N is the number of variables.
23
6 Hardware realizations of quaternary GFSOPs
The following subsections introduce two circuit realizations of canonical quaternary
GFSOP expressions; Universal Logic Modules (ULMs), and Finite Impulse Response
(FIR) Filters. Although the following realizations in subsections 6.1 and 6.2 are for GF(4)
canonical GFSOP forms, analogous realizations can be produced for Galois field of
higher radices equal to 2k where k is a natural number  1 (e.g. 2, 4, 8, 16, 32, 64, …).
6.1 Universal Logic Modules of quaternary GFSOPs
In general, multi-valued Shannon and Davio decision trees (DTs) are multi-level
tree-structured regular circuits with nodes of two level Universal Logic Modules (ULMs)
that are realizations of the corresponding functional expansions. ULMs play an important
role in the realization of circuits that need special regular properties such as
reconfigurable structures, such as FPGAs, with their well-known applications in memorybased ping-pong architectures and parallel processing systems (such as DEC-PERLE
FPGA-based board). Lattices [16] (Pseudo Symmetric Decision Diagrams (PSDDs)) are
another example of such regular structures that are used in: fault detection (circuit
diagnosis; circuit testing), fault localization, and circuit self-repairing. Such properties are
essential in the future advanced technologies for sensitive applications such as spaceoriented applications (e.g. satellite circuits that are immune to galactic/cosmic radiation)
and bio-medical applications (e.g. human-IC interface).
The nonsingular expansions of quaternary Shannon and Davio expansions, can be
realized using a Universal Logic Module (ULM) with control variables corresponding to
the variables of the basis functions (i.e. the variables that are expanded upon). We call it a
universal logic module, because similarly to a multiplexer, all functions of two variables
24
can be realized with two level trees of such modules using constants on the second level
data inputs (See Figure 8). ULMs are complete systems; they can implement all possible
functions of certain number of variables. Analogously to the ternary case [1], the general
ULM that covers the quaternary Shannon and all Davio expansions can be created. Figure
8 illustrates such a realization. Note that by utilizing Figures 6 and 7, all the quaternary
logical addition and multiplication gates in Figure 8 can be converted into the
corresponding binary logical addition and multiplication gates, respectively, similar to the
encoding of a single 4-valued variable into two 2-valued variables presented in [4] (for
simplicity of illustration, the internal 4/2 encoders and 2/4 decoders in Figure 8 are
cancelled and the internal structure is simplified).
GF(2)
GF(4)
A
a1
C
B
A
a2

C
2/4
B
(a)
c1
4/2
4/2
b1
b2
c2
(b)
Figure 6. Realization of GF(4) addition (a) as GF(2) addition (b) (i.e. vector of EXORs).
25
a1
A
4/2
a2
b1
B
.
.
.
.
.
4/2
b2
GF(4)
.
A
B
C

c3
C
2/4
c4
Figure 7. Realization of GF(4) multiplication (a) using GF(2) operations (b)
Where:
4/2
Is a quaternary-to-binary encoder
2/4
Is a binary-to-quaternary decoder
0x
x
x
x’
x’’
”
’
1
1x
.
fi
.
2x
fj
.
.
3x
fk
.
f
+
.
fl
Figure 8. Quaternary ULM that produces quaternary Shannon expansions (equation 6),
and all quaternary Davio expansions (equations 23, 24, 25, and 26).
26
Where:
Is a binary Mux.
.
+
Is a GF(4) logical multiplier
Is a GF(4) logical adder
6.2 FIR filters of quaternary GFSOP canonical forms
Another interesting hardware realization of quaternary GFSOPs are various filter
implementations of which FIR filter is the simplest case. Higher radix extensions of this
kind of filters can be useful for digital signal processing of one-dimensional multi-valued
input/output signals or two-dimensional multi-valued input/output blocks of images (e.g.
4x4, and 8x8 blocks of a still image).
A general FIR filter for single variable GF(4) GFSOP-based expansion can be
produced as in Figure 9. This realization is a direct implementation of the Shannon and
Davio expansions that were obtained in section 3. Such realization can be useful in
various applications such as applications that involve multi-valued input/output biomedical signals.
The addition and multiplication performed in Figure 9 are the Galois field addition
and multiplication operations defined in Tables 1.a and 1.b, respectively. These additions
and multiplications can be implemented using quaternary circuits, or using binary (2valued) circuits as was shown in subsection 6.1.
27
X
Basis1
Generator
Basis2
Generator
C1
*
Basis3
Generator
C2
Basis4
Generator
C3
C4
*
*
*
+
+
+
F
Figure 9. FIR Filter realization of four-valued GF(4) GFSOP canonical forms
Yet, another implementation can be achieved by utilizing a third radix Galois field
multiplication operation. This can be achieved by utilizing the GF(3) multiplication
which is presented in Table 2.b, and noticing the relationship between the multiplication
operation over GF(4) (Table 1.b) and the addition operation over GF(3) (Table 2.a). This
relationship can be stated as follows: Excluding the first row of zeros and the first column
of zeros in the GF(4) multiplication in Table 1.b, then by subtracting one from the
remaining rows and columns, the GF(3) addition operation in Table 2.a is obtained, but
next one must be added to every entry of Table 2.a in order to obtain Table 1.b. This can
be formulated formally as follows:
a

