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1. Physics of Information Technology Crystalline Computation Norm Margolus

2. Crystalline Computation Spatial Computers • Architectures and algorithms for large-scale spatial computation Emulating Physics • Finite-state, locality, invertibility, and conservation laws Physical Worlds • Incorporating comp-universality at small and large scales see: http://people.csail.mit.edu/nhm/cc.pdf

3. Spatial Computers • Finite-state computations on lattices are interesting • Crystalline computers will one day have highest performance • Today’s ordinary computers are terrible at these computations

4. Tom Toffoli and I wanted to see CA-TV Stimulated by new rev rules Rule = a few TTL chips! Later, rule = SRAM LUT LGA’s needed more “serious” machine! CAMs 1 to 6

5. Lattice example //* Four direction lattice gas *// 2D-square-lattice {512 512} create-bit-planes {N S E W} define-step shift-planes {(N,0,-1) (S,0,1) (E,1,0) (W,-1,0)} for-each-site if {N==S && E==W} {xchng(N,E); xchng(S,W)} endif end-site end-step  

6. Lattice example //* Four direction lattice gas *// 2D-square-lattice {512 512} create-bit-planes {N S E W} define-step shift-planes {(N,0,-1) (S,0,1) (E,1,0) (W,-1,0)} for-each-site if {N==S && E==W} {xchng(N,E); xchng(S,W)} endif end-site end-step  

7. Lattice example //* Four direction lattice gas *// 2D-square-lattice {512 512} create-bit-planes {N S E W} define-step shift-planes {(N,0,-1) (S,0,1) (E,1,0) (W,-1,0)} for-each-site if {N==S && E==W} {xchng(N,E); xchng(S,W)} endif end-site end-step 

8. Lattice example //* Four direction lattice gas *// 2D-square-lattice {512 512} create-bit-planes {N S E W} define-step shift-planes {(N,0,-1) (S,0,1) (E,1,0) (W,-1,0)} for-each-site if {N==S && E==W} {xchng(N,E); xchng(S,W)} endif end-site end-step 

9. Lattice example //* Four direction lattice gas *// 2D-square-lattice {512 512} create-bit-planes {N S E W} define-step shift-planes {(N,0,-1) (S,0,1) (E,1,0) (W,-1,0)} for-each-site if {N==S && E==W} {xchng(N,E); xchng(S,W)} endif end-site end-step  

10. Lattice example //* Four direction lattice gas *// 2D-square-lattice {512 512} create-bit-planes {N S E W} define-step shift-planes {(N,0,-1) (S,0,1) (E,1,0) (W,-1,0)} for-each-site if {N==S && E==W} {xchng(N,E); xchng(S,W)} endif end-site end-step  

11. Lattice example //* Four direction lattice gas *// 2D-square-lattice {512 512} create-bit-planes {N S E W} define-step shift-planes {(N,0,-1) (S,0,1) (E,1,0) (W,-1,0)} for-each-site if {N==S && E==W} {xchng(N,E); xchng(S,W)} endif end-site end-step

12. Lattice Gas Machines Better for modeling: • Flexible neighborhoods • Incorporate physics Better for hardware: • Easier communication • Flexible parallelism The lattice gas programming model: alternate steps of movement and interaction.

13. CAM-8 node • maps directly onto programming model (bit fields) • 16-bit lattice sites (use internal dimension to get more) • each node handles a chunk of a larger space

14. CAM-8: discrete hydrodynamics Six direction LGA flow past a half-cylinder. System is 2Kx1K.

15. Crystallization Crystallization using irreversible forces (Jeff Yepez, AFOSR)

16. CAM-8: EM scattering

17. CAM-8: materials simulation 512x512x64 spin system, 6up/sec w rendering 256x256x256 reaction-diffusion (Kapral)

18. Physical sim Logical sim Wavefront sim Microprocessor at right runs at 1KHz CAM-8: logic simulation A simple logic simulation in a gate-array-like lattice dynamics. Wavefront simulation (R. Agin)

19. CAM-8: porous flow • Used MRI data from a rock (Fontainebleau sandstone) • Simulation of flow of oil through wet sandstone agrees with experiment within 10% • 3D LGA’s used for chemical reactions, other complex fluids 64x64x64 simulation (.5mm/side)

20. Local dynamics can be used to find and mark features Bit maps can be “continuously” rotated using 3 shears CAM-8: image processing (Toffoli)

21. 25 Million 16-bit site-updates/sec 8 node prototype arbitrary shifts of all bit fields at no extra cost limited by mem bandwidth still 100x PC! CAM-8 node (1988 technology) 6MB/sec x 16 DRAMs + 16 ASICs 25 MHz 64Kx16

22. 100 x faster per DRAM chip No custom hardware Needed fast FPGA interface to DRAM FPGA node design (1997) RDRAM FPGA 600MB/sec 1 RDRAM + 1 FPGA 150 MHz 64Kx16 pipelined

23. Embedded DRAM (2000) 4 Mbit Block of DRAM: 20 Blocks of DRAM (80 Mbits total): 2K 40K 2K 2K 2K bits / 40 ns 40K bits / 40 ns = 1 Tbit/sec per chip! Each chip would be 1000x faster than 128-DRAM CAM-8!

