Physical Limits of Computing Dr. Mike Frank CIS 6930, Sec.

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Physical Limits of ComputingDr. Mike Frank CIS
6930, Sec. 3753XSpring 2002

• Lecture 28Reversible Scaling Analysis ICost
  Models Leakage-Free LimitMon., Mar. 25

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Administrivia Overview

• Dont forget to keep up with homework!
• We are ?9 out of 14 weeks into the course.
• You should have earned ?64 points by now.
• Course outline
• Part III, Background, Fundamental Limits - done
• Part III, Future of Semiconductor Technology -
  done
• Part IV, Potential Future Computing Technologies
  - done
• Part V, Classical Reversible Computing
• Limits of adiabatics Friction,Leakage,Power
  supplies. - last Mon.
• RevComp theory I Reversible Logic Models - last
  Wed.
• RevComp theory II Emulating Irreversible
  Machines - last Fri. RevComp theory II Bounds
  on Space-Time Overheads - last Fri.
• RevComp scaling analysis I Cost models, w.
  leakage - Mon. 3/25
• RevComp scaling analysis II The low-leakage
  limit. - Wed. 3/27
• (plus 5 more lectures)
• Part VI, Quantum Computing
• Part VII, Cosmological Limits, Wrap-Up

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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 be carried out.

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Components of Cost

• The cost of a computation may be a sum of terms
  for many different components
• Time cost
• Cost to user of having to wait for results
• E.g., missing deadlines, incurring penalties.
• May increase nonlinearly with time for long
  times.
• Spacetime-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 costs
• Cost to rent portion of an enclosing convex hull
  for getting things in out of the system
• Energy, heat, information, people, materials,
  entropy.
• Some examples
• Chip area, power level, cooling capacity, I/O
  bandwidth, desktop footprint, floor space, real
  estate, planetary surface
• Area-time costs scale with the maximum number of
  items that can be sent/received.
• Energy expenditure 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.
• Optimization effort should focus on the cost
  components that are most significant in 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.
• (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 and
  no communication requirements
• Reversible worse than irreversible by a factor of
  sgt1 (adiabatic slowdown factor), times maybe a
  small constant depending on logic style
  used. r,Time ? i,Time s

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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.

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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.

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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 squish the triangles closer
  together to eliminate the wasted spacetime!
• Resulting algorithm is linear time for all n and
  k and dominates Ben89 for time, ops,
  spacetime!

k3n2
k2n3
Emulated time
k4n2
Real time
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On/off power ratio