stem
Reconfigurable
Asynchronous
Logic Automata
(RALA)
wire
XOR
crossover
OR
delete
NAND
copy
AND
wire
David Dalrymple
Erik Demaine
Neil Gershenfeld
Motivation for Theoreticians
 What is the model of computation of the
physical universe?
▪ Computation = local interaction of particles
▪ Particles move around (which takes time)
 Very different from existing models:
▪ Sequential models of computation (RAM, etc.)
▪ Memory hierarchies ~model movement to a
single CPU, but this is an unnecessarily limit
▪ PRAM/UMA ignores communication cost
▪ Most distributed computing ignores geometry
Motivation for Systems
80-core Teraflop
Intel Polaris
 Evolution increasingly multicore
▪ What happens in the limit?
 FPGAs highly successful
▪ What happens in the limit?
XILINX FPGA
 Is there a universal chip architecture
supporting general algorithms nearly as
efficiently as all other chips?
Motivation for Fabrication
 How should computation scale to
physically large objects?
▪ Walls
▪ Displays
▪ Surfaces
▪ 3D print
▪ Robots
Solar furnace in
Odellio, France.
Temperatures
up to 33,000° C.
Ideas Behind RALA
computation
= data = shape
= communication
 Unify computation & data to equal # bits
 Embed into geometric space
 Communicate subject to physical laws
▪ Essentially a general circuit, where
longer wires take longer to carry data
 Globally asynchronous to enable scaling,
but locally synchronous for ease of use
 Reconfigurable to enable programming
Asynchronous Logic Automaton
(ALA)
 Circuit = grid of cells (squares or cubes)
▪ Single-bit stream processor
▪ Up to two inputs
▪ Any number of outputs
 Wires between neighbors
▪ Buffers a single bit (0 or 1)
▪ Can have no bits (X)
 Gate operates when all
inputs are ready (0 or 1)
& outputs are free (X)
OR
X
1
OR
1
NAND
X
1
X
1
AND
X
0
XOR
X
1
1
X
OR
0
X
X
0
ALA
1
0
X
ALA is Easy to Program
(similar to standard digital circuits)
SEA cryptosystem
sorter
multiplier
router
(project with Cisco)
Reconfiguration in RALA
 Stem cell turns data into programs by
reprogramming neighbors & itself
 Format of input stream (from one source):
▪ Direction (3 bits): make neighbor a stem cell
▪ Forward bits to neighbor until it says “done”
▪ Cell type (3 bits): AND, NAND, OR,
XOR, crossover, copy, delete, stem
▪ Input directions (3 bits  2 inputs)
NAND
▪ Output directions (6 bits)
▪ Repeat (or special 000 code  done)
How to Make Anything with RALA
 Circuit rasterization:
NAND,F,L,B
AND,B,R,F
OR,L,N,BR
XOR,R,F,B
forward
OR,B,L,L
OR,R,N,F
right
left
▪ Zig-zag through
OR
Stem
target circuit region
using forwarding
mechanism of stem cell
◦ (requires region to have
a Hamiltonian path)
AND
Stem
OR
Stem
XOR
Stem
NAND
Stem
OR
Stem
▪ Send code to program
cells in reverse order
CAD 
RALA
cad.py
face = circle(0,0,10)
eye = circle(0,0,2)
face = subtract(face,
move(eye,4,3))
face = subtract(face,
move(eye,-4,3))
mouth = circle(0,0,8)
mouth = subtract(mouth,
circle(0,0,6))
mouth = subtract(mouth,
rectangle(-10,10,-2,10))
face = subtract(face,mouth)
CAD  RALA (3D)
How to Make Anything with RALA
 Spanning-tree method:
▪ Perform depth-first
search of spanning tree
of target circuit
▪ Use backtracking ability
of stem-cell forwarding
▪ More general, e.g., if no
Hamiltonian path (or
don’t want to find one)
OR
AND
OR
XOR
NAND
OR
Hierarchical Construction
 Exploit computational power to build
structures faster, e.g., hierarchically
code = 3 * (
right + left +
straight * 2 +
stem(B) +
wire(L,[F,B])
+ null)
Regular Grid Construction
prefix = (right + stem(B) + forward * x) * (m − 1)
unit
unit
unit
+ right + stem(B) + wire(B,R)
+ Theorem:
of
(wire(B,F) * m
(x ×− n1)array
+ wire(B,[F,R])
) unit
* (m −unit1) unit
× y+modules
time * y) * (n −unit
+x
(left
stem(B) +inforward
1) unit unit
n + x y)+ wire(B,L)
vs. O(m n x y) unit unit unit
+ O(m
left ++stem(B)
+ (▪
wire(B,F)
* (yregularity
− 1) + wire(B,[F,L])
) * adder,
(n − 1) shape
E.g. exploit
of memory,
code = right + right
+ left + stem(B)
Replication
+ wire(B,[F,L])
+ wire(R,L)
+ forward
 81-bit string encodes + forward
replicator
universal replicator
program
▪ Replicates given
program infinitely
▪ ~polymerase+ribosome
stem
copy
copy
copy
Cellular Automata
alive = (2 ≤ num_live_neighbors ≤ 3)
 Conway’s Game of Life is Turing-complete
Gosper’s Glider Gun
17141647 Turing machine
RALA vs. Cellular Automata
 Logic automaton:
▪ Designed for digital Boolean logic
▪ Easy to implement standard circuits
 Asynchronous:
▪ Cellular automata require global clock
▪ Impractical/slow for large sizes
 Reconfigurable:
▪ Stem cell converts data into programs
▪ vs. von Neumann gates: enable replication
but not obviously universal “ribosome”
Programmable Matter via RALA
 Cell = one-bit computer (gate)
& one unit of physical space
▪ Communicates with neighbors
 Request new cell on bare face
▪  Can grow a shape
 Convergence of shape,
computation, and
communication
RALA: Convergence of Shape,
Computation, & Communication
 New model for mixing computation, data,
program, & shape (state = space = time = logic)
 Theoretically interesting
▪ Parallel in a new way
▪ Computation embedded into geometry
▪ Programs that generate programs (and so on)
opens new doors for construction, encoding, …
 Practically interesting
▪ Physically realistic (pay for transportation)
▪ Easy to build (next)
stem
Reconfigurable
Asynchronous
Logic Automata
(RALA)
wire
XOR
crossover
OR
delete
NAND
copy
AND
wire
David Dalrymple
Erik Demaine
Neil Gershenfeld