Reversible Computing Theory I: Reversible Logic Models

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Reversible Computing Theory IReversible Logic
Models
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Reversible Logic Models

• It is useful to have a logic-level (Boolean)
  model of adiabatic circuits.
• Can model all logic using maximally pipelined
  logic elements, which consume their inputs.
• A.k.a., input consuming gates.
• Warning In such models Memory requires
  recirculation! Thus, this is not necessarily
  more energy-efficient in practice for all
  problems than retractile (non-input consuming)
  approaches!
• There is a need for more flexible logic models.
• If inputs are consumed, then the input?output
  logic function must be invertible.

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Input-consuming inverter
in
out

• Before After in out in out 0 - -
  1 1 - - 0
• E.g. SCRL implementation

Input arrow indicates inputdata is consumed by
element.
Alternate symbol
in
out
Invertible!
(Symmetric)
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An Irreversible Consuming Gate

• Input-consuming NAND gate Before After
  A B out A B out 0 0 - - - 1 0
  1 - 1 0 - - - 0 1 1 -
• Because its irreversible, it has no
  implementation in SCRL (or any fully adiabatic
  style) as a stand-alone, pipelined logic element!

A
out
B
4 possible inputs, 2possible outputs. At least
2 of the 4 possibleinput cases must lead
todissipation!
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NAND w. 1 input copied?

• Still not invertible Before After A B A
  out A B A out 0 0 - - - -
  0 1 0 1 - - - - 1
  1 1 0 - -
  - - 1 0 1 1 - -
• At least 1 of the 2 transitions to the A0,
  out1 final state must involve energy dissipation
  of order kBT. How much, exactly? See exercise.

Delay buffer
A
A
out
B
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NAND w. 2 inputs copied?

• Finally, invertible! Before After
  A B A B out A B A B out 0 0 -
  - - - - 0 0 1 0
  1 - - - - - 0 1 1
  1 0 - - - - - 1 0
  1 1 1 - - - - -
  1 1 0
• Any function can be made invertible by simply
  preserving copies of all inputs in extra outputs.
• Note Not all output combinations here are legal!
• Note there are more outputs than inputs.
• We call this an expanding operation.
• But, copied inputs can be shared by many gates.

A
A
out
B
B
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SCRL Pipelined NAND
A
B
A
out AB
5T
B
Inverters only neededto restore A, B Can be
shared withother gates that takeA, B as inputs.

• Including inverters 23 transistors
• Not including inverters 7 transistors

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Non-Expanding Gates

• Controlled-NOT (CNOT) or input-consuming
  XOR A B A C 0 0 0
  0 0 1 0 1 1 0 1
  1 1 1 1 0
• Not universal for classical reversible computing.
  (Even together w. all other 1 2 output rev.
  gates.)
• However, if we add 1-input, 1-output
  quantumgates, the resulting gate set is
  universal!
• More on quantum computing in a couple of weeks.

A
A
A
A
B
C A?B
B
C A?B
Can implement w. a diadic gate in SCRL
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Toffoli Gate (CCNOT)
A
A
A B C A B C0 0 0 0 0 00 0 1
0 0 10 1 0 0 1 00 1 1 0
1 11 0 0 1 0 01 0 1 1 0
11 1 0 1 1 01 1 1 1 1 1

• Subsumes AND, NAND, XOR, NOT, FAN-OUT,
• Note that this gate is its own inverse.
• Our first universal reversible gate!

A
AA
B
BB
B
B
C
C
C C?AB
C
(XOR)
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Fredkin Gate

• The first universal reversible logic gate to be
  discovered. (Ed Fredkin, mid 70s)
• B and C are swapped ifA1, else passed
  unchanged.
• Is also conservative, conserves 1s and 0s.
• Thus in theory requires no separate power input,
  even if 1 and 0 have different energy levels!

A B C A B C0 0 0 0 0 00 0 1
0 0 10 1 0 0 1 00 1 1 0
1 11 0 0 1 0 01 0 1 1 0
11 1 0 1 1 01 1 1 1 1 1
A
A
B
B
C
C
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Reversible Computing Theory IIEmulating
Irreversible Machines
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Motivation for this study

• We want to know how to carry out any arbitrary
  computation in a way that is reversible to an
  arbitrarily high degree.
• Up to limits set by leakage, power supply, etc.
• We want to do this as efficiently as possible
• Using as few device ticks as possible
  (spacetime)
• Minimizes HW cost, leakage losses
• Using as few adiabatic transitions as possible
  (ops)
• Minimizes frictional losses
• But, a desired computation may be originally
  specified in terms of irreversible primitives.

