Physics of Information Technology
Crystalline
Computation
Norm Margolus
PIT 05/01/06
Crystalline Computation
Spatial Computers
»
Architectures and algorithms for
large-scale spatial computation
Emulating Physics
»
Finite-state, locality, invertibility,
and conservation laws
Physical Worlds
»
PIT 05/01/06
Incorporating comp-universality at
small and large scales
see: http://people.csail.mit.edu/nhm/cc.pdf
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 example
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

//* Four direction lattice gas *//
Lattice example
PIT 05/01/06

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

//* Four direction lattice gas *//
Lattice example
PIT 05/01/06

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

//* Four direction lattice gas *//
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
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
Lattice Gas Machines
Better for modeling:
• Flexible neighborhoods
• Incorporate physics
The lattice gas programming model:
alternate steps of movement and
interaction.
PIT 05/01/06
Better for hardware:
• Easier communication
• Flexible parallelism
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
PIT 05/01/06
CAM-8: discrete hydrodynamics
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
Six direction LGA flow past a half-cylinder. System is 2Kx1K.
PIT 05/01/06
Crystallization
QuickTi me™ a nd a
Cinep ak decompre ssor
are need ed to se e th is p icture.
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
PIT 05/01/06
256x256x256 reaction-diffusion (Kapral)
CAM-8: logic simulation
QuickTime™ and a
PNG decompress or
are needed to s ee this pic ture.
•
•
•
•
Physical sim
Logical sim
Wavefront sim
Microprocessor
at right runs at
1KHz
A simple logic simulation in a
gate-array-like lattice dynamics.
PIT 05/01/06
Wavefront
simulation
(R. Agin)
QuickTime™ and a
PNG decompressor
are needed to see this picture.
CAM-8: porous flow
64x64x64 simulation (.5mm/side)
PIT 05/01/06
• 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
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!
PIT 05/01/06
6MB/sec x
16 DRAMs
+ 16 ASICs
25 MHz
64Kx16
FPGA node design (1997)
• 100 x faster per
DRAM chip
• No custom
hardware
• Needed fast
FPGA interface
to DRAM
PIT 05/01/06
RDRAM
FPGA
600MB/sec
1 RDRAM
+ 1 FPGA
150 MHz
64Kx16
pipelined
Embedded DRAM (2000)
4 Mbit Block
of DRAM:
20 Blocks of DRAM
(80 Mbits total):
2K
2K
40K
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!
PIT 05/01/06
SPACERAM: A general purpose
crystalline lattice computer
f x , y ,t 1   f x 1, y ,t   f x 1, y ,t   f x , y 1,t
  f x , y 1,t   f x 1, y 1,t   f x 1, y 1,t
  f x 1, y 1,t   f x 1, y 1,t   f x  2, y ,t
  f x  2 , y ,t   f x , y  2 , t   f x , y  2 ,t
  f x , y ,t
PIT 05/01/06
• 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
SPACERAM
. 18µm CMOS, 10 Mbytes DRAM,
1 Tbit/sec to mem, 215 mm2, 20W
PIT 05/01/06
• 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
Lattice computation model
A
B
A
B 
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)
PIT 05/01/06
Mapping the lattice into memory
A
A
B
B
A
A
B 
PIT 05/01/06
B 
Mapping the lattice into memory
CAM-8 approach: group
adjacent into memory words
Shifts can be long but realigning messy  chaining
Can process word-wide
(CAM-8 did site-at-a-time)
PIT 05/01/06
A
B
A
B 
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
PE0
PIT 05/01/06
PE63
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
Crystalline Computation
Spatial Computers
»
Architectures and algorithms for
large-scale spatial computation
Emulating Physics
»
Finite-state, locality, invertibility,
and conservation laws
Physical Worlds
»
PIT 05/01/06
Incorporating comp-universality at
small and large scales
see: http://people.csail.mit.edu/nhm/cc.pdf
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”
In each 3x3 neighborhood,
count the ones, not including
the center:
QuickTime™ and a
PNG decompressor
are needed to see this picture.
256x256 region of a larger grid.
Glider gun inserted near middle.
PIT 05/01/06
If total = 2: center unchanged
If total = 3: center becomes 1
Else: center becomes 0
Conway’s “Game of Life”
256x256 region of a larger grid.
About 1500 steps later.
PIT 05/01/06
• 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
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
PIT 05/01/06
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
PIT 05/01/06
Diffusion rule
Even steps:
a
c
b
d
Odd steps:
a
c
Use 2x2 blockings. Use solid
blocks on even time steps, use
dotted blocks on odd steps.
PIT 05/01/06
b
d
rotate cw or
ccw
c a
b d
or
d b
a c
rotate cw or ccw
c a
b d
or
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).
Diffusion rule
Even steps:
a
c
b
d
Odd steps:
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
a
c
b
d
rotate cw or
ccw
c a
b d
or
d b
a c
rotate cw or ccw
c a
b d
or
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).
PIT 05/01/06
Diffusion rule
Even steps:
a
c
b
d
Odd steps:
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
a
c
b
d
rotate cw or
ccw
c a
b d
