pit 05/01/06 physics of information technology norm margolus crystalline computation
TRANSCRIPT
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
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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
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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 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
QuickTime™ and aCinepak decompressor
are needed to see this picture.
PIT 05/01/06
Lattice Gas Machines
Better for modeling:
• Flexible neighborhoods
• Incorporate physics
Better for hardware:
• Easier communication
• Flexible parallelismThe lattice gas programming model: alternate steps of movement and interaction.
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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
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CAM-8: discrete hydrodynamics
Six direction LGA flow past a half-cylinder. System is 2Kx1K.
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Crystallization
Crystallization using irreversible forces (Jeff Yepez, AFOSR)
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CAM-8: EM scattering
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CAM-8: materials simulation
512x512x64 spin system, 6up/sec w rendering 256x256x256 reaction-diffusion (Kapral)
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CAM-8: logic simulation
• Physical sim
• Logical sim
• Wavefront sim
• Microprocessor at right runs at 1KHz
A simple logic simulation in agate-array-like lattice dynamics.
Wavefront simulation
(R. Agin)
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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 fluids64x64x64 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)
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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
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Embedded DRAM (2000)
2K
2K
2K bits / 40 ns
4 Mbit Blockof 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
tyx
tyxtyxtyx
tyxtyxtyx
tyxtyxtyx
tyxtyxtyxtyx
f
fff
fff
fff
ffff
,,
,2,,2,,,2
,,2,1,1,1,1
,1,1,1,1,1,
,1,,,1,,11,,
τσνμκιηζεδγβα
+
+++
+++
+++
++=
−+−
+−−+−
−+++−
+−++
• 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
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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 mm2, 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
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Mapping the lattice into memory
A
B
A
B
A
B
A
B
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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
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Mapping the lattice into memory
CAM-8: Adjacent bits of bit-field are stored together
=
+
+
+
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Mapping the lattice into memory
SPACERAM: bits stored as evenly spread skip samples.
=
+
+
+
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The SPACERAM chip
module #00
module #01
module #02
module #03
module #19
PE0 PE63
……
…
…
…
…
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
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
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.
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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 behavior256x256 region of a larger grid.
About 1500 steps later.
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Reversibility & other conservations
• Reversibility is conservation of information
• Why does exact conservation seem hard?
The same information is visibleat multiple positions
For rev, one nth of the neighborinfo must be left at the center
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• 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
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Diffusion rule
a b
c d
c a
d b
a b
c d
b d
a c
Even steps: rotate cw or ccw
Odd steps: rotate cw or ccw
orb d
a c
orc a
d b
We “randomly” choose to rotate blocks90-degrees cw or ccw (we actually use afixed sequence of choices for each spot).
Use 2x2 blockings. Use solidblocks on even time steps, usedotted blocks on odd steps.
PIT 05/01/06
QuickTime™ and aCinepak decompressor
are needed to see this picture.
Diffusion rule
a b
c d
c a
d b
a b
c d
b d
a c
Even steps: rotate cw or ccw
Odd steps: rotate cw or ccw
orb d
a c
orc a
d b
We “randomly” choose to rotate blocks90-degrees cw or ccw (we actually use afixed sequence of choices for each spot).
PIT 05/01/06
QuickTime™ and aCinepak decompressor
are needed to see this picture.
Diffusion rule
a b
c d
c a
d b
a b
c d
b d
a c
Even steps: rotate cw or ccw
Odd steps: rotate cw or ccw
orb d
a c
orc a
d b
We “randomly” choose to rotate blocks90-degrees cw or ccw (we actually use afixed sequence of choices for each spot).
PIT 05/01/06
TM Gas rule
a b
c d
c a
d b
a b
c d
b d
a c
Even steps: rotate cw
Odd steps: rotate ccw
Except: 2 ones on diag, nc
Use 2x2 blockings. Use solidblocks on even time steps, usedotted blocks on odd steps.
PIT 05/01/06
TM Gas rule
a b
c d
c a
d b
a b
c d
b d
a c
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
a b
c d
c a
d b
a b
c d
b d
a c
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
a b
c d
c a
d b
a b
c d
b d
a c
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
a b
c d
c a
d b
a b
c d
b d
a c
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
a b
c d
c a
d b
a b
c d
b d
a c
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
a b
c d
c a
d b
a b
c d
b d
a c
Even steps: rotate cw
Odd steps: rotate ccw
Except: 2 ones on diag, nc
Odd step: update dotted blocks
PIT 05/01/06
QuickTime™ and aCinepak decompressor
are needed to see this picture.
TM Gas rule
a b
c d
c a
d b
a b
c d
b d
a c
Even steps: rotate cw
Odd steps: rotate ccw
Except: 2 ones on diag, nc
PIT 05/01/06
QuickTime™ and aCinepak decompressor
are needed to see this picture.
TM Gas rule
a b
c d
c a
d b
a b
c d
b d
a c
Even steps: rotate cw
Odd steps: rotate ccw
Except: 2 ones on diag, nc
PIT 05/01/06
QuickTime™ and aCinepak decompressor
are needed to see this picture.
QuickTime™ and aCinepak decompressor
are needed to see this picture.
TM Gas rule
PIT 05/01/06
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TM Gas rule
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• 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
a b
c d
d c
b a
Except: two ones on diag flip
Even & odd: swap along diags
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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)
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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
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Lattice gas hydrodynamics
Six direction LGA flow past a half-cylinder, with vortex shedding. System is 2Kx1K.
