PARALLEL BIJECTIVE PROCESSING OF REGULAR NANO SYSTOLIC GRIDS VIA CARBON FIELD EMISSION CONTROLLED – SWITCHING

International Journal of Computer Science & Information Technology (IJCSIT) Vol 8, No 3, June 2016
DOI:10.5121/ijcsit.2016.8307 77
PARALLEL BIJECTIVE PROCESSING OF REGULAR
NANO SYSTOLIC GRIDS VIA CARBON FIELD
EMISSION CONTROLLED – SWITCHING
A.N. Al-Rabadia,*
, M.S. Mousab
, R.F. Al-Rabadic
, S. Alnawasrehb
,
B. Altrabshehd
, and M.A. Madanatb
(a) Electrical Engineering Department, Philadelphia University, Jordan &
Computer Engineering Department, The University of Jordan, Amman – Jordan
(b) Department of Physics, Mu'tah University, Al-Karak – Jordan
(c) German Association for International Cooperation (GIZ)
(d) Electrical Engineering Department, Philadelphia University, Jordan
ABSTRACT
New implementations for parallel processing applications using reversible systolic networks and the
corresponding nano and field-emission controlled-switching components is introduced. The extensions of
implementations to many-valued field-emission systolic networks using the introduced reversible systolic
architectures are also presented. The developed implementations are performed in the reversible domain to
perform the required parallel computing. The introduced systolic systems utilize recent findings in field
emission and nano applications to implement the function of the basic reversible systolic network using
nano controlled-switching. This includes many-valued systolic computing via carbon nanotubes and carbon
field-emission techniques. The presented realization of reversible circuits can be important for several
reasons including the reduction of power consumption, which is an important specification for the system
design in several future and emerging technologies, and also achieving high performance realizations. The
introduced implementations for non-classical systolic computation are new and interesting for the design
within modern technologies that require optimal design specifications of high speed, minimum power and
minimum size, which includes applications in adiabatic low-power signal processing.
KEYWORDS
Carbon Nanotubes and Nanotips, Controlled Switching, Field Emission, Reversibility, Systolic Grids.
1. INTRODUCTION
In current and emerging future design techniques, reversible computing will have wider
utilization in synthesizing modern computer circuits and systems [1, 20, 34, 36]. This is due to the
fact that reversible logic leads to the design of computer systems that consume much less power
and occupy much less space [1, 3, 9, 20, 29, 32, 34], where it was shown that if all physical sub-
circuits were designed to be reversible then the whole system composed of inter-connecting these
reversible sub-systems will minimize its operational need for power consumption [29]. For this
* Corresponding Author. This research was performed during sabbatical leave in 2015 – 2016 granted from
The University of Jordan and spent at Philadelphia University.

78
purpose, various important technologies have been investigated to implement reversibility such as
adiabatic CMOS circuits [36], optical computing [4] and quantum computation circuits and
systems [1, 2, 4, 15, 20, 32-34].
Simple and regular interconnections lead to high densities and cheap realizations, and higher
density implies both lower overhead and higher performance for the utilized components [1, 27,
28], where it has been shown that multi-processing plus multi-dimensional pipelining at each
phase of a pipeline can lead to a better computing performance [27, 28]. Systolic systems provide
parallel computation and inexpensive computing power, and provide a model of computation
which captures the important features of parallelism, interconnection sub-systems and pipelining,
which has been utilized in wide range of implementations [2, 27, 28, 38], and provide a good
model of computation for investigating concurrent algorithms for VLSI circuits and systems.
Parallel systolic VLSI processing takes into account issues of control, inter-processor
communications, and I/O processing and interfacing. In systolic systems, multi-dimensional
pipelining overlaps I/O with computing to ensure high throughput with no extra needed control
logic, where communication paths inherently require more energy and space than processing
elements (PEs) and communication between processors is achieved through fixed data paths that
have simple and regular geometries. It has been shown that data flow patterns in systolic
architectures are fundamental in matrix computations [27, 28]. Examples of this include two-way
flow on linearly connected networks which is common to both the solution of linear triangular
systems and matrix-vector multiplication, and three-way flow on hexagonally mesh-connected
networks which is common to both LU-decomposition and matrix multiplication.
Nanotechnology is an important modern scientific field that utilizes analysis and synthesis
techniques from several overlapping science and engineering sectors such as from physics,
chemistry, electronics and biology. Nanowires, nanoparticles, Carbon nanotubes (CNTs) and
Carbon nanofibers are just few examples of such important and widely-used nano systems [4, 5,
8, 11, 14, 16-19, 23, 26, 30, 37, 39]. The CNT is an important emerging technology within
nanotechnology that has wide range of applications in several various fields in technology and
science, where recent examples include TVs based on CNT field-emission that are of much
higher resolution than the available best plasma-based TVs, thinner and consume much less
power, and CNT-based nanocircuits such as CNT-based Field Effect Transistors (FETs) that are
much faster than the available silicon-based FETs and possess high potential for less power
consumption. Recently, carbon nanotubes have attracted much attention not only for their unique
morphologies and relatively small dimensions, but also for their potential applications in several
current and emerging technologies [8, 16-18]. CNT is made from graphite that is shaped in three
forms within nano dimensions: (1) Carbon nanoball (or buckyball) with the shape of a soccer ball,
(2) Carbon nanotube that is shaped as multi-wall CNT (MWCNT) and single-wall CNT
(SWCNT), and (3) Carbon nanocoil. Field electron emission is defined as electron emission
under the influence of the applied electric field from the surface of a cathode which is highly
dependent upon the emitting material work function [10, 21, 23, 25]. In this article, the utilization
of Carbon field emission – based devices that realize a basic building block in modern synthesis
known as the controlled switch [31, 36] is introduced, and the use of the presented Carbon field
emission-based devices in many-valued computations is also shown for the widely-utilized case
of ternary Galois logic. In this new realization, the applied Carbon field emission is obtained
using carbon-based nanotubes [5, 16-19] and carbon-based fiber nano-apex tips [6].
This article introduces reversible parallel processing via field emission-based systolic systems.
Figure 1 illustrates the corresponding layout of the introduced system design method.

79
System Systolic Implementations
Carbon Field-Emission Circuits
Carbon Field-Emission Devices
Carbon-Based Field Emission Technology
Field-Emission Physics
Galois Field Algebra
Figure 1. The introduced and utilized system implementation hierarchy.
The remainder of this article is organized as follows: Basic background on fundamentals of
reversible computing, systolic arrays and carbon-based nanotubes and nanotips are presented in
Section 2. The utilization of the Carbon field emission – based devices in controlled switching
and their application in parity-preserving circuits is introduced in Section 3. The extension of the
utilization of Carbon field emission – based controlled switching to the important case of many-
valued computations is introduced in Section 4. The implementation of controlled switching that
use Carbon field emission – based devices within reversible systolic architectures is introduced in
Section 5. Conclusions and future work are presented in Section 6.
2. BASIC BACKGROUND
This Section presents basic fundamentals and important background material in the fields of
reversible processing, parallel systolic computing and Carbon-based nanotubes and nanotips. This
will be utilized in Section 5 for the reversible parallel processing within systolic arrays using
controlled-switching via field emission – based nano multiplexing devices.
2.1. Reversible Computing
An (N, N) reversible system has the same number of inputs N and outputs N and is a one-to-one
mapping between vector of inputs and vector of outputs, thus the vector of input states is uniquely
reconstructed from the vector of output states [1, 3, 9, 20, 29, 32, 34]. Thus, an (N, N) reversible
map is a bijective function which is both injective (“one-to-one”) and surjective (“onto”). In
reversible computing, "garbage" outputs and inputs are auxiliary outputs and inputs that are
required only for the objective of reversibility. Figure 2 shows important reversible gates that are
utilized in the synthesis of reversible circuits and systems.
{a, c = a ⊕ b} {a, b, d = c ⊕ a ⋅ b} {fr0, fr1, c}
(a) (b) (c)
Figure 2. Important binary reversible primitives: (a) (2, 2) Feynman, (b) (3, 3) Toffoli,
and (c) (3, 3) Fredkin.
Because Galois field (GF) possesses attractive characteristics for several implementations such as
in communications, VLSI, testing and signal processing [1, 34], the performed developments will
be done utilizing the corresponding GF algebraic systems. The addition and multiplication tables
of radix two GF(2) and radix three GF(3) of Galois fields are defined as shown in Figures 3(a)
and 3(b), respectively.
a a
b c
a a
b b
c d
0
1
0
1
c
f0
f1
c
fr0
fr1