or
d b
a c
rotate cw or ccw
c a
b d
or
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).
PIT 05/01/06
TM Gas rule
Even steps:
a
c
rotate cw
b
d
c a
d b
Odd steps:
rotate ccw
a
c
b
d
b 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.
PIT 05/01/06
TM Gas rule
Even steps:
a
c
rotate cw
b
d
c a
d b
Odd steps:
rotate ccw
a
c
b
d
b d
a c
Except: 2 ones on diag, nc
Even step: update solid blocks.
PIT 05/01/06
TM Gas rule
Even steps:
a
c
rotate cw
b
d
c a
d b
Odd steps:
rotate ccw
a
c
b
d
b d
a c
Except: 2 ones on diag, nc
Odd step: update dotted blocks
PIT 05/01/06
TM Gas rule
Even steps:
a
c
rotate cw
b
d
c a
d b
Odd steps:
rotate ccw
a
c
b
d
b d
a c
Except: 2 ones on diag, nc
Even step: update solid blocks
PIT 05/01/06
TM Gas rule
Even steps:
a
c
rotate cw
b
d
c a
d b
Odd steps:
rotate ccw
a
c
b
d
b d
a c
Except: 2 ones on diag, nc
Odd step: update dotted blocks
PIT 05/01/06
TM Gas rule
Even steps:
a
c
rotate cw
b
d
c a
d b
Odd steps:
rotate ccw
a
c
b
d
b d
a c
Except: 2 ones on diag, nc
Even step: update solid blocks
PIT 05/01/06
TM Gas rule
Even steps:
a
c
rotate cw
b
d
c a
d b
Odd steps:
rotate ccw
a
c
b
d
b d
a c
Except: 2 ones on diag, nc
Odd step: update dotted blocks
PIT 05/01/06
TM Gas rule
Even steps:
a
c
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
rotate cw
b
d
c a
d b
Odd steps:
rotate ccw
a
c
b
d
b d
a c
Except: 2 ones on diag, nc
PIT 05/01/06
TM Gas rule
Even steps:
a
c
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
rotate cw
b
d
c a
d b
Odd steps:
rotate ccw
a
c
b
d
b d
a c
Except: 2 ones on diag, nc
PIT 05/01/06
TM Gas rule
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
PIT 05/01/06
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
TM Gas rule
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
PIT 05/01/06
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
Lattice gas refraction
• Half the time: HPP gas rule
everywhere:
Even & odd: swap along diags
a
c
QuickTime™ and a
PNG decompressor
are needed to see this picture.
b
d
d c
b a
Except: two ones on diag flip
• Half the time: HPP gas rule
outside of blue region, ID
rule inside (no change).
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
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
Six direction LGA flow past a half-cylinder, with vortex shedding. System is 2Kx1K.
PIT 05/01/06
Dynamical Ising rule
Gold/silver checkerboard
Even steps: update gold sublattice
Odd steps: update silver sublattice
We divide the space into two
sublattices, updating the gold
on even steps, silver on odd.
PIT 05/01/06
A spin is flipped if exactly 2 of its 4
neighbors are parallel to it. After the
flip, exactly 2 neighbors are still parallel.
Dynamical Ising rule
Even steps: update gold sublattice
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
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
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
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
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
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
Gold/silver 1D lattice
Even steps: update gold sublattice
Odd steps: update silver sublattice
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
A spin is flipped if exactly 2 of its 4
neighbors are parallel to it. After the
flip, exactly 2 neighbors are still parallel.
Bennett’s 1D rule
Even steps: update gold sublattice
Odd steps: update silver sublattice
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
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
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?
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
3D Ising with heat bath
QuickTime™ and a
PNG decompressor
are needed to see this picture.
PIT 05/01/06
If the heat bath is initially much cooler than the
spin system, then domains grow as the spins cool.
2D “Same” rule
Gold/silver checkerboard
Even steps: update gold sublattice
Odd steps: update silver sublattice
We divide the space into two
sublattices, updating the gold
on even steps, silver on odd.
PIT 05/01/06
A spin is flipped if all 4 of its neighbors
are the same. Otherwise it is left
unchanged.
2D “Same” rule
Even steps: update gold sublattice
Odd steps: update silver sublattice
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
A spin is flipped if all 4 of its neighbors
are the same. Otherwise it is left
unchanged.
PIT 05/01/06
2D “Same” rule
Even steps: update gold sublattice
Odd steps: update silver sublattice
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
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
QuickTime™ and a
Motion JPEG B decompressor
are needed to see this picture.
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
PIT 05/01/06
Reversible aggregation rule
Gold/silver checkerboard
Even steps: update gold sublattice
g x
xh x
Odd steps: update silver sublattice
xh x
g x
We update the gold sublattice,
then let gas and heat diffuse,
then update silver and diffuse.
When a gas particle diffuses next to exactly
one crystal particle, it crystallizes and emits
a heat particle. The reverse also happens.
