Investigation of Reusable
and Reversible Logic Gate
with LGA
Final project for CSE599D/EE590A, fall 2013
Tsung-Wei Huang
Department of Electrical Engineering, University of Washington
Abstract
• Title: Investigation of Reusable and Reversible Logic Gate with LGA
• Abstract:
First, we construct some logic gates in LGA model using the
scheme of billiard-ball model. We find out that the limit of the LGA
model may result in some undesired properties of these gates.
Therefore, rather than number of particle to build some logic
structures, we use time-encoding method. It is shown that this
method can produce reusable and reversible logic gate, which does
not cost energy in the thermodynamic point of view. Finally, we
investigate the properties of the logic structures and gates, and
proposed some ideas which may make the construction easier. Since
the construction of a logic gate is not easy (NP problem), an efficient
way to calculate the output of a logic structure is necessary for
building a non-trivial logic gate.
Outline
• LGA logic gates in the scheme of billiard-ball model
• The intrinsic limit of LGA model
• Trajectory
• Time-encoding Boolean value under LGA model
• Orbit and path
• Orbit level representation
• Logic gates
• Orbit level
• Layout and LGA simulations – NOT gate
• Properties of logic structures and gates
• Reusable and reversible
• Vector representation
• Challenges in designing logic gates
• Conclusion
LGA logic gates in the scheme of
billiard-ball model
Presence of particle -> 1
No particle -> 0
• I1
• I2
• I3
• O1=I1(I2I3)
• O2=I2(I3I1)
• O3=I3(I1I2)
O3
I2
I1
O1
O2
I3
3 particles collision gates (NOT, NAND, Switch)
NAND gate
NOT gate
1
O1
I1
C1
I1
0
O2
SWITCH gate
1
O1
1
I1
O1
C1
I2
4, 5 particles collision gates (COPY, AND, OR)
COPY gate
1
AND gate
O1
I1
1
1
1
O2
I1
OR gate
1
I2
1
I1
O1
1
O1
12
1
However, back reflection sometimes occur,
and extra control bit is required.
The intrinsic limit of LGA model
• Define the trajectory of a particle as the path the particle can run
through when there are no other particles.
=>All of the “deterministic” collisions cannot switch any particles
from their original path, which is different from that in billiard ball
model.
T2
T2
O3
T1
O1’
I2
I1
O1
O2
I3
T1
I2
I1
O2
T3
O1
O2’
T3
Intrinsic limit: trajectory
• Closed trajectory: number of particles is constant
n(t+1) = n(t)
• Open trajectory: the number of output equals to
the number of input if there is no reflection
no = ni
1
1
• As a result, if there is no collision from opposite
direction (which produce random outcome), the
numbers of particles cannot be used to conduct
logic calculation (i.e. a particle for 1, and 0 particle
for 0).
Time encoding Boolean value under
LGA model
• Since the number of output particles is equal to the
number of input particles, we can only change the
time of arrival of an output particle.
(1,1,0)
gate
(1,0,1)
Time encoding: orbit and path
• Orbit (green line): pairs of particles move between sites
on their orbits.
• If the distances between each pairs are 2nΔ for some integers
n, then every pairs do not interfere with other pairs.
• Path (blue line): one bit particle moving from left to
right.
• Bit particle can only collide with pair particles on the sites
shared with orbit(s).
The time needed to move to the next site is Δ.