80
+ 0 1 ∗∗∗∗ 0 1
0 0 1 0 0 0
1 1 0 1 0 1
(a) (b)
Figure 3. Galois field addition and multiplication tables: (a) radix two and (b) radix three.
We define the single-variable function of 1-Reduced Post Literal (1-RPL) [1] as:



≠
=
=
ix
ix
xi
,0
,1 . (1)
For instance, {0
x, 1
x, 2
x} are the zero, first and second polarities of the 1-Reduced Post Literal,
respectively. Also, let us define the ternary shifts of variable x as {x, x', x"} to be the zero, first
and second shifts, respectively, i.e., x = x + 0, x’ = x +1 and x” = x + 2, where x ∈ {0, 1, 2}. The
ternary Shannon expansion over GF(3) for a ternary single-variable function is produced as:
f = 0
x f0 + 1
x f1 + 2
x f2, (2)
where f0 is the cofactor of f with respect to the variable x of value 0, f1 is the cofactor of f with respect to the
variable x of value 1, and f2 is the cofactor of f with respect to the variable x of value 2. It was presented [1]
that the 1-Reduced Post Literals defined in Equation (1) are related in terms of powers to the shifts of
variables as follows:
0
x = 2(x)2
+ 1, (3)
0
x = 2(x')2
+ 2(x'), (4)
0
x = 2(x")2
+ x", (5)
1
x = 2(x)2
+ 2(x), (6)
1
x = 2(x')2
+ x', (7)
1
x = 2(x")2
+ 1, (8)
2
x = 2(x)2
+ x, (9)
2
x = 2(x')2
+ 1, (10)
2
x = 2(x")2
+ 2(x"). (11)
After the substitution of Equations (3) - (11) in Equation (2), and after the minimization of the
terms according to the axioms of the Galois field, one obtains the following Equations:
f = 1⋅ f0 + x ⋅ (2f1+f2) + 2(x)2
(f0+f1+f2), (12)
f = 1⋅ f2 + x' ⋅ (2f0+f1) + 2(x')2
(f0+f1+f2), (13)
f = 1⋅ f1 + x" ⋅ (2f2+f0) + 2(x")2
(f0+f1+f2). (14)
Ternary fundamental single-variable Shannon and Davio expansions are presented in Equations
(2) and (12) - (14), respectively. These equations can be also rewritten in matrix-based forms as:
f = [ 0
x 1
x 2
x]










100
010
001












f
f
f
2
1
0
, (15)
f = [1 x x2
]










222
120
001












f
f
f
2
1
0 , (16)
f = [1 x’ (x’)2
]










222
012
100












f
f
f
2
1
0 , (17)
f = [1 x” (x”)2
]










222
201
010












f
f
f
2
1
0
. (18)
The extension for two or more variables of the many-valued Shannon and Davio decompositions
is achieved utilizing the corresponding Kronecker product of the transform matrix and Kronecker
+ 0 1 2 ∗∗∗∗ 0 1 2
0 0 1 2 0 0 0 0
1 1 2 0 1 0 1 2
2 2 0 1 2 0 2 1

81
product of the vector of basis functions. To introduce the method for creating reversible many-
valued Shannon and Davio decompositions [1], Definitions (1) – (2) are presented.
Definition 1. The matrix that is obtained from the permutations of many basis functions of the
same type of the corresponding spectral transform is called Generalized Basis Functions Matrix
(GBFM).
Definition 2. From all possible GBFM, the matrices that produce reversible decompositions are
called Reversible Generalized Basis Functions Matrices (RGBFM).
Example 1. For the following ternary Shannon transform over GF(3), [ ]




















=
2
1
0
210
100
010
001
f
f
f
cccf
,
the following is one possible GBFM












ccc
ccc
ccc
012
120
210
. Yet, as will be shown, this GBFM is not
reversible since it does not generate a reversible expansion. One possible Reversible Shannon
GBFM (RSGBFM) that leads to a reversible expansion is












ccc
ccc
ccc
102
021
210
. As can be observed here, a
necessary constraint to generate the reversible many-valued Shannon decomposition [1] is that
the elements in any row or column of the utilized RSGBFM are orthogonally permuted compared
to the other elements in the corresponding positions of the other rows or columns.
This reversibility condition can be explained by means of tables as follows: since the Shannon
matrix is orthogonal, then the following (3, 3) ternary Shannon expansion in Equation (19a):










=






























==
2
1
0
2
1
0
210
021
102
100
010
001
][
r
r
r
in
f
f
f
f
f
f
ccc
ccc
ccc
fRSGBFMf
rr , (19a)
is reversible subjected to input c is produced in the output, for which an altered form of Equation
(19a) is:












=






































=
c
f
f
f
c
f
f
f
ccc
ccc
ccc
f
r
r
r
2
1
0
2
1
0
210
021
102
1000
0100
0010
0001
1000
0
0
0
r
. (19b)
As an example, we are using in this case Shannon set of basis functions {0
c, 1
c, 2
c} that appear in
the third row, but Shannon set of basis functions {0
c, 1
c, 2
c} can appear in any of the rows of the
corresponding RGBFM. The ternary Shannon expansion in Equation (19) is reversible as shown
in Table 1 using the permutation of cofactors.
Table 1. Permutation of cofactors as reversibility proof of the ternary Shannon expansion in Equation (19).
Thus, one may note that due to the uniqueness in the 1-RPL selection, the resulting function
values (i.e., cofactors f0, f1 and f2) are always distinct for different 1-RPL values. The same
method can be performed for any configuration that applies RSGBFM for any radix.
Example 2. The following are all possible permutations of the RGBFM for the Shannon matrix










100
010
001 to produce the corresponding reversible GF(3) Shannon expansions [1] given that input c is
produced in the output.
RPL
Function 0
c 1
c 2
c
Equation (19): f = f1 f2 f0

82










=
































=
2
1
0
2
1
0
102
021
210
100
010
001
r
r
r
f
f
f
f
f
f
ccc
ccc
ccc
f
r , (20)










=
































=
2
1
0
2
1
0
021
102
210
100
010
001
r
r
r
f
f
f
f
f
f
ccc
ccc
ccc
f
r , (21)










=
































=
2
1
0
2
1
0
102
210
021
100
010
001
r
r
r
f
f
f
f
f
f
ccc
ccc
ccc
f
r , (22)










