1. Fredkin/Toffoli Templates for Reversible Logic Synthesisby
Dmitri Maslov
Dmitri Maslov
Gerhard Dueck
Michael Miller
ICCAD, November 11, 2003, San Jose, CA

2. Outline Synthesis Procedure
Basic definitions
Synthesis Procedure
The Templates (definition, classification, application)
Results
ICCAD, San Jose, CA
November 11, 2003
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3. Basic Definitions
Definition. Multiple output Boolean function is called reversible iff:
is a bijection.
In reversible logic fan-outs and feed-back conventionally are not allowed, thus any network is a cascade.
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November 11, 2003
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4. Basic Definitions … Toffoli type gates NOT Toffoli CNOT (Feynman)
Generalized Toffoli
Toffoli type gates
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November 11, 2003
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5. Basic Definitions Fredkin gates are the controlled SWAPs.
Fredkin gates can be effectively simulated by a Toffoli gate and two CNOTs.
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November 11, 2003
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6. Synthesis Procedure The Basic Algorithm
Assume the function is given in a truth table as a reversible specification.
Start creating the cascade from its end: transform the output to the form of the input.
Transform the output pattern to the form of input in lexicographical order.
While working with the pattern of higher order do not affect patterns with lower order.
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November 11, 2003
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7. Synthesis Procedure in 000 001 010 011 100 101 110 111 out 100 110 101
Final circuit 1
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November 11, 2003
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8. Synthesis Procedure Further Improvements
Bidirectional modification: while synthesizing a network the gates can be assigned at both sides.
Output permutation.
Control input reduction: there may be more than one possible assignment of controls.
Apply templates.
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November 11, 2003
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9. Templates as a Simplification Tool
As 3 gates can be rewritten by a sequence of 2, some s gates can be rewritten as a sequence of k gates (k<s).
Network simplification approach.
preprocessing: find as many rewriting rules as possible.
simplify by matching rewriting rules and rewriting the circuit.
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November 11, 2003
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10. Templates as a Simplification Tool
Problems in such naive approach.
The number of rewriting rules is very large:
a. For s=3, k=2 and n=3 (number of lines) the number of rewriting rules is 180 (using Toffoli gates only).
b. Many rewriting rules are redundant.
c. The number of non-redundant rewriting rules only grows exponentially on n.
2. Very often a rewriting can be applied only when certain gates are moved.
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November 11, 2003
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11. The Templates: Definition
Observation 1. If one has a rewriting rule
then the gates in it
satisfy equation
Observation 2. If we have an identity
then for any parameter p,
is a valid rewriting rule.
Observation 3. If , then
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November 11, 2003
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12. The Templates: Definition
A size m template is a cascade of m gates which realizes the identity function. Any template of size m should be independent of smaller size templates, i.e. application of smaller templates does not decrease the number of gates in a size m template.
Given G0G1…Gm-1, a template of size m, its application for parameter p, is:
for
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November 11, 2003
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13. The Templates: Definition
Example. Template ABCDEFG. p=4. Starting gate B. Direction: backward. F F E E B A G C D G D A C B m2
How many rewriting rules are there in one template?
Parameter p: m/2 choices. Starting gate i: m choices. Directions of application: 2 (forward, backward).
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November 11, 2003
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14. The Templates: Classification
A class of Toffoli-Fredkin templates is defined as an identity with certain conditions on the form of gates in it.
Box gate:
- Line with the box is either NOT or SWAP.
- All the boxes on the same line are of the same type.
- If a line with a box has a NOT or a SWAP on it, the box is necessarily substituted with NOT.
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November 11, 2003
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15. The Templates: Classification
Class 1: duplication deletion.
Class 2: passing rule.
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November 11, 2003
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16. The Templates: Classification
Group: semi-passing
Group: link
Group: Fredkin definition
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November 11, 2003
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17. The Templates: Classification
Class 3: main template.
Class 4: fttftt template.
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November 11, 2003
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18. The Templates: Application
In a program realization:
use size 4 templates to move gates – moving rule. Semi-passing templates are used to pass the gate that does not change.
for other templates apply smaller templates first (in a sense, smaller template does a more general transformation).
- given a template, match it by trying both directions, starting with any gate and trying to move other gates by the moving rule.
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November 11, 2003
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19. The Templates: Application
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20. The Templates: Application
semi-passing
passing
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November 11, 2003
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21. Results j
optimal
synthesized 1 18 2
184 3
1318
1290 4
6474
5680 5
17695
13209 6
14134
13914 7
496
5503 8
512 9
WA:
5.134
5.437
Number of reversible functions of size 3 using a specified number of Toffoli-Fredkin gates.
More than 77% of synthesized functions are optimal.
Average gate count of the synthesized Toffoli-Fredkin circuit is less than optimal gate count for Toffoli network.
ICCAD, San Jose, CA
November 11, 2003
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22. Benchmark synthesis results
Name
Size*
Number of gates
rd32 4 4T
rd53 7
12T
mod5 5 9T
hwb5
24F
hwb6 6
65F
hwb7
166F
ham3 3
4FT
ham7
24T
ham15 15
138T
* - when the function is not reversible, it was synthesized in its minimal reversible specification.
ICCAD, San Jose, CA
November 11, 2003
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23. END Fredkin/Toffoli Templates for Reversible Logic Synthesis
Thanks for Your attention!