### Avraham Guttman

```Reversible Computation
Avraham Guttman
Topics in Biological Physics
Prof. Elisha Moses, Dr. Roy Bar-Ziv
R. Landauer, IBM J. Res. Dev. 3, 183 (1961)
Bennett, C. H., IBM J. Res. Dev. 17, 525 (1973)
R. Landauer, Physica Scripta. Vol. 35, 88-95, 1987
Feynman, Lectures on Computation, 1999
Reversible Computation - Outline
• Thermodynamic Reversibility
• Physics of Logic Reversibility
0
• mRNA Transcription
1
0
1
Energy
kT2ln 2
Lossloss
 kT ln
T
0
1
Memory Tape
0
1
0
0
Single Bit Operation
0
1
0
1
Carnot Engine
http://galileoandeinstein.physics.virginia.edu/more_stuff/flashlets/carnot.htm
Carnot Cycle
Thermodynamically Reversible
AND - Irreversible gate
Loss of Information
Physically Loss of Information
0
1
T
Loss of
kT ln(
V final
Vinitial
) free energy
von Neumann – Landauer Limit
In each irreversible bit operation:
Energyloss  kT ln 2
Information Erasure
Erasure
Irreversible Process
Loss of energy
Infinite tape:
No need for erasure
For N erased bits:
Nonphysical
Loss  N  kT ln 2
Physical Reversible Copy
Copier
Model
X
Bennetts’ Three-Tape
Reversible Computation
1. Algorithm steps while saving
2. Copying the output
3. Retracing the algorithm steps
Standard
History
Copy
0
1
1
0
R1-1 R2-1 R3-1
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
R01 R02 R03 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
0
1
R3 R2 R1
0
1
0
1
Loss  kT ln 2
mRNA Transcription
from DNA to protein
mRNA Transcription
RNA (N bases) +X-S-P-P-P
RNA (N+1 bases) +P-Pi
(Triphosphate)
(Pyrophosphate)
Loss  10 kT
2
Transistor:
Loss  104 kT
Reversible Computation - Outline
• Thermodynamic Reversibility
• Physics of Logic Reversibility
0
• mRNA Transcription
1
0
1
Energy
loss
kT2ln 2
Loss
 kTYou
ln
Thank
T
0
1
```