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Fault Tolerant and Online Testability in Reversible Logic
Synthesis
Sajib Kumar MitraDepartment of Computer Science and Engineering
University of [email protected]
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Overview
• Background StudyReversible Logic Fault Tolerant MethodOnline Testability
• Online Testable Fault Tolerant Circuit• Full Adder Circuit
Existing DesignProposed Design
• Performance Analysis• About Authors• Conclusion
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Reversible Logic
• Unique mapping between input and output vectors which governs to have same number of input-output lines of any reversible circuit
• Recovers heat dissipation unlikely irreversible logic and uses low power CMOS technology
• Feedback and Fan-out are not allowed• Single unit able to compute more than one
operation
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Reversible Logic (cont…)
Reversible Circuit
I1
I2
I3
In
O1
O2
O3
On
An (n x n) Reversible Circuit
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Reversible Logic (cont…)
Reversible Circuit
I1
I2
I3
In
O1
O2
O3
On
Unique mapping of Reversible Circuit
O1
O2
O3
.
.
.On
Output VectorOutput Vector
I1I2I3...In
Input VectorInput
Vector
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Reversible Logic (cont…)
I1
I2
I3
In
O1
O2
O3
On
Gat
e 1
Gat
e 2
Gat
e 3
.
.
.
Gat
e S
An (n x n) Reversible Circuit Architecture
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Reversible Logic (cont…)
Reversible EX-OR operation and 2x2 Feynman Gate
FGA
B Q=A B
P=A
OU
TP
UT
VE
CT
OR
(P
, Q
)
3
2
1
0
2
1
INP
UT
VE
CT
OR
(A
, B
)
0
3
(a) Feynman Gate (b) Unique Mapping between Input and Output
Vector
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Reversible Logic (cont…)
Popular 3x3 Reversible Gates
F2GA
BC
A
A C
A B
(c) Feynman Double Gate
PGA
BC
AA B
AB C
(a) Peres Gate
FRGA
BC
A
A’C AB
A’B AC
(e) Fredkin Gate
TGA
BC
A
AB C
B
(b) Toffoli Gate
NFTA
BC
AC’ B’CAC’ BC
A B
(d) New Fault Tolerant Gate
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Reversible Logic (cont…)
4x4 Reversible Gates
D
ABC
P = AQ = R = A B CS = (A B)C AB D
MTSG A B
D
ABC
P = AQ = A’C’ B’R = A’C’ B’ DS = ( A’C’ B’)D AB C
ABC
P = A
D
Q = A BR = AB CS = AB’ D
MIG TSG
(a) Modified IG Gate (b) TSG Gate
(c) Modified TSG Gate
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Reversibility prevents Bit Loss but not able to detect Bit
Error or Fault in Circuit
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Fault Tolerant Method
Bit Error means the alteration of the value of output bits because of internal fault of digital circuit.
Input Output
A B A A B
0 0 0 0
0 1 0 1
1 0 1 1
1 1 1 0
FG0
1
0
(a) Reversible EX-OR Operation
0
What is the meaning of Bit Error?
Bit Error in Reversible Circuit
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Fault Tolerant Method (cont…)
Bit Error means the alteration of the value of output bits because of internal fault of digital circuit.
Input Output
A B A A B
0 0 0 0
0 1 0 1
1 0 1 1
1 1 1 0
FG0
1
0
(a) Reversible EX-OR Operation
0
What is the meaning of Bit Error?
Bit Error in Reversible Circuit
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Fault Tolerant Method (cont…)
Preserves same parity between Input and Output vectors over one to one mapping of reversible circuit
Reversible Circuit
I1
I2
I3
In
O1
O2
O3
On
EV
EN
Par
ity
EV
EN
Par
ity
Parity Preservation of Reversible Circuit
Reversible Circuit
I1
I2
I3
In
O1
O2
O3
OnO
DD
Par
ity
OD
D P
arit
y
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Fault Tolerant Method (cont…)
Let, Iv and Ov are input and output vectors of a reversible circuit, so the relation is Iv↔Ov.