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General-Case vs. Special-Case

• Wed like to know two kinds of things
• For arbitrary general-purpose computations,
• How to automatically emulate them in a fairly
  efficient reversible way,
• w/o needing new intelligent/creative design work
  in each case?
• Topic of todays lecture
• For various specific computations of interest,
• What are the most efficient reversible
  algorithms?
• Or at least, the most efficient that we can find?
• Note These may not necessarily look anything
  like the most efficient irreversible algorithms!
• More on this point later

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The Landauer embedding

• The obvious embedding of irreversible ops into
  expanding reversible ones leads to a linear
  increase in space through time. (Landauer 61)
• Or, increase in width of an input-consuming
  circuit

Expandingoperations(e.g., AND)
Desiredoutput
Garbagebits
input
Circuit depth, or time ?
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Lecerf Reversal

• Lecerf (63) was interested in the group-theory
  question of whether an iterated permutation of
  items would eventually return to initial item.
• Proved undecidable by reducing Turings halting
  problem to this question, w. a reversible TM.
• Reversible TM reverses direction instead of
  halting.
• Returns to initial state iff irreversible TM
  would halt.
• Only problem with thisNo useful output data!

Desiredoutput
f
f ? 1
Garbage
Copy ofInput
Input
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The Bennett Trick

• Bennett (73) pointed out that you could simply
  fan-out (reversibly copy) the desired output
  before reversing.
• Note O(T) storage is still temporarily needed!

Desired output
f
f ? 1
Copy ofInput
Input
Garbage
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Improving Spacetime Efficiency

• Bennett 73 transforms a computation taking
  spacetime ST to one taking ?(ST2) spacetime in
  the worst case.
• Can we do better?
• Bennett 89 Described a technique that takes
  spacetime
• Actually, can generalize slightly and arrange for
  exponent on T to be 1?, where ??0 (very slowly)
• Lange, McKenzie, Tapp 97 Space ?(S) is even
  possible, if you use time ?(exp(?(S)))
• Not any more spacetime-efficient than Bennett.

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Reversible Climbing Game

• Suppose a guy armed with a hammer, N spikes, a
  rope is trying to climb acliff, while obeying
  the following rules.
• Question How high can he climb?
• Rules
• Standing on the ground or on a spike, he
  caninsert remove a spike 1 meter higher up.
• He can raise lower himself betweenspikes the
  ground using his rope.
• He cant insert or remove a spike whiledangling
  from a higher spike!
• Maybe not enough leverage/stability?

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Analogy w. Emulation Problem

• Height on cliff represents
• How many steps of progress havewe made through
  the irreversiblecomputation?
• Number of spikes represents
• Available memory of reversible machine.
• Spike in cliff at height H represents
• Using a unit of memory to record the state of
  the irreversible machine after H steps.
• Adding/removing a spike at height H1if there is
  a spike is at height H represents
• Computing/uncomputing state at H1 steps given
  state at H.

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Lets Climb!
0. Standing on ground.
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How high can we climb?

• Using only N spikes, and the strategy
  illustrated, we can climb to height 2N?1 (wow!)
• Li Vitanyi (Theorem) This is the optimal
  strategy for this game.
• Open question
• Are there more efficient general reversiblization
  techniques that are not based on this game model?

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Pebble Game Representation
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Triangle representation
k 2n 3
k 3n 2
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Analysis of Bennett Algorithm

• n of recursive levels of algorithm
• k of lower-level iterations to go forward 1
  higher-level step
• Tr of reversible lowest-level steps
  executed 2(2k?1)n
• Ti of irreversible steps emulated kn
• So, n logk Ti, and so Tr 2(2k?1)log Ti/log k
  2elog(2k?1)log(Ti)/log k 2Tilog(2k ?1)/log k

(n1 spikes)
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Linear-Space Emulation
(Lange, McKenzie, Tapp 97)
Unfortunately, the tree may have 2S nodes!
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Can we do better?

• Bennett 73 takes order-T time, LMT 97 takes
  order-S space.
• Can some technique achieve both, simultaneously?
• Theorem (Frank Ammer 97) The problem of
  iterating a black-box function cannot be done in
  time T space S on a reversible machine.
• Proof really does cover all possible algorithms!
• The paper also proves loose lower bounds on the
  extra space required by a linear-time simulation.
• Results might also be extended to the problem of
  iterating a cryptographic one-way function.
• Its not yet clear if this can be made to work.