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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 4neighbors are parallel to it. After theflip, exactly 2 neighbors are still parallel.
We divide the space into twosublattices, updating the goldon even steps, silver on odd.
PIT 05/01/06
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are needed to see this picture.
Dynamical Ising rule
Even steps: update gold sublattice
Odd steps: update silver sublattice
A spin is flipped if exactly 2 of its 4neighbors are parallel to it. After theflip, exactly 2 neighbors are still parallel.
PIT 05/01/06
QuickTime™ and aCinepak decompressor
are needed to see this picture.
Dynamical Ising rule
Even steps: update gold sublattice
Odd steps: update silver sublattice
A spin is flipped if exactly 2 of its 4neighbors are parallel to it. After theflip, 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 4neighbors are parallel to it. After theflip, 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 4neighbors are parallel to it. After theflip, exactly 2 neighbors are still parallel.
At each site in a 1D space, we put2 bits of state. We’ll call one the“gold” bit and one the “silver” bit.We update the gold bits on evensteps, and the silver on odd steps.
PIT 05/01/06
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Bennett’s 1D rule
Even steps: update gold sublattice
Odd steps: update silver sublattice
A spin is flipped if exactly 2 of its 4neighbors are parallel to it. After theflip, 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 4neighbors are parallel to it. After theflip, 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 thespin system, then domains grow as the spins cool.
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2D “Same” rule
Even steps: update gold sublattice
Odd steps: update silver sublattice
A spin is flipped if all 4 of its neighborsare the same. Otherwise it is leftunchanged.
Gold/silver checkerboard
We divide the space into twosublattices, updating the goldon even steps, silver on odd.
PIT 05/01/06
QuickTime™ and aCinepak decompressor
are needed to see this picture.
2D “Same” rule
Even steps: update gold sublattice
Odd steps: update silver sublattice
A spin is flipped if all 4 of its neighborsare the same. Otherwise it is leftunchanged.
PIT 05/01/06
QuickTime™ and aCinepak decompressor
are needed to see this picture.
2D “Same” rule
Even steps: update gold sublattice
Odd steps: update silver sublattice
A spin is flipped if all 4 of its neighborsare the same. Otherwise it is leftunchanged.
PIT 05/01/06
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QuickTime™ and aCinepak decompressor
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3D “Same” rule
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xx
Reversible aggregation rule
xh
xxh xg
Even steps: update gold sublattice
Odd steps: update silver sublattice
When a gas particle diffuses next to exactlyone crystal particle, it crystallizes and emitsa 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
QuickTime™ and aCinepak decompressor
are needed to see this picture.
Reversible aggregation rule
for more info, see cond-mat/9810258
xx xh
xxh xg
Even steps: update gold sublattice
Odd steps: update silver sublattice
When a gas particle diffuses next to exactlyone crystal particle, it crystallizes and emitsa heat particle. The reverse also happens.
g
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Adding forces irreversibly
3D momentum conserving crystallization.
becomes:
Particles six sites apart alongthe lattice attract each other.
PIT 05/01/06
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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.
b c da c d ab
xx xhg
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Conservations
b c da c d ab
xx xhg
Homework Problem #3:• Make up a CA rule that
is reversible and has an exact conservation.
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
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
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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
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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
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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 arandom initial pattern. All activitydies 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 reversibleBilliard Ball Logic Gate
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Billiard Ball Logic
Fixed mirrors allow signals to be routed around.
Mirrors allow signals to cross without interaction.
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A BBM CA rule
2x2 blockings.The solid blocksare used at eventime steps, thedotted blocks atodd steps.
A BBMCA collision:
BBMCA rule.Single one goesto opposite corner,2 ones on diagonalgo to other diag, noother cases change.
PIT 05/01/06
The “Critters” rule
Use 2x2 blockings. Use solidblocks on even time steps, usedotted blocks on odd steps.
This rule is appliedboth to the even andthe odd blockings.
We show all cases:each rotation of a caseon the left maps to thecorresponding rotationof the case on the right.
Note that the number ofones in one step equals the number of zeros inthe next step.
PIT 05/01/06
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The “Critters” rule
This rule is appliedboth to the even andthe odd blockings.
We show all cases:each rotation of a caseon the left maps to thecorresponding rotationof the case on the right.
Note that the number ofones in one step equals the number of zeros inthe next step.Reversible “Critters” rule, started from
a low-entropy initial state (2Kx2K).
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“Critters” is universal
A BBMCA collision:
Critters “glider” collision:
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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!
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UCA with momentum conservation
Hard sphere collision Soft sphere collision
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UCA with momentum conservation
Can shrink balls to points! Soft sphere collision
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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 lattice 3D Cubic lattice
PIT 05/01/06
Macroscopic universalityWith exact microscopic control of every bit, the SSM model lets us compute reversibly and withmomentum 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!
€
E∑ = ′ E ∑E i
r v i∑ = ′ E i ′
r v i∑
€
12 miv
2∑ = 12 ′ m i∑ ′ v 2
mi∑ = ′ m i∑mi
r v i∑ = ′ m i ′
r v i∑
€
(sincer p = γm
r v = γmc 2 ×
r v /c 2)
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 alongthe lattice attract each other.
PIT 05/01/06
QuickTime™ and aCinepak decompressor
are needed to see this picture.
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