A
B
C
D
Represent the structure in orbit level
A
B
C
D
In vector:
Orbit = [ 0, 1, 0, -1]
Path = [ 1, 0, 0, 0, 0, 0]
A
B
C
collision site
D
non-collision site
Logic gate: NOT (orbit level)
A
B
C
t = 0Δ
1
0
t = 8Δ
A
B
C
0
1
Logic gate: NOT (operation detail)
t = 0Δ
1
t = 0Δ
0
Logic gate: NOT (operation detail)
t = 1Δ
1
t = 1Δ
0
Logic gate: NOT (operation detail)
t = 2Δ
1
t = 2Δ
0
Logic gate: NOT (operation detail)
t = 3Δ
1
t = 3Δ
0
Logic gate: NOT (operation detail)
t = 4Δ
1
t = 4Δ
0
Logic gate: NOT (operation detail)
t = 5Δ
1
t = 5Δ
0
Logic gate: NOT (operation detail)
t = 6Δ
1
t = 6Δ
0
Logic gate: NOT (operation detail)
t = 7Δ
1
t = 7Δ
0
Logic gate: NOT (operation detail)
t = 8Δ
0
t = 8Δ
1
Logic gate: NOT
-reusable and reversible
t = 0Δ
1
0
t = 8Δ
0
The gate returns to its initial state after operation.
1
Logic gate: NOT (layout)
Layout using Δ=16
t=0
input=0 (top gate)
Input=1 (bottom gate)
t=8*Δ=128
output=1 (top gate)
output=0 (bottom gate)
Logic gate: NOT (layout)
-sequential input
Layout using Δ=16
t=0
input=[0, 1] (top gate)
Input=[1, 0] (bottom gate)
t=8*Δ=128
output=[1] (top gate)
output=[0] (bottom gate)
t=2*(8*Δ)=256
output=[1, 0] (top gate)
output=[0, 1] (bottom gate)
Possibility to connect more than
one path using orbits
Properties of the logic structures and
gates: reusable and reversible
• Reusable:
• There are no garbage particles in these designing.
• If a gate can return to its initial state after some steps,
then it is reusable.
• Reversible:
• Since every step is deterministic, any structure build
from orbits and paths are reversible.
• In the thermodynamic point of view, these
properties imply the gate does not dissipate energy,
while the need to erase garbage particles in
“number of particle” gate makes it cost energy.
Vector representation of paths
and orbits
• We can map a site to an entry of a path vector and orbit vector.
• Path vector “p”:
• “1” for particle moving to the output end
• “-1” for particle moving to the input end
• “0” for no particle
• Orbit vector “o”:
•
•
•
•
“1” for pair moving counterclockwise
“-1” for pair moving clockwise
“i” for 2 pairs meeting on the same site
“0” for no pair
• Moving rule:
•
•
•
•
All “1” will move to its next site of the vector
All “-1” will move to its previous site of the vector
If “1” and “-1” move to the same site, assign the value “i” to the site
All “i” will transform into a “1” at its next site and a “-1” at its previous site
Vector representation of paths
and orbits (cont.)
• Collision rule:
• Define CCW/CW site as the site shared by a path vector
entry p[i] and an orbit vector entry o[j] on which a
CCW/CW pair is moving to the output end
• CCW site:
• if p[i] == -o[j], then p[i] = -p[i] and o[j] = -o[j]
• CW site:
• if p[i] == o[j], then p[i] = -p[i] and o[j] = -o[j]
CCW site
CW site
Challenges in designing logic gates
• Since the complexity of the state and outcome of a
time-encoding gate grows exponentially with the
number of sites in the gate (NP problem), it is not
easy to design a logic gate by trial and error.
• Furthermore, since the operation is non-linear, the
vector representation does facilitate the computer
simulation, but does not reduce the computation.
• However, it might be possible to use computer to
randomly generate some logic structure under
some constrains to get some non-trivial logic gates.
Conclusion
• The principle of particle collision in LGA model results
in the limit of constructing a logic gate, making the LGA
model differs from billiard-ball model.
• It is impossible to build a reusable “number-of-particle”
logic gate if every collision used in the gate is
deterministic.
• An alternative way is to build a “time-encoding” logic
gate, which can be reusable and reversible under LGA
model.
• The complexity of the state and outcome of a timeencoding gate is too large. Therefore, an efficient way
to calculate the state and output of a logic structure is
required.