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One-Way Functions

• are invertible functions f such that f is easy
  to compute (e.g., takes polynomial time) but f ?1
  is hard to compute (e.g., takes exponential
  time).
• A simple example
• Consider f(p,q) pq with p,q prime.
• Multiplication of integers is easy.
• Factoring is hard (except using quantum
  computers).
• The one-way-ness of this function is essential
  to the security of the RSA public-key
  cryptosystem.
• No function has yet been proven to be one-way.
• However, certain kinds of one-way functions are
  known to exist if P ? NP.

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Elements of Frank-Ammer Proof

• Consider a chain of bit-strings (size S each)
  that is incompressible by a certain compressor.
• This is easily proven to exist. (See next slide.)
• Machines job is to follow this chain from one
  node to the next by using a black-box function.
• The compressor can run a reversible machine
  backwards, to reconstruct earlier nodes in the
  chain from later machine configurations.
• If the reversible machine only uses order-S space
  in its configurations, then the chain is
  compressible!
• Contradicts choice of incompressible chain QED.

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Existence of Incompressibles

• A decompressor or description system s0,1
  maps any bit-string description d to the
  described string x.
• Notation fD means a unary operator on D, fD?D
• x is compressible is s iff ?d s(d)x, dltx
• Notation b means the length of bit-string b in
  bits.
• Theorem Every decompressor has an incompressible
  input of any given length ?.
• Proof There are 2? length-? bit-strings, but
  only shorter descriptions.

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Cost-Efficiency Analysis

• Cost EfficiencyCost Measures in
  ComputingGeneralized Amdahls Law

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Cost-Efficiency

• Cost-efficiency of anything is min/,
• The fraction of actual cost that really needed
  to be spent to get the thing, using the best
  poss. method.
• Measures the relative number of instances of the
  thing that can be accomplished per unit cost,
• compared to the maximum number possible
• Inversely proportional to cost .
• Maximizing means minimizing .
• Regardless of what min actually is.
• In computing, the thing is a computational task
  that we wish to carry out.

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Components of Cost

• The cost of a computation may generally be a
  sum of terms for many different components
• Time-proportional (or related) costs
• Cost to user of having to wait for results
• E.g., missing deadlines, incurring penalties.
• May increase nonlinearly with time for long
  times.
• Spacetime-proportional (or related) costs
• Cost of raw physical spacetime occupied by
  computation.
• Cost to rent the space.
• Cost of hardware (amortized over its lifetime)
• Cost of raw mass-energy, particles, atoms.
• Cost of materials, parts.
• Cost of assembly.
• Cost of parts/labor for operation maintenance.

• Cost of SW licenses

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More cost components

• Continued...
• Area-time proportional (or related) costs
• Cost to rent a portion of an enclosing convex
  hull for getting things in out of the system
• Energy, heat, information, people, materials,
  entropy.
• Some examples incurring area-time proportional
  costs
• Chip area, power level, cooling capacity, I/O
  bandwidth, desktop footprint, floor space, real
  estate, planetary surface
• Note that area-time costs also scale with the
  maximum number of items that can be
  sent/received.
• Energy expenditure proportional (or related)
  costs
• Cost of raw free energy expenditure (entropy
  generation).
• Cost of energy-delivery system. (Amortized.)
• Cost of cooling system. (Amortized.)

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General Cost Measures

• The most comprehensive cost measure includes
  terms for all of these potential kinds of costs.
• comprehensive Time SpaceTime AreaTime
  FreeEnergy
• Time is an non-decreasing function
  f(?tstart?end)
• Simple model Time ? ?tstart?end
• FreeEnergy is most generally
• Simple model FreeEnergy ? ?Sgenerated
• SpaceTime and AreaTime are most generally
• Simple model
• SpaceTime ? Space ? Time
• AreaTime ? Area ? Time

Max ops thatcould be done
Max items thatcould be I/Od
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Generalized Amdahls Law

• Given any cost that is a sum of components, tot
  1 n,
• There are diminishing proportional returns to be
  gained from reducing any single cost component
  (or subset of components) to much less than the
  sum of the remaining components.
• ? Design-optimization effort should concentrate
  on those cost components that dominate total cost
  for the application of interest.
• At a design equilibrium, all cost components
  will be roughly equal (unless externally driven)

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Reversible vs. Irreversible

• Want to compare their cost-efficiency under
  various cost measures
• Time
• Entropy
• Area-time
• Spacetime
• Note that space (volume, mass, etc.) by itself as
  a cost measure is only significant if either
• (a) The computer isnt reusable, so the cost to
  build it dominates operating costs, or
• (b) I/O latency ? V1/3 affects other costs.

