PIT 05/01/06 Physics of Information Technology Norm Margolus Crystalline Computation.

PIT 05/01/06 Physics of Information Technology Norm Margolus Crystalline Computation

PIT 05/01/06 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 Crystalline Computation see: http://people.csail.mit.edu/nhm/cc.pdf

PIT 05/01/06 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

PIT 05/01/06 CAMs 1 to 6 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!

PIT 05/01/06 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

PIT 05/01/06 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

PIT 05/01/06 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

PIT 05/01/06 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.

PIT 05/01/06 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

PIT 05/01/06

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

PIT 05/01/06 Crystallization Crystallization using irreversible forces (Jeff Yepez, AFOSR)

PIT 05/01/06 CAM-8: EM scattering

PIT 05/01/06 CAM-8: materials simulation 512x512x64 spin system, 6up/sec w rendering 256x256x256 reaction-diffusion (Kapral)

PIT 05/01/06 CAM-8: logic simulation Physical sim Logical sim Wavefront sim Microprocessor at right runs at 1KHz A simple logic simulation in a gate-array-like lattice dynamics. Wavefront simulation (R. Agin)

PIT 05/01/06 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)

PIT 05/01/06 CAM-8: image processing Local dynamics can be used to find and mark features Bit maps can be “continuously” rotated using 3 shears (Toffoli)

PIT 05/01/06 CAM-8 node (1988 technology) 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! 6MB/sec x 16 DRAMs + 16 ASICs 25 MHz 64Kx16

PIT 05/01/06 FPGA node design (1997) 100 x faster per DRAM chip No custom hardware Needed fast FPGA interface to DRAM 600MB/sec 1 RDRAM + 1 FPGA 150 MHz 64Kx16 pipelined FPGARDRAM

PIT 05/01/06 Embedded DRAM (2000) 2K 2K bits / 40 ns 4 Mbit Block of DRAM: 40K 2K 40K bits / 40 ns = 1 Tbit/sec per chip! 20 Blocks of DRAM (80 Mbits total): Each chip would be 1000x faster than 128-DRAM CAM-8!

PIT 05/01/06 SPACERAM: SPACERAM: A general purpose crystalline lattice computer Large scale prob with spatial regularity 1 Tbit/sec ≈ 10 10 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

PIT 05/01/06 SPACERAM 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 http://people.csail.mit.edu/nhm/isca.pdf. 18µm CMOS, 10 Mbytes DRAM, 1 Tbit/sec to mem, 215 mm 2, 20W

PIT 05/01/06 Lattice computation model 1D example: the rows are bit-fields and columns sites Shift bit-fields periodically and uniformly (may be large) Process sites independently and identically (SIMD) A B   A B

PIT 05/01/06 Mapping the lattice into memory A B   A B A B   A B

PIT 05/01/06 Mapping the lattice into memory CAM-8 approach: group adjacent into memory words Shifts can be long but re- aligning messy  chaining Can process word-wide (CAM-8 did site-at-a-time) A B   A B

PIT 05/01/06 Mapping the lattice into memory CAM-8: Adjacent bits of bit-field are stored together =+++=+++

PIT 05/01/06 Mapping the lattice into memory SPACERAM: bits stored as evenly spread skip samples. =+++=+++

PIT 05/01/06 The SPACERAM chip module #00 module #01 module #02 module #03 module #19 PE0PE63 … … … … … …

PIT 05/01/06 Hardware Evolution 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

PIT 05/01/06 Why emulate physics? Comp must adapt to microscopic physics Comp models may help us understand nature Rich dynamics Start with locality: Cellular Automata

PIT 05/01/06 Conway’s “Game of Life” If total = 2: center unchanged If total = 3: center becomes 1 Else: center becomes 0 In each 3x3 neighborhood, count the ones, not including the center: 256x256 region of a larger grid. Glider gun inserted near middle.

PIT 05/01/06 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.

PIT 05/01/06 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 n th of the neighbor info must be left at the center

PIT 05/01/06 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 Adding conservations

PIT 05/01/06 Diffusion rule ab cd ca db ab cd bd ac Even steps: rotate cw or ccw Odd steps: rotate cw or ccw or bd ac ca db 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.