=
































=
2
1
0
2
1
0
021
210
102
100
010
001
r
r
r
f
f
f
f
f
f
ccc
ccc
ccc
f
r , (23)










=
































=
2
1
0
2
1
0
210
102
021
100
010
001
r
r
r
f
f
f
f
f
f
ccc
ccc
ccc
f
r , (24)










=
































=
2
1
0
2
1
0
210
021
102
100
010
001
r
r
r
f
f
f
f
f
f
ccc
ccc
ccc
f
r . (25)
For the case of GF(2) two-valued XOR logic, there are two reversible Shannon primitives as
follows:






=

















=
1
0
1
0
10
01
r
r
f
f
f
f
cc
cc
f
r
, (26)






=

















=
1
0
1
0
10
01
r
r
f
f
f
f
cc
cc
f
r
. (27)
Reversible many-valued Davio expansions have been also created [1]. For example, for the
following reversible ternary Shannon expansion:
































=
2
1
0
102
021
210
100
010
001
f
f
f
ccc
ccc
ccc
f
r , (28)
there exist three Davio types for each row of the RSGBFM. Utilizing the derivation of the ternary
Davio expansion (as in Equations (12) – (14)), the following are the D0-type expansions for the
first – third rows of the above RSGBFM












ccc
ccc
ccc
102
021
210
, respectively:
fD0,row0 = [1 c c2
]










222
120
001












f
f
f
2
1
0 , (29)
fD0,row1 = [1 c c2
]










222
012
100












f
f
f
2
1
0 , (30)
fD0,row2 = [1 c c2
]










222
201
010












f
f
f
2
1
0 . (31)
Equation (32) shows the ternary reversible Davio0 decomposition for the corresponding transform
matrix










222
012
100
in Equation (30). Note that we have two other choices of










222
120
001 and










222
201
010
transform matrices in Equations (29) and (31).

83
)32(
)222()2(
)222()2(
)222()2(
)32(
)222)(1()2)(2(
)222()2(
)222)(21()2)(1(
)32(
222
2
121
1
2111
222
012
100
121
1
2111
0,2
0,1
0,0
210
2
021
210
2
102
210
2
210
210
2
102
210
2
102
210
2
102
210
10
2
2
2
2
2
1
0
2
2
2
0
c
f
f
f
fffcffcf
fffcffcf
fffcffcf
b
fffccffcf
fffcffcf
fffccffcf
a
fff
ff
f
ccc
cc
ccc
f
f
f
ccc
cc
ccc
f
Dr
Dr
Dr
D










=












+++++
+++++
+++++
=












++++++++
+++++
++++++++
=










++
+












+++
+++
=
































+++
+++
=
r
In general, there are many reversible gates that are used in the reversible synthesis of logic
circuits and systems, where Figure 4 shows important ternary GF-based reversible gates.
(a) (b) (c)
Figure 4. Ternary GF(3) reversible primitives: (a) (4, 4) Shannon (also known as Fredkin) primitive for
Equation (21), (b) (2, 2) Feynman primitive, and (c) (3, 3) Toffoli primitive.
For example, Table 2 shows the corresponding reversibility proof for the Toffoli primitive shown
in Figure 4(c). Completely reversible digital circuits and systems will significantly minimize
power consumption by implementing three constraints: (a) logical reversibility where the vector
of input states can always be uniquely reconstructed from the vector of output states, (b) physical
reversibility where the physical switch operates bidirectionally (backwards as well as forwards),
and (c) ideal-like switches that have no parasitic resistances.
Table 2. Reversibility of GF(3) (3, 3) Toffoli gate where inputs are in order {a, b, c} and outputs are in
order {a, b, d = c +3 a ∗3 b}.
2.2. Parallel Processing Systolic Arrays
Systolic arrays are special-purpose, cost-effective and high-performance systems that have wide
range of applications in scientific computing. Systolic arrays are single-instruction multiple-data
(SIMD) machines in which each processing element (PE) is only capable of performing a simple
000 000 100 100 200 200
001 001 101 101 201 201
002 002 102 102 202 202
010 010 110 111 210 212
011 011 111 112 211 210
012 012 112 110 212 211
020 020 120 122 220 221
021 021 121 120 221 222
022 022 122 121 222 220
0 1 2 0 1 2 0 1 2
c
fr0 fr1 fr2
f0 f1 f2
c
a
b
a
c
a
b
c
a
b
d

84
single operation [2, 27, 28]. In systolic computing, computations are synchronized by a global
clock signal, and data flows in a rhythmic fashion from the host through the array where data
items are pumped out from a memory. In systolic arrays, regular data flow is kept up in the
network where every PE regularly pumps data each time performing a short computation.
A systolic system consists of a set of interconnected cells, each capable of performing some
simple operation. Properly designed parallel structures communicate only with their nearest
neighbors since valuable time is usually lost when communicating between modules that are far
apart. Because regular, simple control and communication structures have substantial advantages
in implementations, cells in a systolic system are typically interconnected to form a systolic tree
or a systolic array. In systolic systems, communication with the outside world occurs only at the
“boundary” cells and information flows between cells in a pipelined fashion. The basic operation
of a systolic array is achieved by replacing a single PE with an array of PEs. The gain in
processing speed (millions of operations/second) is achieved as the number of pipeline stages has
been increased n-times which is equal to n-PEs. Other several advantages of systolic arrays
include regular and simple flows of data and control, repetitive use of simple and uniform
processing cells, and the important characteristic of expansionability.
Orthogonally connected, linearly connected and hexagonally connected PEs are different forms of
mesh-connected systolic arrays. For example, one-dimensional linear arrays are suitable for FIR
filter, discrete Fourier transform and convolution; two-dimensional square arrays are suitable for
graph algorithms and dynamic programming; two-dimensional hexagonal arrays are suitable for
DFT and matrix arithmetic; trees are suitable for searching algorithms; and triangular arrays are
suitable for formal language recognition and the inversion of triangular matrix. The globally
structured systolic array can speed-up computations only if the I/O bandwidth is high, where I/O
is still the bottleneck in modern systolic systems.
The following sub-sections present important examples of parallel processing using various
systolic architectures that highly accelerate intensive arithmetic – based computations which are
common in scientific computations.
2.2.1. Hexagonal (band matrix – band matrix multiplication) Kung systolic array
Since several scientific and engineering computations involve band matrices, band matrices find
important and wide implementations in several fields of scientific computing. This is also
important because a dense matrix can be viewed as a band matrix that have the maximum
bandwidth.
Each pulsation in a systolic array consists of the following operations: (1) shift, (2) multiply, and
(3) add. Basic processing cells that are used in the construction of systolic arithmetic arrays are
the add-multiply cells. This kind of cells has three inputs {a, b, c} and three outputs {a = a, b = b,
d = c + a ∗ b}. One may view the processing cell boundaries as having six interface registers that
are attached at I/O ports. All of these registers are clocked for synchronous data transfer between
adjacent cells. The add-multiply operation is required to perform the inner product within matrix-
matrix multiplication, matrix inversion, and dense matrix LU decomposition.
It has been shown that hexagonally-connected processing elements can maximize the
performance of the corresponding matrix multiplication, where pipelined three data streams flow
through the utilized array. Figure 5 shows the multiplication of two band matrices [A] and [B]
using two-dimensional hexagonal systolic array.

