1. Physical Limits of Computing Dr. Mike Frank CIS 6930, Sec. #3753X Spring 2002 Lecture #28 Reversible Scaling Analysis I: Cost Models & Leakage-Free Limit Mon., Mar. 25

2. Administrivia & Overview Don’t forget to keep up with homework!Don’t forget to keep up with homework! –We are  9 out of 14 weeks into the course. You should have earned  ~64 points by now.You should have earned  ~64 points by now. Course outline:Course outline: –Part I&II, 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.Limits of adiabatics: Friction,Leakage,Power supplies. - last Mon. RevComp theory I: Reversible Logic Models - last Wed.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 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. 3/25RevComp scaling analysis I: Cost models, w. leakage - Mon. 3/25 RevComp scaling analysis II: The low-leakage limit. - Wed. 3/27RevComp scaling analysis II: The low-leakage limit. - Wed. 3/27 (plus ~5 more lectures…)(plus ~5 more lectures…) –Part VI, Quantum Computing –Part VII, Cosmological Limits, Wrap-Up

3. Cost-Efficiency Analysis Cost Efficiency Cost Measures in Computing Generalized Amdahl’s Law

4. Cost-Efficiency Cost-efficiency of anything is % $ = $ min /$,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 possiblecompared to the maximum number possible –Inversely proportional to cost $. –Maximizing % $ means minimizing $. Regardless of what $ min actually is.Regardless of what $ min actually is. In computing, the “thing” is a computational task that we wish to be carried out.In computing, the “thing” is a computational task that we wish to be carried out.

5. Components of Cost The cost $ of a computation may be a sum of terms for many different components: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 resultsCost 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 of raw physical spacetime occupied by computation. –Cost to rent the space. Cost of hardware (amortized over its lifetime)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 parts/labor for operation & maintenance. Cost of SW licenses

6. More cost components Continued...Continued... –Area-time costs: Cost to rent portion of an enclosing convex hull for getting things in & out of the systemCost to rent portion of an enclosing convex hull for getting things in & out of the system –Energy, heat, information, people, materials, entropy. Some examples: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.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 raw free energy expenditure (entropy generation). Cost of energy-delivery system. (Amortized.)Cost of energy-delivery system. (Amortized.) Cost of cooling system. (Amortized.)Cost of cooling system. (Amortized.)

7. General Cost Measures The most comprehensive cost measure includes terms for all of these potential kinds of costs.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(  t start  end )$ Time is an non-decreasing function f(  t start  end ) –Simple model: $ Time   t start  end $ FreeEnergy is most generally$ FreeEnergy is most generally –Simple model: $ FreeEnergy   S generated $ SpaceTime and $ AreaTime are most generally:$ SpaceTime and $ AreaTime are most generally: –Simple model: $ SpaceTime  Space  Time$ SpaceTime  Space  Time $ AreaTime  Area  Time$ AreaTime  Area  Time Max # ops that could be done Max # items that could be I/O’d

8. Generalized Amdahl’s Law Given any cost that is a sum of components, $ tot = $ 1 + … + $ n,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.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)At a “design equilibrium,” all cost components will be roughly equal (unless externally driven)

9. Reversible vs. Irreversible Want to compare their cost-efficiency under various cost measures: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:Note that space (volume, mass, etc.) by itself as a cost measure is only significant if either: –(a) The computer isn’t reusable & so the cost to build it dominates operating costs. –(b) I/O latency  V 1/3 affects other costs. Or, for some applications, one quantity might be minimized while another one (space, time, area) is constrained by some hard limit.

10. Time Cost Comparison For computations with unlimited power/cooling and no communication requirements:For computations with unlimited power/cooling and no communication requirements: –Reversible worse than irreversible by a factor of ~s>1 (adiabatic slowdown factor), times maybe a small constant depending on logic style used. $ r,Time  $ i,Time · s

11. Time Cost Comparison, cont. For parallelizable power-limited applications:For parallelizable power-limited applications: –With nonzero leakage: $ r,Time  $ i,Time / R on/off g Worst-case computations: g  0.4Worst-case computations: g  0.4 Best-case computations: g = 0.5.Best-case computations: g = 0.5. For parallelizable area-limited, entropy-flux- limited, best case applications:For parallelizable area-limited, entropy-flux- limited, best case applications: – with leakage  0: $ r,Time  $ i,Time · d 1/2 –where d is system’s physical diameter.

12. Time cost comparison, cont. For entropy-flux limited, parallel, heavily communication-limited, best case applications:For entropy-flux limited, parallel, heavily communication-limited, best case applications: –with leakage approaching 0: $ r,Time  $ i,Time 3/4 –where $ i,Time scales up with the space requirement V as $ i,Time  V 2/9 –so the reversible speedup scales with the 1/18 power of system size.

13. Bennett 89 alg. is not optimal k = 2 n = 3 k = 3 n = 2 Just look at all the spacetime it wastes!!!

14. Parallel “ Frank02” algorithm We can simply squish the triangles closer together to eliminate the wasted spacetime!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!Resulting algorithm is linear time for all n and k and dominates Ben89 for time, #ops, & spacetime! Real time Emulated time k=2 n=3 k=3 n=2 k=4 n=2

15. On/off power ratio