PIT 05/01/06 Diffusion rule ab cd ca db ab cd bd ac Even steps: rotate cw or ccw Odd steps: rotate cw or ccw or bd ac ca db We “randomly” choose to rotate blocks 90-degrees cw or ccw (we actually use a fixed sequence of choices for each spot).

PIT 05/01/06 TM Gas rule ab cd ca db ab cd bd ac Even steps: rotate cw Odd steps: rotate ccw Except: 2 ones on diag, nc Use 2x2 blockings. Use solid blocks on even time steps, use dotted blocks on odd steps.

PIT 05/01/06 TM Gas rule ab cd ca db ab cd bd ac Even steps: rotate cw Odd steps: rotate ccw Except: 2 ones on diag, nc Even step: update solid blocks.

PIT 05/01/06 TM Gas rule ab cd ca db ab cd bd ac Even steps: rotate cw Odd steps: rotate ccw Except: 2 ones on diag, nc Odd step: update dotted blocks

PIT 05/01/06 TM Gas rule ab cd ca db ab cd bd ac Even steps: rotate cw Odd steps: rotate ccw Except: 2 ones on diag, nc Even step: update solid blocks

PIT 05/01/06 TM Gas rule ab cd ca db ab cd bd ac Even steps: rotate cw Odd steps: rotate ccw Except: 2 ones on diag, nc

PIT 05/01/06 TM Gas rule

PIT 05/01/06 Half the time: HPP gas rule everywhere: Half the time: HPP gas rule outside of blue region, ID rule inside (no change). Lattice gas refraction ab cd dc ba Except: two ones on diag flip Even & odd: swap along diags

PIT 05/01/06 2D ILGA for EM Fields Integer lattice gas version of TLM Like HPP, but with 4-bit integer particle counts for each direction Conserve E and H fields, momentum and energy http://people.csail.mit.edu/nhm/EM2D.pdf (IEEE MICROWAVE AND GUIDED WAVE LETTERS, VOL. 9, NO. 1, JAN 1999)

PIT 05/01/06 3D LGA for EM Fields Based on TDFD model of K. S. Yee and expanded TLM Unit cell is 8 sites, two of which aren’t used Six components of E and H occupy separate sites http://people.csail.mit.edu/nhm/EMLGA.pdf (J Comp Phys 151, 816–835,1999)

PIT 05/01/06 Lattice gas hydrodynamics Six direction LGA flow past a half-cylinder, with vortex shedding. System is 2Kx1K.

PIT 05/01/06 Dynamical Ising rule Even steps: update gold sublattice Odd steps: update silver sublattice Gold/silver checkerboard A spin is flipped if exactly 2 of its 4 neighbors are parallel to it. After the flip, exactly 2 neighbors are still parallel. We divide the space into two sublattices, updating the gold on even steps, silver on odd.

PIT 05/01/06 Dynamical Ising rule Even steps: update gold sublattice Odd steps: update silver sublattice A spin is flipped if exactly 2 of its 4 neighbors are parallel to it. After the flip, exactly 2 neighbors are still parallel.

PIT 05/01/06 Dynamical Ising rule Even steps: update gold sublattice Odd steps: update silver sublattice A spin is flipped if exactly 2 of its 4 neighbors are parallel to it. After the flip, exactly 2 neighbors are still parallel. Homework Problem #1: Show that the waves running along the boundary obey the wave equation exactly Hint: The wave equation’s solutions consist of a superposition of right- and left-going waves

PIT 05/01/06 Bennett’s 1D rule Even steps: update gold sublattice Odd steps: update silver sublattice Gold/silver 1D lattice A spin is flipped if exactly 2 of its 4 neighbors are parallel to it. After the flip, exactly 2 neighbors are still parallel. At each site in a 1D space, we put 2 bits of state. We’ll call one the “gold” bit and one the “silver” bit. We update the gold bits on even steps, and the silver on odd steps.

PIT 05/01/06 Bennett’s 1D rule Even steps: update gold sublattice Odd steps: update silver sublattice A spin is flipped if exactly 2 of its 4 neighbors are parallel to it. After the flip, exactly 2 neighbors are still parallel.