85
Utilizing Figure 5, one can understand the operation of the two-dimensional hexagonal systolic
array by following the corresponding data flow via moving transparencies of the band matrices
over the entire network.
(a)
(b)
Figure 5. Two-dimensional hexagonal systolic array: (a) Kung cell and (b) Kung systolic array.
Utilizing Figure 5, the multiplication of band matrices [A]⋅[B] = [C], and the associated definition
of bandwidths, are given as:
where w1 = 3 + 2 - 1 = 4, w2 = 2 + 3 – 1 = 4, and w3 = w1 + w2 – 1 = 4 + 4 – 1 = 7. Each front data
level in the three data flow streams in Figure 5 is called a wave front. From the lower side of
Figure 5, the initial values of the elements of input [C] array are all zeros, and the resulting values
of the elements of output [C] array at the upper side are obtained as follows:
.,,,,0
,,,
,,,,
,,,
,,,,
,,0,,,,
5555455435535555454544354345453435333535232515
5455445434535454454444344324424444343433243234
3423242224241214435433535343443343234243
433433332332133133332323221321232312131113
32535232432242423233223212313232232222122122
221212111251214241213211313121221121212112111111
bababacbababacbabacbacc
bababacbabababacbababac
babacbacbabacbababac
babababacbababacbabac
bacbabacbababacbababac
babaccbacbabacbabacbabac
++=++=+===
++=+++=++=
+==+=++=
+++=++=+=
=+=++=++=
+===+=+=+=
The distribution of PEs in the shown hexagonal systolic array is obtained as: # PEs in top-left =
w1 = 4, # PEs in top-right = w2 = 4, # PEs in bottom = w3 = 7. The total # PEs = 4 ⋅ 4 = 16.
2.2.2. Band matrix - vector multiplication systolic array
Since a dense matrix can be viewed as a band matrix that have the maximum bandwidth,
transforming vectors using band matrices are important in several implementations. This type of
multiplication is illustrated as follows:
a
d = c + a⋅b
b c a
b
PE
b21
b12
b13
b22 b23
b32
c11
c21 c12
c31 c22 c13
c41 c32 c23 c14
c42 c33 c24
c52 c43 c34 c25
b33
b24
b43
b11
a11a21a31
a12
a22a32a42
a23a33
a34
C

86
















=
































5
4
3
2
1
5
4
3
2
1
53
45444342
34333231
232221
1211
......00
0
0
00
000
y
y
y
y
y
x
x
x
x
x
a
aaaa
aaaa
aaa
aa
,
where the bandwidth w is computed as w = 3 + 2 - 1 = 4. Figure 6 illustrates the band matrix-
vector multiplication using one-dimensional systolic array, where the following operations occur:
.
,,,
5454443432424
434333232131332322212122121111
xaxaxaxay
xaxaxaxayxaxaxayxaxay
+++=
+++=++=+=
Using this type of arrays, it has been shown that the computation of n components of y are
performed in (2⋅n + w) time units as compared to O(n⋅w) which is needed on a uni-processor.
yi
(k+1)
= yi
(k)
+ aik⋅xki
(a) (b)
Figure 6. Band matrix – vector multiplication systolic array: (a) cell and (b) systolic architecture.
2.3. Carbon Nanotubes and Nanotips
This sub-section presents important background on Carbon-based nanotubes and nanotips and
their corresponding characteristics. The presented Carbon-based nanotubes and nanotips will be
utilized in building controlled-switching that will be later used for constructing the corresponding
reversible nano systolic networks in Section 5.
2.3.1. Carbon nanotubes characteristics
Carbon nanotube, which is a cylindrical graphite sheet, is geometrically formed in two distinct
shapes that affect CNT characteristics: (a) straight CNT and (b) twisted CNT. Figure 7 shows
different forms of CNTs and an important application in electronic devices using CNT as a
channel in FET transistors.
(a) (b) (c) (d)
Figure 7. Various formations and shapes of CNTs: (a) Carbon nanoball which has the shape of a soccer
ball, (b) single-wall CNT (SWCNT), (c) multi-wall CNT (MWCNT) array image using Scanning Electron
Microscopy (SEM) where these MWCNTs have been grown utilizing Chemical Vapor Deposition (CVD),
and (d) the implementation of CNT as a channel in FET transistors.
The emerging CNT technology has been applied in several new implementations [4, 5, 8, 11, 14,
16-19, 23, 26, 30, 37]. This includes TVs that are based on the field emission of CNTs, CNT-
based FETs that are faster and consume less power than the available silicon FETs, Carbon
a31
a42
a43
a21
a32
a22
a33
a11
a12
a23
a34
yi
xi

87
nanocoils that are used as nanosprings in nano dynamic systems and as inductors in nanofilters,
CNT-based rings, new composite materials, CNT probes, drug delivery systems, CNT data
storage devices capable of storing 1015
bytes/cm2
, nano lithography, and CNT gears. Table 3
illustrates several of the important CNT characteristics. For example, CNTs are very good field
emitters and facilitate the miniaturization of electronic devices. The dynamic and static
characterizations of field emitters consisting of a single CNT - both multi-walled and single-
walled - were illustrated [24] and the comparison to a field emitter consisting of an etched single
tungsten crystal was also investigated.
Table 3. Carbon nanotube characteristics.
Characteristic Single-Walled CNT Comparing Examples
Size
diameter of 0.6 - 1.8 nm
Electron-based beam
lithography can generate lines
50 nm wide and few nm thick
Density 1.33 – 1.40 g/cm3 Aluminum: 2.7 g/cm3
Strength
≈ 45·109
Pascals
High-strength steel alloys break
at ≈ 2·109
Pascals
Resilience
Can be bent at large angles and re-
straightened without damage
Metals and carbon fibers
fracture at grain boundaries
Current Capacity ≈ 1·109
A/cm2 Copper: ≈ 1·106
A/cm2
Field Emission
Can activate phosphors at
1 – 3 V if electrodes are spaced 1
micron apart
Molybdenum tips need
≈ 50 – 100 V/micrometer
with very short lifetimes
Heat Transmission
≈ 6,000 W/m·K at room
temperature
Almost pure diamond
transmits ≈ 3,320 W/m·K
Thermal Stability
Stable up to 2,800 C in vacuum
and 750 C in air
Metal wires in microchips
melt at ≈ 600 – 1,000 C
Cost ≈ 1,500 $/g Gold: ≈ 40 $/g
Quantum Electron Spin Preservation is very high
Preservation is low in
regular conductors
Power Consumption Very low Higher in metal wires
Performance (Speed) ≥ 1·1012
Hz nanoswitch ≥ 1,000 times as fast as µP
Resistance (Electron
Scattering)
Almost none Comparatively high
Energy Band Gaps
Easily tunable: almost zero like a
metal, as high as a Silicon, and
almost anywhere in between
No other recognized material
can be adjusted so simply
2.3.2. Carbon nanotube - based multiplexing devices
Figure 8 shows the design of a logic inverter. This device functions in the same way as a regular
CMOS-based inverter, where it uses two nanotube-based FETs [18].
In Figure 8, silicon is used as back-gate in the device, silicon dioxide is used as an insulator, gold
is used as conductors (electrodes), and PMMA is utilized as a cover that protects anything
beneath it from being exposed to oxygen since oxygen is used to obtain p-CNTFET from the
corresponding n-CNTFET.