GF ( 4 )
0 , if a  0 or b  0

b 
[( a  1)  (b  1)]  1 , otherwise
Where:  is either GF(3) addition or mod-3 addition
+ is either a shift up operation or mod-3 addition
- is a shift down operation.
For simplicity of implementation, shift-up operation is used instead of mod-3
addition for + operation, and mod-3 addition is used instead of GF(3) addition for the 
28
operation. Figures 10 and 11 illustrate an implementation of the GF(4) addition and
multiplication operations. Similar approaches can be used to realize addition and
multiplication operators for higher radices of Galois fields.
+
0
1
2
0
1
2
0 1
1 2
2 0
2
0
1
* 0
0
1
2
Table 2.a. GF(3) addition
1
2
0 0
0 1
0 2
0
2
1
Table 2.b. GF(3) multiplication
a1
a2
a
a
b
c
+
b1
+
b
+
b2
c
c1
c2
(a)
(b)
(c)
Figure 10. Implementation of three input single GF(4) addition gate (a) as double twoinput tree-structured GF(4) addition gates (b) and consequently as quadruple two-input
tree-structured vector of GF(2) EXORs (c).
29
a
a
b
c
*
b
*
*
c
(a)
(b)
a
-1
b
-1
3
0
1
+1
0
a+b
(c)
Figure 11. The implementation of a three input single GF(4) logical multiplication gate
(a) using double two-input tree-structured GF(4) logical multiplication gates (b), and the
implementation of a single GF(4) logical multiplication gate utilizing mod-3 addition and
shift operations (c).
Where:
-1
Is a -1 shifter
+1
Is a +1 shifter
3
Is a modulo3 addition
Is a comparator to value 0
Is a multiplexer
The physical implementation of the circuit in Figure 11 can be implemented
utilizing various choices of technologies such as Pass Transistor Logic (PTL) that have
30
recently found popularity within the CMOS technologies due to circuit optimization
issues [2, 29, 30, 31]. Such an implementation will be realized in our forthcoming paper.
Other future technologies such as single electron transistors, reversible logic, and
Josephson junction technologies require efficient realization of multi-level Shannon and
Davio canonical expressions and Galois gates.
7 Conclusion
In this contribution, we have presented the important case of quaternary canonical
Shannon and Davio forms over GF(4) and the corresponding quaternary S/D (QS/D) trees
and quaternary Inclusive Forms (QIFs). An extension of the quaternary case to the case
of arbitrary radix of Galois fields is a direct and systematic process following the same
methodology used for ternary case in [1], and the quaternary case presented in this work.
Counting the number of Inclusive Forms (IFs) can play a critical part in the efficient
searching for minimum GFSOP. Thus, we provided a general formula to calculate the
number of IFs per variable order for an arbitrary Galois field radix and arbitrary number
of variables, that includes the formulas in [1, 3] as special cases. The important case of
expansions of functions using pairs of variables can be produced using the general
procedure for expansions over Linearly Independent (LI) logic [20]. This can be done by
the recursive expansions of a multi-variable function over bases of two variables. The
advantage of such expansions is the regular usage of multiplexers (data selectors) with
two control variables and four data inputs (data functions) [20]. The choice of functions
with two variables is important in many applications, as in the case of Universal Logic
Modules (ULMs) for pairs of variables [20]. Thus we introduced a generic method to
calculate the number of IFs for an arbitrary Galois field radix and functions of two
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variables; the IFn,2 Triangles. This method allows for fast calculation of the number of
IFs without the need to use the general formulas that were derived in this paper and in [1,
3]. The IFn,2 Triangle of Coefficients shows a close relationship to the well known
Pascal Triangle; an issue that helps in reusing the already existing algorithms to
generate the IFn,2 Triangles. Binary realization of a general quaternary Universal Logic
Module (ULM) that produces quaternary Shannon and all Davio canonical expressions is
presented. This sort of realization has the advantage as most of the industrial systems
produced so far are of binary nature (i.e. two-valued radix). Also, mapping from
quaternary logic to binary logic does not produce any don’t cares (as the ternary case
does for instance), so unique realization of the ULM exist for specific encoding. Future
technologies will all require the satisfaction of two conditions: reversibility and
regularity, which are both guidances of our research, as illustrated in this paper and
especially in its following papers.
8 References
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35