PIT 05/01/06 Bennett’s 1D rule Even steps: update gold sublattice Odd steps: update silver sublattice A spin is flipped if exactly 2 of its 4 neighbors are parallel to it. After the flip, exactly 2 neighbors are still parallel. Homework Problem #2: Simulate Bennett’s 1D rule and measure the average width of the 5 smallest “gliders” and their speed. How is speed related to width?

PIT 05/01/06 3D Ising with heat bath If the heat bath is initially much cooler than the spin system, then domains grow as the spins cool.

PIT 05/01/06 2D “Same” rule Even steps: update gold sublattice Odd steps: update silver sublattice A spin is flipped if all 4 of its neighbors are the same. Otherwise it is left unchanged. Gold/silver checkerboard We divide the space into two sublattices, updating the gold on even steps, silver on odd.

PIT 05/01/06 2D “Same” rule Even steps: update gold sublattice Odd steps: update silver sublattice A spin is flipped if all 4 of its neighbors are the same. Otherwise it is left unchanged.

PIT 05/01/06 3D “Same” rule

PIT 05/01/06 x x Reversible aggregation rule xh x xg Even steps: update gold sublattice Odd steps: update silver sublattice When a gas particle diffuses next to exactly one crystal particle, it crystallizes and emits a heat particle. The reverse also happens. We update the gold sublattice, then let gas and heat diffuse, then update silver and diffuse. g for more info, see cond-mat/9810258 Gold/silver checkerboard

PIT 05/01/06 Reversible aggregation rule for more info, see cond-mat/9810258 x x xhxh xxhxhxg Even steps: update gold sublattice Odd steps: update silver sublattice When a gas particle diffuses next to exactly one crystal particle, it crystallizes and emits a heat particle. The reverse also happens. g

PIT 05/01/06 Adding forces irreversibly 3D momentum conserving crystallization. becomes: Particles six sites apart along the lattice attract each other.

PIT 05/01/06 Adding forces irreversibly Crystallization using irreversible forces (Jeff Yepez, AFOSR)

PIT 05/01/06 Conservations summary 1.the data are rearranged without any interaction, or 2.the data are partitioned into disjoint groups of bits that change as a unit. Data that affect more than one such group don’t change. To make conservations manifest, we employ a sequence of steps, in each of which either Conservations allow computations to map efficiently onto microscopic physics, and also allow them to have interesting macroscopic behavior. bcda cdab x x xhg

PIT 05/01/06 Conservations bcda cdab x x xhg Homework Problem #3: Make up a CA rule that is reversible and has an exact conservation.

PIT 05/01/06 Review: Why emulate physics? Comp must adapt to microscopic physics Comp models may help us understand nature Rich dynamics Started with locality (Cellular Automata).

PIT 05/01/06 Review: Conway’s “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 Observation: It’s hard to create (or discover) conservations in conventional CA’s. 256x256 region of a larger grid. Activity has mostly died off.

PIT 05/01/06 Physical Worlds Some regular spatial systems: 1.Programmable gate arrays at the atomic scale 2.Fundamental finite-state models of physics 3.Rich “toy universes” All of these systems must be computation universal

PIT 05/01/06 Computation Universality If you can build basic logic elements and connect them together, then you can construct any logic function -- your system can do anything that any other digital system can do! It doesn’t take much. Can construct CA that support logic. Can discover logic in existing CAs (eg. Life) Universal CA can simulate any other Logic circuit in gate-array-like CA

PIT 05/01/06 Computation Universality If you can build basic logic elements and connect them together, then you can construct any logic function -- your system can do anything that any other digital system can do. It doesn’t take much. Can construct CA that support logic. Can discover logic in existing CAs (eg. Life) Universal CA can simulate any other Logic circuit in gate-array-like CA

PIT 05/01/06 What’s wrong with Life? Glider guns in Conway’s “Game of Life” CA. Streams of gliders can be used as signals in Life logic circuits. One can build signals, wires, and logic out of patterns of bits in the Life CA

PIT 05/01/06 What’s wrong with Life? One can build signals, wires, and logic out of patterns of bits in the Life CA Life is short! Life is microscopic Can we do better with a more physical CA? Life on a 2Kx2K space, run from a random initial pattern. All activity dies out after about 16,000 steps.