24. SPACERAM: A general purpose crystalline lattice computer • Large scale prob with spatial regularity • 1 Tbit/sec ≈ 1010 32-bit mult-accum / sec (sustained) for finite difference • Many image processing and rendering algorithms have spatial regularity • Brute-force physical simulations (complex MD, fields, etc.) • Bit-mapped 3D games and virtual reality (simpler and more direct) • Logic simulation (physical, wavefront) • Pyramid, multigrid and multiresolution computations

25. The Task: Large-scale brute force computations with a crystalline-lattice structure The Opportunity: Exploit row-at-a-time access in DRAM to achieve speed and size. The Challenge: Dealing efficiently with memory granularity, commun, processing The Trick: Data movement by layout and addressing The Result: Essentially perfect use of memory and communications bandwidth, without buffering SPACERAM . 18µm CMOS, 10 Mbytes DRAM, 1 Tbit/sec to mem, 215 mm2, 20W http://people.csail.mit.edu/nhm/isca.pdf

26. Lattice computation model 1D example: the rows are bit-fieldsand columns sites A B Shift bit-fields periodically and uniformly (may be large)  A B  Process sites independently and identically (SIMD)

27. Mapping the lattice into memory A A B B  A  A B  B 

28. Mapping the lattice into memory CAM-8 approach: group adjacent into memory words A B Shifts can be long but re-aligning messy  chaining  A B  Can process word-wide (CAM-8 did site-at-a-time)

29. Mapping the lattice into memory CAM-8: Adjacent bits of bit-field are stored together = + + +

30. Mapping the lattice into memory SPACERAM: bits stored as evenly spread skip samples. = + + +

31. module #00 … module #01 … module #02 … … module #03 … module #19 PE0 PE63 The SPACERAM chip

32. Virtual Processor Space-efficient parallelism Static routing (systolic) Lattice Gas Machines Simpler & more general Ideal for physical modeling Virtual SIMD programming General Lattice Computer Perfect memory embedding Finer-grained virtual-SIMD Computing Crystals Architecture in software Easier to design/build/test Hardware Evolution

33. Crystalline Computation Spatial Computers • Architectures and algorithms for large-scale spatial computation Emulating Physics • Finite-state, locality, invertibility, and conservation laws Physical Worlds • Incorporating comp-universality at small and large scales see: http://people.csail.mit.edu/nhm/cc.pdf

34. Why emulate physics? • Comp must adapt to microscopic physics • Comp models may help us understand nature • Rich dynamics • Start with locality: Cellular Automata

35. Conway’s “Game of Life” In each 3x3 neighborhood, count the ones, not including the center: If total = 2:center unchanged If total = 3:center becomes 1 Else:center becomes 0 256x256 region of a larger grid. Glider gun inserted near middle.

36. Conway’s “Game of Life” • Captures physical locality and finite-state But, • Not reversible (doesn’t map well onto microscopic physics) • No conservation laws (nothing like momentum or energy) • No interesting large-scale behavior 256x256 region of a larger grid. About 1500 steps later.

37. Reversibility & other conservations • Reversibility is conservation of information • Why does exact conservation seem hard? The same information is visible at multiple positions For rev, one nth of the neighbor info must be left at the center

38. Adding conservations • With traditional CA’s, conservations are a non-local property of the dynamics. • Simplest solution: redefine CA’s so that conservation is a manifestly local property • CA regular computation in space & time • Regular in space: repeated structure • Regular in time: repeated sequence of steps

39. Diffusion rule Even steps:rotate cw or ccw a b c a b d or c d d b a c Odd steps:rotate cw or ccw a b c a b d or c d d b a c We “randomly” choose to rotate blocks 90-degrees cw or ccw (we actually use a fixed sequence of choices for each spot). Use 2x2 blockings. Use solid blocks on even time steps, use dotted blocks on odd steps.

40. Diffusion rule Even steps:rotate cw or ccw a b c a b d or c d d b a c Odd steps:rotate cw or ccw a b c a b d or c d d b a c We “randomly” choose to rotate blocks 90-degrees cw or ccw (we actually use a fixed sequence of choices for each spot).

41. Diffusion rule Even steps:rotate cw or ccw a b c a b d or c d d b a c Odd steps:rotate cw or ccw a b c a b d or c d d b a c We “randomly” choose to rotate blocks 90-degrees cw or ccw (we actually use a fixed sequence of choices for each spot).

42. TM Gas rule Even steps:rotate cw a b c a c d d b Odd steps:rotate ccw a b b d c d a c Except:2 ones on diag, nc Use 2x2 blockings. Use solid blocks on even time steps, use dotted blocks on odd steps.

43. TM Gas rule Even steps:rotate cw a b c a c d d b Odd steps:rotate ccw a b b d c d a c Except:2 ones on diag, nc Even step: update solid blocks.

44. TM Gas rule Even steps:rotate cw a b c a c d d b Odd steps:rotate ccw a b b d c d a c Except:2 ones on diag, nc Odd step: update dotted blocks

45. TM Gas rule Even steps:rotate cw a b c a c d d b Odd steps:rotate ccw a b b d c d a c Except:2 ones on diag, nc Even step: update solid blocks

46. TM Gas rule Even steps:rotate cw a b c a c d d b Odd steps:rotate ccw a b b d c d a c Except:2 ones on diag, nc Odd step: update dotted blocks

47. TM Gas rule Even steps:rotate cw a b c a c d d b Odd steps:rotate ccw a b b d c d a c Except:2 ones on diag, nc Even step: update solid blocks

48. TM Gas rule Even steps:rotate cw a b c a c d d b Odd steps:rotate ccw a b b d c d a c Except:2 ones on diag, nc Odd step: update dotted blocks

49. a b c a c d d b TM Gas rule Even steps:rotate cw Odd steps:rotate ccw a b b d c d a c Except:2 ones on diag, nc