88
Figure 8. Inverter gate using two nanotube-based FETs.
Figure 9 illustrates solid-state CNT-based multiplexer [5]. In Figure 9, CNT is used as a channel
in the corresponding FET transistor. Variable C is control input, selected inputs are A and B, and
F is the output. The device operates as a controlled-switch by switching between inputs A and B
depending on the specified value of control C [5].
Figure 9. The device structure of solid-state inter-molecular CNT multiplexer.
2.3.3. Carbon nanotubes - based field emitters
Field electron emission is the electron emission under the influence of the applied electric field (3
x 109
V/m) from a cathode surface which is highly dependent upon the material work function.
The general form of Fowler-Nordheim-type formulae [22] can be obtained as follows [21]:
J = λLaφ-1
F2
EXP[ -VF b φ3/2
/ F ] (33)
Equation (33) is used in all cases of field emission processes, where J is the local emission
current density, a and b are the first and the second Fowler – Nordheim constants, νF is the
barrier form correction factor where it accounts for the particular shape of the potential barrier
model, and λL is the local pre-exponential correction factor where it takes into account all of the
other factors that influence the emission. Factors νF and λL depend on the implemented field F.
To achieve the static modeling of CNT field emitters, field emission CNTs shown in Figures
10(b)-10(c) [24] were manufactured by Xintek, Inc. The copper anode is at the right and the CNT
emitter is mounted on a tungsten wire which is attached to the copper cylinder at the left as shown
in Figure 10(a). Carbon nanotube M-4 has a single MWCNT as the emitter, and carbon nanotube
C-3 has a single SWCNT as the emitter. The utilized CNTs were shaped in bundles of 10-30 nm
diameters, but in each carbon nanotube the field emission is performed from the single CNT at
the end of the bundle where the most intense electric field exists.

89
(a) (b) (c)
Figure 10. Field emission from CNT: (a) field emission CNTs made by Xintek, Inc. and (b) - (c) Scanning
Electron Microscopy (SEM) images of CNT emitters using JEOL Ltd. model JEM 6300.
The I-V DC characteristics were measured for these CNTs, as well as a field emitter tube from
Leybold Didactic GmbH, which has an etched tungsten single crystal as the emitter. The data
from the DC measurements were reduced by the Fowler-Nordheim analysis which is based on the
following simplified form of the Fowler-Nordheim equation:
J = A E2
e (-B/E)
, (34)
where J and E are the magnitudes of the current density and the electric field intensity, A = 1.541
x 10-6
/Φ, and B = 6.831 x 109
Φ3/2
. The work function values are Φ = 4.5 eV for tungsten and was
set to Φ = 4.9 eV for graphene. In order to apply the Fowler-Nordheim equation to the I-V DC
data, the following equation was also utilized which is correct for provided CNT:
I = CV2
e (-D/V)
. (35)
To get the equations for parameters S and R, which are utilized to characterize the used field
emitters, Equations (34) and (35) can be combined to obtain:
S = CD2
/AB2
, (36)
R ≡ V/E = D/B, (37)
where parameter S refers to the effective emitting area which would be the physical area of the
emitter if the current density were uniform over a fixed area and zero elsewhere, and parameter R
refers to the effective radius of curvature of the emitter but also includes the effects of local
intensification of the electric field caused by elongation of the emitter or the reduction of the field
which may be caused by shielding due to adjacent structures. The Fowler-Nordheim plots of the
I-V DC data were performed as shown in Figure 11.
0 .0 0 0 9 5 0 .0 0 1 0 0 0 .0 0 1 0 5 0 .0 0 1 1 0 0 .0 0 1 1 5 0 .0 0 1 2 0
-3 2
-3 1
-3 0
-2 9
-2 8
-2 7
ln(I/V
2
)
1 /V
Figure 11. The Fowler-Nordheim plot for experimented CNT that have a single SWCNT as emitter.
The values of parameter R, which is the effective radius of emitter curvature, were found to vary
within 77-110 nm for the utilized carbon nanotubes with CNT emitters. This suggests that values
of the local electric field at the emitting sites were as high as 14 V/nm in some of these
measurements. Others studying the field emission from various CNTs have provided approximate
values for the electric field by dividing the applied voltage by the distance between the anode and

90
the emitting tip, noting that this field would be intensified by the shape of the CNT, but not
estimating the local electric field at the emitting sites [35]. The Fowler-Nordheim analysis gave a
value of 91 nm for the effective radius of emitter curvature in the Leybold tube, suggesting that
the local electric field was as high as 5 V/nm in some of the performed measurements [24].
Current densities as high as 109
and 1012
A/m2
may be drawn from a tungsten emitter in steady-
state and pulsed operation, respectively, and the corresponding values of the applied static field
are 4.7 and 8.6 V/nm which may be considered as limiting field strengths for tungsten under these
applied conditions.
The Fowler-Nordheim analysis also showed that the parameter S, which is the effective emitting
area, varied within 81-230 nm2
for the carbon nanotubes with CNT emitters. If the current density
was uniform, this would correspond to circular emitting spots having radii of approximately 5-9
nm. Others have used Lorenz microscopy to observe the emitting sites for MWCNT field
emission. It has been found that one or more sites having radii of several nm, and the
corresponding data was in good agreement with other conducted experiments. The Fowler-
Nordheim analysis also showed that the effective emitting area for the tungsten tip in the Leybold
tube would correspond to a hemisphere with a radius of 290 nm. This result, and the value of 91
nm for the effective radius of emitter curvature in the Leybold tube, were in good agreement with
the radius of 100-200 nm which was indicated by Leybold.
Conducted experiments and simulations had shown that optical heterodyning (i.e., photomixing)
in laser-assisted field emission could be utilized as a new multi-octave bandwidth terahertz or
microwave source [7, 12, 13, 25].
2.3.4. Carbon fiber nanotips - based field emitters
Because of the technological significance of carbon fibers, there has been a rising interest with
regards to investigating the mechanism of field electron emission from these fibers under the
effect of an implemented electric field. Several advantages are obtained by using carbon fibers as
cathodes such as high current stability, long emitter durability, simplicity of emitter
manufacturing, and offering the ability to work in a relatively high pressure environment (10-6
mbar). These utilized cathodes are made either from carbon fibers or from other carbon-based
materials. Carbon fiber emitters can be achieved by electrolytic etching technique, where a 0.1
Molarit of sodium hydroxide solution (250 ml of distal water with 2g of NaOH) is used [6]. This
etching process could be controlled by choosing a suitable etching current. The etching process is
started after dipping the tip in the solution by about 2 mm and using a power supply to increase
the voltage until a certain initial current of about 30 µA is reached. The chosen etching current
produces sharp tips at the liquid surface which are afterwards being ultrasonically cleaned using
an ultrasonic cleaning device and mounted in a standard FEM microscope with a tip screen
distance of 10 mm. The anode is formed as phosphored screen to allow recording of the emission
images.
The FEM was evacuated to ultra high vacuum (UHV) conditions using rotary pump that produce
pressures of about 10-3
mbar and a diffusion pump system in addition to a liquid nitrogen (LN2)
trap that lead to finally reaching a base pressure of about 10-9
mbar. Then, the tips received
sample conditioning which consists of an initial baking of the system for 12 hours at 170 °C, a
follow up baking of the system for 12 hours, thermal relaxation for 12 hours at 170 °C, and
finally cooling the sample by liquid nitrogen while studying emission behavior [6]. This allowed
the recording of the effects of these conditioning processes on carbon fiber tips.