PIT 05/01/06 Billiard Ball Logic Simple reversible logic gates can be universal Turn continuous model into digital at discrete times! (A,B)  AND(A,B) isn’t reversible by itself Can do better than just throw away extra outputs Need to also show that you can compose gates Fredkin’s reversible Billiard Ball Logic Gate

PIT 05/01/06 Billiard Ball Logic Fixed mirrors allow signals to be routed around. Mirrors allow signals to cross without interaction.

PIT 05/01/06 A BBM CA rule 2x2 blockings. The solid blocks are used at even time steps, the dotted blocks at odd steps. A BBMCA collision: BBMCA rule. Single one goes to opposite corner, 2 ones on diagonal go to other diag, no other cases change.

PIT 05/01/06 The “Critters” rule Use 2x2 blockings. Use solid blocks on even time steps, use dotted blocks on odd steps. This rule is applied both to the even and the odd blockings. We show all cases: each rotation of a case on the left maps to the corresponding rotation of the case on the right. Note that the number of ones in one step equals the number of zeros in the next step.

PIT 05/01/06 The “Critters” rule This rule is applied both to the even and the odd blockings. We show all cases: each rotation of a case on the left maps to the corresponding rotation of the case on the right. Note that the number of ones in one step equals the number of zeros in the next step. Reversible “Critters” rule, started from a low-entropy initial state (2Kx2K).

PIT 05/01/06 “Critters” is universal A BBMCA collision: Critters “glider” collision:

PIT 05/01/06 UCA with momentum conservation Hard sphere collision Hard-sphere collision conserves momentum Can’t make simple CA out of this that does Problem: finite impact parameter required Suggestion: find a new physical model!

PIT 05/01/06 UCA with momentum conservation Hard sphere collision Soft sphere collision

PIT 05/01/06 UCA with momentum conservation Can shrink balls to points! Soft sphere collision

PIT 05/01/06 UCA with momentum conservation SSM rule: rotations also act like this. All other cases remain unchanged. This is a Lattice Gas: movement and interaction steps alternate. Can shrink balls to points!

PIT 05/01/06 UCA with momentum conservation SSM rule: rotations also act like this. All other cases remain unchanged. This is a Lattice Gas: movement and interaction steps alternate. Add mirrors at lattice points to guide balls.

PIT 05/01/06 UCA with momentum conservation Add mirrors at lattice points to guide balls. SSM rule with mirrors

PIT 05/01/06 UCA with momentum conservation Add mirrors at lattice points to guide balls. Mirrors allow signals to cross without interacting.

PIT 05/01/06 SSM collisions on other lattices Triangular lattice3D Cubic lattice

PIT 05/01/06 Macroscopic universality With exact microscopic control of every bit, the SSM model lets us compute reversibly and with momentum conservation, but an interesting world should have macroscopic complexity! Relativistic invariance would allow large-scale structures to move: laws of physics same in motion This would allow a robust Darwinian evolution Requires us to reconcile forces and conservations with invertibility and universality.

PIT 05/01/06 Relativistic conservation  Non-relativistically, mass and energy are conserved separately  Simple lattice gasses that conserve only m and mv are more like rel than non-rel systems! Non-relativistic: Relativistic: (energy) (mass) (mom) (energy) (mom)

PIT 05/01/06 Can we add macroscopic forces? 3D momentum conserving crystallization. becomes: Particles six sites apart along the lattice attract each other.

PIT 05/01/06 Can we add macroscopic forces? Crystallization using irreversible forces (Jeff Yepez, AFOSR)

PIT 05/01/06 Universality Universality is a low threshold that separates triviality from arbitrary complexity More of the richness of physical dynamics can be captured by adding physical properties: »Reversible systems last longer, and have a realistic thermodynamics. »Reversibility plus conservations leads to robust “gliders” and interesting macroscopic properties & symmetries. We know how to reconcile universality with reversibility and relativistic conservations

PIT 05/01/06 Use CA hardware to try to do computation better Use CA models to try to understand the world better CA’s are new worlds that we can explore and study Conclusions see: http://people.csail.mit.edu/nhm/cc.pdf

PIT 05/01/06 Physics of Information Technology Norm Margolus Crystalline Computation.

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