Or, for some applications,one quantity might be
minimizedwhile another one (space, time,
area)is constrained by some hard limit.
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Time Cost Comparison

• For computations with unlimited power/cooling
  capacity, and no communication requirements
• Reversible is worse than irreversible by a factor
  of sgt1 (adiabatic slowdown factor), times maybe
  a small constant depending on the logic style
  used. r,Time ? i,Time s

38
Time Cost Comparison, cont.

• For parallelizable, power-limited applications
• With nonzero leakage r,Time ? i,Time /
  Ron/offg
• Worst-case computations g ? 0.4
• Best-case computations g 0.5.
• For parallelizable, area-limited,
  entropy-flux-limited, best-case applications
• with leakage ? 0 r,Time ? i,Time / d 1/2
• where d is systems physical diameter.
• (see transparency)

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Time cost comparison, cont.

• For entropy-flux limited, parallel, heavily
  communication-limited, best case applications
• with leakage approaching 0 r,Time ? i,Time3/4
• where i,Time scales up with the space
  requirement V as i,Time ? V2/9
• so the reversible speedup scales with the 1/18
  power of system size.
• not super-impressive!

(details later)
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Bennett 89 alg. is not optimal
k 2n 3
k 3n 2
Just look at all the spacetime it wastes!!!
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Parallel Frank02 algorithm

• We can simply scrunch the triangles closer
  together to eliminate the wasted spacetime!
• Resulting algorithm is linear time for all n and
  k and dominates Ben89 for time, spacetime,
  energy!

k3n2
k2n3
Emulated time
k4n1
Real time
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Setup for Analysis

• For energy-dominated limit,
• let cost equal energy.
• c energy coefficient, r r(min) leakage
  power
• i energy dissipation per irreversible
  state-change
• Let the on/off ratio Ron/off r(max)/r(min)
  Pmax/Pmin.
• Note that c ? itmin i (i / r(max)),
  so r(max) ? i2/c
• So Ron/off ? i2 / cr(min) i2 / cr

43
Time Taken

• There are n levels of recursion.
• Each multiplies the width of the base of the
  triangle by k.
• Lowest-level triangles take time ctop.
• Total time is thus ctopkn.

k4n1
Width 4 sub-units
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Number of Adiabatic Ops

• Each triangle contains k (k ? 1) 2k ? 1
  immediate sub-triangles.
• There are n levels of recursion.
• Thus number of adiabatic ops is c(2k ? 1)n

k3n2
52 25little triangles(adiabaticoperations)
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Spacetime Usage

• Each triangle includes the spacetime usage of all
  k ? 1 of its subtriangles,
• Plus,additional spacetime units, each
  consisting of 1 storage unit, for time
  topkn?1

k5n1
1 state of irrev. mach. Being stored
1
2
Time top kn-1
3
Resulting recurrence relationST(k,0) 1 (or
c)ST(k,n) (2k?1)ST(k,n?1) (k2?3k2)kn?1/2
123 units
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Reversible Cost

• Adiabatic cost plus spacetime cost r a r
  (2k-1)nc/t ST(k,n)rt
• Minimizing over t gives r 2(2k-1)n
  ST(k,n) c r1/2
• But, in energy-dominated limit, c r ? i2 /
  Ron/off,
• So r 2i (2k-1)n ST(k,n) / Ron/off1/2

47
Tot. Cost, Orig. Cost, Advantage

• Total cost i for irreversible operation
  performed at end of algorithm, plus reversible
  cost, gives tot i 1 2(2k-1)n
  ST(k,n) / Ron/off1/2
• Original irreversible machine performing kn ops
  would use cost orig ikn, so,
• Advantage ratio between reversible irreversible
  cost,

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Optimization Algorithm

• For any given value on Ron/off,
• Scan the possible values of n (up to some limit),
• For each of those, scan the possible values of k,
• Until the maximum R(i/r) for that n is found
• (the function only has a single local maximum)
• And return the max R(i/r) over all n tried.

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Spacetime blowup
Energy saved
k
n
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Asymptotic Scaling

• The potential energy savings factor scales as
  R(i/r) ? Ron/off0.4,
• while the spacetime overhead goes only as
  R(i/r) ? R(i/r)0.45, or Ron/off0.18.
• E.g., with an Ron/off of 109, you can do
  worst-case computation in an adiabatic circuit
  with
• An energy savings of up to a factor of 1,200 !
• But, this point is 700,000 less
  hardware-efficient, if Frank02 algorithm is used
  for the emulation.