91
To record the emission behavior, the applied voltage from extra high tension (EHT) to the tip was
slowly increased until the emission current increased to about one microampere, and then the
voltage was slowly reduced until the emission current disappeared. Within this range, a linear
Fowler-Nordheim plot is expected. Figure 12 presents the scanning electron micrograph of a
produced very sharp carbon fiber tip at about 10,000 x magnification.
Figure 12. Scanning electron micrograph of a very sharp carbon fiber tip at 10,000 × magnification.
The apex radii of carbon fiber tips have been measured as the average of the graphically best-
fitting circles. The latter ones are utilized to interpret the experimental data and extract relevant
information such as the apex radii. During the experiments, electronic emission images have been
recorded by a standard digital camera to study the stability of the emission current and spatial
distribution. For practical applications, stability as well as brightness are important factors for
judging the quality of the used electron source. During the sample conditioning treatment, it was
found that there were statistical variations in the electronic emission of the various tip
microemitters under the corresponding UHV conditions. One of the very sharp carbon fiber tips
has been tested during sample conditioning treatment [6] where the apex radius of this tip was
around 57 nm. Figures 13-16 show the emission characteristics that are derived as the I-V
characteristics and the corresponding Fowler-Nordheim plots.
Figure 13. The I-V characteristics (left) and FN plot (right) of very sharp tip after initial baking for 12 hours at 170°C.
Figure 14. The I-V characteristics (left) and FN plot (right) of very sharp tip after follow-up baking for 12 hours at 170°C.
Figure 15. The I-V characteristics (left) and FN plot (right) of very sharp tip after thermal relaxation process for 12 hours.

92
Figure 16. The I-V characteristics (left) and FN plot (right) of very sharp tip during cooling process.
After initial baking, as was shown in Figure 13, the FN plot shows that emission current was
unstable and a non-linear behavior. The follow up baking, as shown in Figure 14, shows a
disconnected plot because the voltage drops from 66 V to 5.5 V where this drop is due to
hysterical current. As shown in Figures 15 and 16, the emission current stability becomes much
higher after the sample conditioning.
3. CARBON FIELD EMISSION – BASED CONTROLLED SWITCHING
This Section introduces the Carbon field emission – based controlled switches and their extension
to the case of many-valued Galois logic.
3.1. Two-to-One Controlled Switching
Utilizing the previously experimented and observed characterizations and operations of Carbon
field emission that were presented in sub-section 2.3, Figure 17 introduces the Carbon field
emission – based primitive that realizes the 2-to-1 controlled switch. The applied electric field
intensity (E), or equivalently the voltage (V) or work function (Φ), are utilized to implement the
input control signal that is used to control the electric function of the device.
The description of the operation of the Carbon field emission – based device shown in Figure
17(b) is as follows: by the implementation of the high voltage control signal (HV), the voltage
difference between the facing anode and Carbon cathodes is altered. This change will cause the
Carbon cathode with (HV) control signal to be field emitting while Carbon cathode with
complementary ( VH ) control signal to be without field emission. When the voltage difference is
reversed, Carbon cathode with complementary ( VH ) control signal will be field emitting while
Carbon cathode with (HV) control signal will be without field emission. Therefore, the device in
Figure 17(b) realizes the function of 2-to-1 controlled switching G = ac + bc'.
(a) (b) (c)
Figure 17. The implementation of the operation of 2-to-1 controlled switching using Carbon field emission:
(a) 2-to-1 multiplexing, (b) controlled switching via CNT field emission, and (c) 2-to-1 multiplexer block
diagram, where HV is the source with high-voltage and VH is the opposite (i.e., complementary) source.
The distance d needed between the facing anode and cathodes – as shown in the experimental
results - must be around 10 mm. If this distance is not applied, beam distortion will exist that will
affect the collected current at the facing anode screen.
a
b
c
2-1
CS
G
.
.
Control: c
Input1: a
Input2: b
G
HVHV
- -
R1
R2

93
Since the equations that relate the electric field intensity (E), voltage (V), distance (d), current
density (J), and work (energy) function (Φ), are as follows:
Φ = e⋅V, (38)
V = E⋅d, (39)
d = V / E, (40)
J = I / (a/Ω) = I ⋅ (Ω/a), (41)
where e is the electron charge ≅ 1.602 ⋅ 10-19
C, a is the tip area and Ω is the emission angle, then
the following represents the equation that models current value on the anode screen:
2VV
Bd
e
2Ωd
aA
I










−
=
(42)
Observing Equation (42), one can note the proportionality relation between the current value (I)
and both of the voltage difference (V) and tip area (a), and the inverse relation with the emission
angle (Ω), where A = 1.541 x 10-6
/Φ and B = 6.831 x 109
Φ3/2
. For instance, the values of the
work function are Φ = 4.5 eV for tungsten and Φ = 4.9 eV for CNT, where the usual value used
for electric field intensity E is ≥ 3⋅109
V/m, the distance d between the anode screen and CNT
cathode is ≅ 10 mm, and V ≅ 3⋅107
V.
3.2. Generalization to Many-to-One Controlled Switching
The synthesis of many-to-one (m-to-1) controlled switching via Carbon field emission can be
performed utilizing the basic two-to-one controlled switching from Figure 17(b). For instance, for
the case of three-valued logic, two devices from Figure 17(b) are required to implement the
function of 3-to-1 controlled switching as shown in Figure 18.
Figure 18. The three-to-one controlled switching utilizing Carbon field emission, where devices {D1, D2}
can be implemented using the switching device from Figure 17(b).
For the more general case of m-valued logic, and to realize the function of an m-to-1 controlled
switching, one requires (m-1) of 2-to-1 controlled switches as shown in Figure 19.
Figure 19. The realization of (m-to-1) controlled switching, where devices {D1,..., D(m-1)} can be the Carbon field
emission – based controlled switching from Figure 17(b).
Example 3. Using carbon field emission – based controlled switching, this example shows the
implementation of the important parity-preserving reversible controlled-swap gate. For error
.
.
.
D1
Control Signal #1:
Condition #1 ≡ “0”
Control Signal #(m-1):
…
.
.
.
.
.
.
D2
D3
D(m-1)
Control Signal #1:
Control Signal #2:
D1
D2