But to be a Reversible Fault Tolerant circuit, itself must preserve following equation:
where Iv={I1, I2, I3, …, In} and Ov={O1, O2, O3, …, On}
nn OOOIII 2121
Input Parity = Output Parity
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Fault Tolerant Method (cont…)
Parity Preservation over reversibility between Input and Output vectors can be realized from the Truth Table of Fredkin Gate as shown below:
Fredkin Gate and Corresponding Truth Table
Input Output
A B C P Q R
0 0 0 0 0 0
0 0 1 0 0 1
0 1 0 0 1 0
0 1 1 0 1 1
1 0 0 1 0 0
1 0 1 1 1 0
1 1 0 1 0 1
1 1 1 1 1 1
A
BC
P=A
R=A’C AB
Q=A’B ACFRG
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Fault Tolerant Method (cont…)
Input Output
A B C P Q R
0 0 0 0 0 0
0 0 1 0 0 1
0 1 0 0 1 0
0 1 1 0 1 1
1 0 0 1 0 0
1 0 1 1 1 0
1 1 0 1 0 1
1 1 1 1 1 1
A
BC
P=A
R=A’C AB
Q=A’B ACFRG
Fredkin Gate and Corresponding Truth Table
RQPCBA
Now Verify the following equation:
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Fault Tolerant Method (cont…)
1
01
1FRG 1
0EV
EN
EV
EN
Fault detection of FRG gate
RQPCBA
Verify the following equation:
1
01
1FRG 1
1EV
EN
OD
D
Fault exist in CircuitNo Fault exist in Circuit
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Fault Tolerant Method (cont…)
But the minimum dimension of Fault Tolerant gates is 3. Why?
Finally Reversible Gate which preserves same parity between input and output vectors is called Fault Tolerant Gate or Parity Preserving Gate
A A’
Input Output
A A’
0 1
1 0
1x1 Reversible Gate Never be a Fault Tolerant Gate
NOT operation
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Fault Tolerant Method (cont…)
But the minimum dimension of Fault Tolerant gates is 3. Why?
Input Output
A B P Q
0 0 0 0
0 1 0 1
1 0 1 0
1 1 1 1
2x2 Reversible Gates have no any significance as Fault Tolerant Gate
2x2Reversible
Gate
A
B
A
B
Input Output
A B P Q
0 0 0 0
0 1 1 0
1 0 0 1
1 1 1 1
Input Output
A B P Q
0 0 1 1
0 1 1 0
1 0 0 1
1 1 0 0
Input Output
A B P Q
0 0 1 1
0 1 0 1
1 0 1 0
1 1 0 0
2x2Reversible
Gate
A
B
B
A
2x2Reversible
Gate
A
B
A’
B’
2x2Reversible
Gate
A
B
B’
A’
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Fault Tolerant Method (cont…)
A
BC
P=A
R=A’C AB
Q=A’B ACFRG
(b)
F2GA
BC
A
A C
A B
(a)
NFTA
BC
Q=AC’ B’CR=AC’ BC
P=A B
(c)
Existing 3x3 and 4x4 Fault Tolerant Gates
ABC
P = A
D
Q = A BR = AB CS = AB’ D
MIG
(d)
A
B
C
B C D
D
PPHCG B C AB D AC D A
(e)
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Online Testability
• Built-In Self Testing method• Detects bit-error at outputs of any circuit in run
time• Reversible gates able to adopt testability feature
by deducing output and corresponding input bits• To be online testable an (n x n) reversible gates
must preserve the following properties:
where Iv={I1, I2, I3, …, In} and Ov={O1, O2, O3, …, On}
121 nnn OOOIO
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Online Testability (cont…)
F2GA
BC
U=A
W=A C
V=A B
WO
BCBAAC
VUCO
Let
3
3
,
To be online testable an (n x n) reversible gates must have the following properties:
121 nnn OOOIO
3x3 F2G is not online Testable Gates
But F2G can be Testable by deducing extra an input and corresponding output line.