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Various Cost Measures

• Entropy - advantage as per previous analysis
• Area times time - scales w. entropy generated
• Performance, given area constraint -
• In leakage-free limit, advantage proportional to
  d1/2
• With leakage, whats the max advantage? (See hw)
• NOW
• Are there any performance/cost advantages from
  adiabatics even when there is no cost or
  constraint for entropy or for area?
• YES, for flux-limited computations that require
  communications. Lets see why

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Perf. scaling w. of devices

• If alg. is not limited by communications needs,
• Use irreversible processors spread in a 2-D
  layer.
• Remove entropy along perpendicular dimension.
• No entropy removal rate limits,
• so no speed advantage from reversibility.
• If alg. requires only local communication,latency
  ? cyc. time, in an NDNDND mesh,
• Leak-free reversible machine perf. scales better!
• Irreversible tcyc ?(ND1/3)
• Reversible tcyc ?(ND1/4) ?(ND1/12) faster!
• To boost reversibility speedup by 10, one must
  consider 1036-CPU machines (1.7 trillion moles
  of CPUs!)
• 1.7 trillion moles of H atoms weighs 1.7 million
  metric tons!
• A 100-m tall hill of H-atom sized CPUs!

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Lower bound on irreversible time

• Simulate Nproc ND3 cells for Nsteps ND steps.
• Consider a sequence of ND update steps.
• Final cell value depends on ND4 ops in time T.
• All ops must occur within radius r cT of cell.
• Surface area A ? T2, rate Rop ? T2 sustainable.
• Nops ? Rop T ? T3 needs to be at least ND4.
• ? T must be ?(ND4/3) to do all ND steps.
• Average time per step must be ?(ND1/3).
• Any irreversible machine (of any technology or
  architecture) must obey this bound!

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Irreversible 3-D Mesh
55
Reversible 3-D Mesh
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Non-local Communication

• Best computational task for reversibility
• Each processor must exchange messages with
  another that is ND1/2 nodes away on each cycle
• Unsure what real-world problem demands this
  pattern!
• In this case, reversible speedup scales with
  number of CPUs to only the 1/18th power.
• To boost reversibility speedup by 10, only
  need 1018 (or 1.7 micromoles) of CPUs
• If each was a 1-nm cluster of 100 C atoms, this
  is only 2 mg mass, volume 1 mm3.
• Current VLSI Need cost level of 25B before
  you see a speedup.

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Ballistic Machines

• In the limit if cS ? 0, the asymptotic benefit
  for 3-D meshes goes as ND1/3 or Nproc1/9.
• Only need a billion devices to multiply
  reversible speedup by 10.
• With 1 nm3 devices, a cube 1 ?m on a side
  (bacteria size) would do it!
• Does Drexlers rod logic have low enough cS and
  small enough size to attain this prediction?
• (Need to check.)

58
Minimizing volume via folding

• Allows previoussolutions to bepacked in
  min.volume.
• Volume scalesproportionallyto mass.
• No change inspeed or entropy flux.

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Cooling Technologies
60
Irreversible Max Perf. Per Area
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Reversible Entropy Coeffs.
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Rev. vs. Irrev. Comparisons
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Sizes of Winning Rev. Machines
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Some Analytical Challenges

• Combine Frank 02 emulation algorithm,
• Analysis of its energy and space efficiency as a
  function of n and k,
• And plug it into the analysis for the 3-D meshes,
  to see
• What are the optimal speedups for arbitrary mesh
  computations on rev. machines, as a function of
• Ron/off, device volume, entropy flux limit,
  machine size.
• And, does perf./hw improve, and if so, how much?

65
Reversible Processor Architecture
66
Why reversible architectures?

• What about automatic emulation algorithms?
• E.g. Ben73, Ben89, LMT, Frank02.
• Transform an irreversible alg. to an equiv.
  revble one.
• But, these do not yield the most cost-efficient
  reversible algorithms for all problems!
• E.g., log(RE(i./r)/Ron/off) may be only 0.4
  rather than 0.5.
• Finding the best reversible algorithm requires a
  creative algorithm discovery process!
• An optimally cost-efficient general-purpose
  architecture must allow the programmer to specify
  a custom reversible algorithm that is specific to
  his problem.