94
correction purpose, parity checking is one of the most widely used and oldest techniques that are
utilized for error detection within digital circuits and systems. It can be shown that reversible
logic gates having an equal number of inputs and outputs are sufficient for the parity preservation
if a reversible circuit at each gate is parity-preserving, where parity-preserving reversible gates
refer to reversible gates for which the parity of the output matches that of the input. The selection
from such gates depends on the specifications for the required circuit design process.
Fredkin gate [1, 20, 34] is a basic building block in reversible computing, where several
realizations were performed to implement Fredkin gate such as in optical, electrical and nano
mechanical technologies. As previously stated, parity-preserving reversible gates can be utilized
for the detection of existing faults. Given the fact that reversible gates have the same number of
input and output vectors, a sufficient requirement for parity preservation - in the course of
reversible computation with such gates - is that each gate must be parity-preserving, i.e., have the
same parity for the input and output vector.
For the binary Galois radix, let I = (a; b; c) and O = (P; Q; R) be the input and output vectors of a
(3, 3) reversible gate, respectively. We call a reversible gate to be a Parity-Preserving Reversible
Logic Gate (PPRLG) if the gate satisfies the parity-based equation: a ⊕ b ⊕ c = P ⊕ Q ⊕ R,
where ⊕ is the Boolean XOR, i.e., GF(2) addition.
Figure 20 shows two widely used PPRLGs which are based on the fundamental Fredkin primitive
(also known as the two-valued reversible Shannon primitive). In Figure 20, both of the reversible
gates are parity-preserving because they satisfy the parity equation a ⊕ b ⊕ c = P ⊕ Q ⊕ R.
(a) (b)
Figure 20. Representations for important reversible primitives: (a) reversible GF(2) Shannon1 (i.e.,
Fredkin1) and (b) reversible GF(2) Shannon2 (i.e., Fredkin2). Carbon field emission - based realization
shown in Figure 17(b) is used to realize both of the 2-to1 controlled switches.
For the general case of m-valued logic, an m-valued Galois PPRLG satisfies the equation a +m b
+m … +m c = P +m Q +m … +m R, where m = pk
, p is a prime number, k is a natural number of
value k ≥ 1, and {a, b, …, c} and {P, Q, …, R} are the corresponding input and output vectors of
an m-valued (m + 1, m + 1) reversible gate, respectively. As an example, for the important
specific case of the third radix Galois field, let I = (a; b; c; d) and O = (P; Q; R; L) be the input
and output vectors of a ternary (4, 4) reversible gate, respectively. A ternary PPRLG is a gate that
satisfies the parity-based GF3 equation of a +3 b +3 c +3 d = P +3 Q +3 R +3 L. For instance,
g0 g1 c g1r0 g1r1 c
0 0 0 0 0 0 0 0
1 0 0 1 0 0 1 1
2 0 1 0 0 1 0 2
3 0 1 1 1 0 1 5
4 1 0 0 1 0 0 4
5 1 0 1 0 1 1 3
6 1 1 0 1 1 0 6
7 1 1 1 1 1 1 7
g0 g1 c g2r0 g2r1 c
0 0 0 0 0 0 0 0
1 0 0 1 0 0 1 1
2 0 1 0 1 0 0 4
3 0 1 1 0 1 1 3
4 1 0 0 0 1 0 2
5 1 0 1 1 0 1 5
6 1 1 0 1 1 0 6
7 1 1 1 1 1 1 7
0 1 0 1
g0 g1
cc
g2r0 g2r1
0 1 0 1
g0 g1
c
g1r0 g1r1

95
Equation (43) characterizes the operation of the 3-valued Fredkin gate (also known as 3-valued
reversible Shannon):










=
































=
2
1
0
2
1
0
210
021
102
100
010
001
r
r
r
g
g
g
g
g
g
ccc
ccc
ccc
g
r (43)
where i
c is 1-RPL and c ∈ {0, 1, 2}. An alternative form of Equation (43) is given in Equation
(44) for which it is explicitly shown that input c is produced in the output.














=










































=
c
g
g
g
c
g
g
g
ccc
ccc
ccc
g
r
r
r
2
1
0
2
1
0
210
021
102
1000
0100
0010
0001
1000
0
0
0
r (44)
The reversible Shannon expansion in Equation (44) is represented using the 3-to-1 controlled
switching as illustrated in Figure 21, where Table 4 illustrates the corresponding reversibility
proof.
Figure 21. Realization of the reversible expansion in Equation (44) using three multiplexing devices.
Carbon field emission - based realization shown in Figure 18 is used to realize this 3-to-1 controlled switch.
Table 4. Reversibility proof of the (4, 4) primitive from Figure 21 and Equation (44).
4. MANY-VALUED COMPUTING USING CARBON FIELD EMISSION-BASED
DEVICES
The previously developed GF(3) Carbon field emission–based device will be used in this section
for implementing many-valued computing. This section demonstrates the utilization of Carbon-
based controlled switching for the ternary case, but implementations for higher Galois radices
follow the same illustrated method.
Inputs (I) Outputs (O)
g0 g1 g2 c gr0 gr1 gr2 c
g0 g1 g2 0 g1 g2 g0 0
g0 g1 g2 1 g2 g0 g1 1
g0 g1 g2 2 g0 g1 g2 2
0 1 2 0 1 2 0 1 2
c
gr0 gr1 gr2
g0 g1 g2
c

96
(a) (b)
Figure 22. The Carbon - based implementation of Galois arithmetic operations: (a) controlled switch that
can be implemented using the device from Figure 17(b), and (b) circuit that uses controlled-switching to
implement GF(3) addition and multiplication tables.
Figure 22 shows a controlled switch-based circuit that implements GF(3) addition and
multiplication tables, where Figure 22(a) can be implemented using the 2-to-1 Carbon field
emission – based device from Figure 17(b). For the internal nano interconnects in Figure 22(b),
they can be implemented – as one alternative – using CNTs as shown in Figure 23, where the
symbol = means a metallic CNT that is used as a nanowire.
In Figure 22(b), variables {A, B} ∈ {0, 1, 2} and inputs Ck (k = 0, 1, 2, 3) are 2-valued control
variables ∈ {0, 1}. Note that Figure 22(b) implements GF(3) addition and multiplication
operations by using the appropriate values of Ck that select variable inputs {A, B} and constant
inputs {0, 1, 2}. Table 5 illustrates an example for implementing the corresponding GF(3)
addition and multiplication tables using Figure 22(b).
(a) (b)
Figure 23. The realization of nano interconnects using CNT: (a) image from TEM of a bundle of SWCNTs
catalyzed by Ni/Y mixture, and (b) developing CNT wires on catalysts, where CNT meshes
are illustrated on which the metal catalyst is covered.
The circuit in Figure 22(b) can be used in the system design where GF(3) addition and
multiplication operations are applied. A simple example is presented as follows.
Table 5. An example for the implementation of GF(3) addition and multiplication tables using Figure 22(b),
where + means GF(3) addition, * means GF(3) multiplication, Ck (+) means that control variable Ck
implements the ternary addition operation, and Ck (*) means that control variable Ck implements the
ternary multiplication operation.
A B C G
a b 0 a
a b 1 b
A B C0 (+) C1 (+) C2 (+) C3 (+) C0 (*) C1 (*) C2 (*) C3 (*) + *
0 0 0 0 0 0 0 0 0 0 0 0
0 1 1 0 0 0 0 0 0 0 1 0
0 2 1 0 0 0 0 0 0 0 2 0
1 0 0 0 0 0 1 0 0 0 1 0
1 1 0 0 0 1 0 0 0 0 2 1
1 2 0 1 0 0 1 0 0 0 0 2
2 0 0 0 0 0 1 0 0 0 2 0
2 1 0 1 0 0 0 0 0 0 0 2
2 2 0 0 1 0 0 0 1 0 1 1
A 0
B 1
C
G
A
B
0
1
2
C0 C1 C2 C3
0
1 0
1 K
0
1
0
1

97
Example 4. Three-valued function K = I4I1 +3 I2I3I1 +3 I2I3 can be implemented as follows using
the addition and multiplication operations that are realized using Figure 22(b) and Table 5.
Figure 24. A GF(3) circuit to implement the three-valued function K = I4I1 +3 I2I3I1 +3 I2I3, where Table 5
shows the implementation of GF(3) addition and multiplication tables using Figure 22(b), and the symbol =
means a metallic CNT that is used as a nanowire.
Example 5 shows the system-level design of an Arithmetic and Logic Unit (ALU) circuit by
illustrating the implementation of a 2-digit multiplier using the introduced Carbon field emission
- based devices.
Example 5. For a 2-digit m-valued multiplication performed using mod-multiplication, Figure 25
shows maps for the ternary multiplication and carry out functions, and Figure 26 shows the
corresponding 3-valued circuit realization.
(a) (b)
Figure 25. Ternary multiplication for ternary 2-digit multiplier: (a) multiply and (b) carry out.
Figure 26. Logic circuit of a ternary 2-digit multiplier.
The corresponding GF(3) addition and multiplication operations in Figure 26 could be
implemented using Table 5 that specifies input values to Figure 22(b). Further implementation of
the general m-valued N-bit full ALU, which is the main functional unit in the microprocessor data
path, that includes the realization of all arithmetic sub-units (addition, subtraction, multiplication,
division) and all logic sub-units (NOT, AND, OR, XOR), can be performed from the utilization
of the Carbon-based controlled switch device previously introduced by using the same method
which was used in the realization of the 2-digit multiplier shown in Figure 26.
x y 0 1 2
0 0 0 0
1 0 1 2
2 0 2 1
x y 0 1 2
0 0 0 0
1 0 0 0
2 0 0 1
*
*
*
* + K
+
0
*
+
B0 A0
*
+
B1 A0
0
++++ 0
0
Cout S3 S2 S1 S0
*
+
B0 A1
*
+
B1 A1
0