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Online Testability (cont…)
F2GA
BC
A
A C
A BTestable F2G
A
BC
U=A
P Q=P A B C
V=A BW=A C
QO
CBAPCABAAP
WVUPO
Let
3
3
,
To be online testable an (n x n) reversible gates must have the following properties:
121 nnn OOOIO
Testing OutputTesting Input
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Online Testability (cont…)
Testable F2G
A
BC
U=A
P Q=P A B C
V=A BW=A C
Testable F2G
1
00
U=1
0
V=1W=1Q=1
Verification of Testable F2G at Runtime
QO
WVUPO
Let
3
3
11110
,
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Online Testability (cont…)
Testable F2G
A
BC
U=A
P Q=P A B C
V=A BW=A C
Testable F2G
1
00
U=1
0
V=1W=0Q=1
Verification of Testable F2G at Runtime
QO
WVUPO
Let
3
3
00110
,
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Online Testability (cont…)
Testable F2G
A
BC
U=A
P Q=P A B C
V=A BW=A C
Testable F2G
1
00
U=1
0
V=1W=0Q=1
Verification of Testable F2G at Runtime
Testable R2
D
EF
X=D
R S=R D E F
Y=E
Z=F
0
:
R
WF
VE
UD
Assignment
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Online Testability (cont…)
Testable F2G
ABC0
Testable R2
Q 0 S
XYZ
UVW
""
""
ErrorBitelse
ErrorBitNOthenSQIf
Testable F2G
A
BC
U=A
P Q=P A B C
V=A BW=A C
Testable R2
D
EF
X=D
R S=R D E F
Y=E
Z=F
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Online Testability (cont…)
Testable Reversible
F2G
ABCPR
Q
XYZ
S
Testable F2G
ABCP
Testable R2
Q R S
XYZ
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Online Testability (cont…)
Testable Reversible
Cell(TRC)
ABCPQ
R
XYZ
S
Operational Outputs
Testing OutputsTesting Inputs
(Constant Value)
Operational Inputs
TRC is a Cascading Block not Gate
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Online Testability (cont…)
Testable Reversible
Cell(TRC)
I1I2I3
In+2
O1O2O3
On+2
Conversion of nxn Reversible Gate into (n+2)x(n+2) reversible Cell
DeducibleTestable
GateDR
I1I2I3
In+1
O1O2O3
On+1
nxn Reversible
GateR
I1I2
In
O1O2
On
Testable R based on following Law:
121 nnn OOOIO
Testable Reversible Cell by using Cascading Attachment
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Reversible Fault Tolerant Full Adder Circuit
• Full Adder circuit produces Sum and Cout as following equations respectively:
• Full Adder can be realized by using only one MTSG gate as follows:
ABCBAC
CBASum
inout
in
)(
D
ABC
P = AQ = R = A B CS = (A B)C AB D
MTSG A B
0
ABCin A B Cin
(A B)Cin AB
MTSG
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Reversible Fault Tolerant Full Adder Circuit (cont…)
0
ABCin A B Cin
(A B)C AB
MTSG
Pros o Garbage = 2o Gate = 1o Quantum Cost = 6
Cons Neither Fault Tolerant nor Online testable
You have to make Fault Tolerant and Online Testable circuit by using fault Tolerant Gates. So
start now…
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Existing Fault Tolerant Fault Tolerant Adder Circuit
By using MIG…
ABC
P = A
D
Q = A BR = AB CS = AB’ D
MIG
ABCD G
MIGSum
MIGCin
CoutG
G
Design is Fault tolerant but uses higher dimensional reversible Gates
To make online testable, circuit has to increase an extra input-output line
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Proposed Design of Fault Tolerant Full Adder
Uses 3x3 Fault Tolerant gates Easily adoptable to online testable full adder Minimum number of Garbage, 3 Preferable for Carry Look Ahead adder
CinFRG
F2GF2G
F2G
0
A
B
0
G
G
GA B Cin
AB BCin ACin
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Proposed Design of Online Testable Fault Tolerant Full Adder
F2GTRC
PPIRC
FRGTRC
PPIRC
F2GTRC
PPIRC
01
01
01
TESTING INPUT
1 1 1
e
Cout S
a3 a2 a1
BLOCK ERROR TESTING OUTPUT
F2GTRC
PPIRC
01
1
a4
0
A
B0
0
Cin
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Performance Analysis
Fault Tolerant
Full Adder
Total Gates Total
Garbage
Quantum Cost
3x3 4x4
Proposed [b]
4 0 3 11
Existing [a] 0 2 3 14Cin
FRG
F2GF2G
F2G
0
A
B
0
G
G
GA B Cin
AB BCin ACin
(b)
ABCD G
MIGSum
MIGCin
CoutG
G
(a)
Table 1: Comparison between proposed and existing design
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About Author
Ahsan Raja Chowdhury received his B.Sc .and MS degrees in Computer science and Engineering from the University of Dhaka, Bangladesh, in 2004 and 2006, respectively. He worked with the Department of Computer Science and Engineering, Northern University, Bangladesh, from 2004 to 2007 as faculty member. He is the faculty member of the Department of Computer Science and Engineering,
Sajib Kumar Mitra is an MS student of Dept. of Computer Science and Engineering, University of Dhaka, Dhaka, Bangladesh. His research interests include Electronics, Digital Circuit Design, Logic Design, and Reversible Logic Synthesis.
Md. Faisal Hossain has completed his undergraduate from Dept. of Computer Science and Engineering, University of Dhaka, Dhaka, Bangladesh. His research interest includes Logic Design, especially Reversible Logic Design.
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Thanks To All