67
Reversibility Affects All Levels

• As Ron/off increases cost of device manuf.
  declines (while the cost of energy stays high),
• Maximizing overall cost-efficiency requires an
  increasingly large fraction of all bit-ops be
  done adiabatically.
• Maximizing the efficiency of the resulting
  algorithms, in turn, requires reversibility in
• Logic design
• Functional units
• Instruction set architectures
• Programming languages
• High-level algorithms

(unless a perfect emulator is found)
Increasing requirementfor degree of
reversibility
Pro-gram-mingmodel
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All Known Reversible Architectures

• Ed Barton (MIT class project, 1978)
• Conservative logic, w. garbage stack
• Andrew Ressler (MIT bachelors thesis, 1979 MIT
  masters thesis, 1981)
• Like Bartons, but more detailed. Paired
  branches.
• Henry Baker (1992)
• Reversible pointer automaton machine
  instructions.
• J. Storrs JoSH Hall (1994)
• Retractile-cascade-based PDP-10-like
  architecture.
• Carlin Vieri (MIT masters thesis, 1995)
• Early Pendulum ISA, irrev. impl., full VHDL
  detail.
• Frank Rixner (MIT class project, 1996)
• Tick VLSI schematics layout of Pendulum
  subset, w. paired branches
• Frank Love (MIT class project, 1996)
• FlatTop Adiabatic VLSI impl. of programmable
  reversible gate array
• Vieri (MIT Ph.D. thesis, 1999)
• Fully adiabatic VLSI implementation of Pendulum
  w. paired branches

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Reversible Programmable Gate-Array Architectures
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(as of May 99)
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Photo of packaged FlatTop chip
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A Bouncing BBMCA Ball
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A BBMCA Fredkin Gate
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Reversible von Neumann Architectures
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Reversible Architecture Issues

• Instruction-Set Architecture (ISA) Issues
• How to define irrev. ops (AND, etc.) reversibly?
• How to do jumps/branches reversibly?
• What kind of memory interface to have?
• What about I/O?
• How to permit efficient reversible algorithms?
• Should the hardware guarantee reversibility?
• Microarchitectural issues
• Register file interface
• Reversible ALU operations
• Shared buses
• Program counter control

88
The Trivial Cases

• Many typical instructions already reversible
• NOT a
• Set register a to its bitwise logical complement,
  a a
• NEG a
• Set a to its twos complement negation
• a -a or a a 1
• INC a
• Increment a by 1 (modulo 2?).
• ADD a b
• Add register b into register a (a (a b)
  mod 2?)
• XOR a b
• Exclusive-or b into a (a a ?
  b)
• ROL a b
• Rotate bits in register a left by positions
  given by b.

89
The Nontrivial Cases

• Other common instructions are not reversible
• CLR a
• Clear register a to 0.
• LD a b
• Load register a from addr. pointed to by b.
• LDI a 3
• Load immediate value 3 into register a.
• AND a b
• Set a to the bitwise AND of a and b
• JMP a
• Jump to the instruction pointed to by a.
• SLL a b
• Shift the bits in a left by b bits, filling with
  0s on right.

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Irreversible Data Operations

• How to do expanding ops reversibly?
• E.g., AND a b - Prior value of a is lost.
• Approach 1 Garbage Stack approach.
• Based on Landauers embedding.
• Push all data that would otherwise be destroyed
  onto a special garbage stack hidden from pgmr.
• Can unwind computation when finished to recover
  stack space. (Lecerf 63/Bennett 73 approach)
• Problems Large garbage stack memory needed.
• Limits computation length.
• Leaves programmer no opportunity to choose a more
  efficient reversible algorithm!

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Illustrating Garbage Stack

• Let ? mean reversible move, ? mean reversible
  copy, ? a reversible uncopy.

Garbage StackMemory (GSM)
AND a b
implemented by...
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0
Garbage StackPointer (GSP)
1. t ? a2. a ? t b3. t ? GSMGSP
1
46
3
17
2
0
3
0
4
...
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Programmer-Controlled Garbage

• Put extra data in a programmer-manipulable
  location.
• What if destination location isnt empty?
• Signal an error, or
• Use an op that does something reversible anyway
• Provide undo operations to accomplish
  unexpanding inverses of expanding ops.
• 1st method Errors on non-empty destination
• AND A B C -If (A0) A?BC else error
• UNAND A B C -If (ABC) A?BC else error
• 2nd method Use always-reversible store ops.
• ANDX A B C - A ? A ? (B C) (self-undoing!)

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Pendulum - packaged die photo
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