98
5. REVERSIBLE SYSTOLIC ARRAYS VIA CARBON FIELD - EMISSION
MULTIPLEXING
This section introduces the synthesis of m-valued Galois functions using Carbon field emission –
based reversible nano systolic arrays utilizing any of two methods: (1) reusing the classical add-
multiply PE cells by showing their m-valued Galois logic reversibility and thereafter reusing the
whole systolic structure since the connection of reversible PEs will produce by necessity a
reversible systolic circuit (cf. Figure 28), and (2) direct mapping of classical irreversible systolic
circuits into reversible systolic circuits by interconnecting the reversible counterpart of the
irreversible PEs. Moreover, the implementation of the m-valued logic functions using the
reversible systolic arrays is performed through two methods as shown in Figure 27.
Figure 27. Two methods to realize reversibly m-valued Galois logic by using reversible function
decompositions and reversible systolic arrays.
(a) (b)
(c)
Figure 28. Reversible GF(2) Kung systolic array: (a) reversible (3, 3) Toffoli gate,
(b) reversible (3, 3) Kung cell, and (c) reversible Kung systolic.
a
d = c +2 a ∗2 b
b
c
a
b
PE
a a
b b
c d = c +2 a ∗2 b
b21
b12
b13
b22 b23
b32
c11
c21 c12
c31 c22 c13
c41 c32 c23 c14
c42 c33 c24
c52 c43 c34 c25
b33
b24
b43
b11a11a21a31
a12a22a32a42
a23a33
a34
C
m-valued Reversible
Decompositions
m-valued
Reversible
Systolic Arrays
Galois m-valued
Function

99
Add-multiply cells are basic PEs used in the construction of arithmetic systolic arrays. Figure
28(a) introduces the binary (two-valued) (3, 3) reversible Toffoli gate that implements reversibly
- in the second radix Galois logic - the classical Kung add-multiply cell shown in Figure 28(b).
Since the interconnection of reversible PEs will produce by necessity a reversible systolic circuit,
Figure 28(c) implements the two-dimensional reversible Kung systolic array over GF(2) by
interconnecting the corresponding GF(2) (3, 3) reversible Toffoli gates.
The systolic implementation using GF(2) matrix-based elements, that can be obtained through the
corresponding GF(2) reversible Shannon function decomposition, is obtained as follows:
















+
+
=
















⋅
















00000
00000
0000
0000
00000
00000
00000
0000
0000
00000
00000
00000
000
000
00000
1
0
0
1
1
1
0
0
1
0
01
10
efef
efef
f
f
ee
ee (45)
where 1
0
0
1
321
1
0
0
22132022
0
33
1
32
1
23
0
22 ,,,,,,, efefcefefcfbfbeaeaeaea +=+======= .
For the systolic implementation of m-valued functions, and as an example, the needed matrix-
based elements for the GF(3) reversible Shannon and Davio decompositions are, respectively:
















++
++
++
=
















⋅
















00000
0000
0000
0000
00000
00000
0000
0000
0000
0000
00000
00
00
00
00000
0
2
2
1
1
0
2
0
1
2
0
1
2
2
1
1
0
0
0
2
1
0
210
021
210
efefef
efefef
efefef
f
f
f
f
eee
eee
eee (46)
Where:
.,,
,,,,,,,,,,,,,
2
1
1
0
0
2
432
0
1
2
0
1
332
2
1
1
0
0
23
043233123013
2
44
1
43
0
42
0
33
2
32
1
31
2
23
1
22
0
21
efefefcefefefcefefefc
fbfbfbfbeaeaeaeaeaeaeaeaea
++=++=++=
=============
















++++++++
+++++
++++++++
=
















++
+
⋅
















+++
+++
00000
00)222)(1()2)(2(00
00)222()2(00
00)222)(21()2)(1(00
00000
00000
0000
0022200
00200
0000
00000
01120
001
002111
00000
210
2
102
210
2
102
210
2
102
2
210
10
2
2
2
2
fffeeffef
fffeffef
fffeeffef
f
fff
ff
f
eee
ee
eee
(47)
Where:
).222)(1()2)(2(),222()2(
),222)(21()2)(1(
,),222(,2,,1
,1,2,,,1,1,1,1
210
2
10243210
2
10233
210
2
10223
24321033102321344
2
4342
2
333231
2
232221
fffeeffefcfffeffefc
fffeeffefc
fbfffbffbfba
eeaeaeaeaaeeaeaa
++++++++=+++++=
++++++++=
=++=+===
++=+====++=+==
The realization of a many-variable function using the methods shown in Equations (45) – (47)
follow the matrix-based expansion using the Kronecker (tensor) product and GF(3) Toffoli PE.
The temporal complexity of the array in Figure 28 is T = 3n + min(w1, w2) and spatial complexity
for the count of auxiliary (garbage) outputs is 212 ww ⋅ .
The generalized case of m-valued logic is performed similarly to Figure 28 except that the
arithmetic operations will be done within the general case of m-valued Galois field and the
controlled-switching implementation will utilize the corresponding multiplexing architecture. For
example, the implementation of the fundamental cell in Figure 28(b) for the case of GF(3) is
performed by the utilization of Table 5 for the implementation of the Galois arithmetic GF(3)
addition and multiplication operations using Figure 22(b) that utilizes the serial interconnects of
several 2-to-1 basic controlled-switching elements, where each of these 2-to-1 controlled
switching elements can be directly implemented using the technique shown previously in Figure
17 utilizing Carbon field emission–based devices.

100
6. CONCLUSIONS AND FUTURE WORK
In this article, the synthesis and operation of reversible nano systolic architectures for the
implementation of general m-valued Galois functions using Carbon field emission – based nano
devices are introduced.
The implemented systolic architectures utilize nano controlled-switching realizations to
implement in the reversible space the corresponding nano reversible systolic systems using switch
logic for which logic circuits are implemented as combination of switches rather than a
combination of the more costly logic gates as in gate logic. Also, since controlled switching is of
fundamental importance in modern logic design, the presented design methods can have a wide
spectrum of implementations in a large variety of nano-based arithmetic-intensive modern
applications.
Future work will include items such as the fabrication and test of the new Carbon field emission–
based controlled switching devices and their corresponding integrated application in full system-
level nano reversible systolic architectures for real-time arithmetic-intensive implementations.
Also, future work will include the investigation of implementing m-valued logic functions in
three-dimensional Carbon field-emission nano systolic architectures.
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architectures and their quantum circuits,” Journal of Circuits, Systems, and Computers, Vol. 17, No.
4, pp. 729-771, 2008.
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