PIT 05/01/06
for more info, see cond-mat/9810258
Reversible aggregation rule
Even steps: update gold sublattice
g x
xh x
Odd steps: update silver sublattice
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
xh x
g x
When a gas particle diffuses next to exactly
one crystal particle, it crystallizes and emits
a heat particle. The reverse also happens.
PIT 05/01/06
for more info, see cond-mat/9810258
Adding forces irreversibly
becomes:
Particles six sites apart along
the lattice attract each other.
PIT 05/01/06
3D momentum conserving crystallization.
Adding forces irreversibly
QuickTi me™ a nd a
Cinep ak decompre ssor
are need ed to se e th is p icture.
Crystallization using irreversible forces (Jeff Yepez, AFOSR)
PIT 05/01/06
Conservations summary
To make conservations manifest, we employ a sequence of
steps, in each of which either
1.
2.
the data are rearranged without
any interaction, or
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.
a b c d
g x
b c d a
xh x
Conservations allow computations to map efficiently onto
microscopic physics, and also allow them to have interesting
macroscopic behavior.
PIT 05/01/06
Conservations
Homework Problem #3:
• Make up a CA rule that is
reversible and has an
exact conservation.
PIT 05/01/06
a b c d
g x
b c d a
xh x
Crystalline Computation
Spatial Computers
»
Architectures and algorithms for
large-scale spatial computation
Emulating Physics
»
Finite-state, locality, invertibility,
and conservation laws
Physical Worlds
»
PIT 05/01/06
Incorporating comp-universality at
small and large scales
see: http://people.csail.mit.edu/nhm/cc.pdf
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
QuickTime™ and a
PNG decompress or
are needed to s ee this pic ture.
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
QuickTime™ and a
PNG decompressor
are needed to see this picture.
Logic circuit in gate-array-like 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
Glider guns in Conway’s “Game of
Life” CA. Streams of gliders can be
used as signals in Life logic circuits.
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?
PIT 05/01/06
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
Life on a 2Kx2K space, run from a
random initial pattern. All activity
dies out after about 16,000 steps.
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.
PIT 05/01/06
Mirrors allow signals to
cross without interaction.
A BBM CA rule
2x2 blockings.
The solid blocks
are used at even
time steps, the
dotted blocks at
odd steps.
A BBMCA collision:
PIT 05/01/06
BBMCA rule.
Single one goes
to opposite corner,
2 ones on diagonal
go to other diag, no
other cases change.
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.
Use 2x2 blockings. Use solid
blocks on even time steps, use
dotted blocks on odd steps.
PIT 05/01/06
Note that the number of
ones in one step equals
the number of zeros in
the next step.
The “Critters” rule
This rule is applied
both to the even and
the odd blockings.
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
Reversible “Critters” rule, started from
a low-entropy initial state (2Kx2K).
PIT 05/01/06
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.
“Critters” is universal
Critters “glider” collision:
A BBMCA collision:
PIT 05/01/06
UCA with momentum conservation
• 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!
Hard sphere collision
PIT 05/01/06
UCA with momentum conservation
Hard sphere collision
PIT 05/01/06
Soft sphere collision
UCA with momentum conservation
Can shrink balls to points!
PIT 05/01/06
Soft sphere collision
UCA with momentum conservation
Can shrink balls to points!
PIT 05/01/06
SSM rule: rotations also
act like this. All other
cases remain unchanged.
This is a Lattice Gas:
movement and interaction
steps alternate.
UCA with momentum conservation
Add mirrors at lattice
points to guide balls.
PIT 05/01/06
SSM rule: rotations also
act like this. All other
cases remain unchanged.
This is a Lattice Gas:
movement and interaction
steps alternate.
UCA with momentum conservation
Add mirrors at lattice
points to guide balls.
PIT 05/01/06
SSM rule with mirrors
UCA with momentum conservation
Add mirrors at lattice
points to guide balls.
PIT 05/01/06
Mirrors allow signals to
cross without interacting.
SSM collisions on other lattices
Triangular lattice
PIT 05/01/06
3D Cubic lattice
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-relativistic:

1
2
2
(energy)
miv 2   12 m
v

i
 m   m
r
r
 m v   mv 
i
i i
i
(mass)
i i
(mom)
Relativistic:
 E   E 
r
r
 E v   E v 
(energy)
(mom)
r
r
r 2
2
(since p  mv  mc  v /c )
i i
PIT 05/01/06
i i
 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!
Can we add macroscopic forces?
becomes:
Particles six sites apart along
the lattice attract each other.
PIT 05/01/06
3D momentum conserving crystallization.
Can we add macroscopic forces?
QuickTi me™ a nd a
Cinep ak decompre ssor
are need ed to se e th is p icture.
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
Conclusions
•
•
•
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
see: http://people.csail.mit.edu/nhm/cc.pdf