cellular automata and amorphous computing melanie mitchell portland state university and santa fe...

Cellular Automata and Amorphous Computing

Melanie Mitchell

Portland State University and Santa Fe Institute

Complex Systems Summer School

Friday June 20, 2008

Copyright © 2008 by Melanie Mitchell

What are cellular automata?

Game of Life

http://www.bitstorm.org/gameoflife/

Review (?) of Computation Theory

• Hilbert’s problems

• Turing machines

• Universal Turing Machines

• Uncomputability of the halting problem

Turing machine

Example of Turing machine rule set

Two fundamental theorems of computation theory:

1. There exists a universal Turing machine

2. There is no Turing machine that can solve the halting problem.

Interesting problem: Given an initial configuration, can you calculate analytically how many steps Life will run for before it reaches a fixed configuration?

Universal Computation in the Game of Life

What is the feasibility of using this kind of universal computation in practice?

Von Neumann’s self-reproducing automaton


John von Neumann1903–1957


• After his key role in designing the first electronic computers, von Neumann became interested in links between computers and biology



In the last years of his life, von Neumann worked on the “logic” of self-reproduction and devised the first instance of a self-reproducing “machine” (in software, finally implemented in 1990s).



Von Neumann’s design is complicated, but some of its key ideas can be captured by a simpler problem:


Von Neumann’s design is complicated, but some of its key ideas can be captured by a simpler problem:

Design a computer program that will print out a copy of itself.

A candidate self-copying program


program copy


program copy

print( “program copy”);


program copy


print( “ print(“program copy”);”);


program copy



print(“ print( “ print(“program copy”);”);”);


program copy



print(“ print( “ print(“program copy”);”);”);

“A machine can’t reproduce itself; to do so it would

have to contain a description of itself, and that

description would have to contain a description of itself,

and so on ad infinitum”.

Some commands we will need in our programming language


• mem

– the memory location of the instruction currently being executed


• mem


1 2 3 4 5

computermemory

program test print(“Hello, world”); print(“Goodbye”);end


• mem


1 2 3 4 5

computermemory


mem = 2


• line(n)

– the string of characters in memory location n


• line(n)


1 2 3 4 5



• line(n)


print(line(2));

will print

print(“Hello, world”);

1 2 3 4 5



• loop until condition

– loops until the condition is true




x = 0;

loop until x = 4

{

print(“Hello, world);

x = x+1;

}




x = 0;

loop until x = 4

{

print(“Hello, world);

x = x+1;

}

Hello, worldHello, worldHello, worldHello, world

Output:

A self-copying program

1 program copy2 L = mem + 1;3 print(“program copy”);4 print(“ L = mem + 1;”);5 loop until line[L] =“end”6 {7 print(line[L]);8 L = L+1;9 }10 print(“end”);11 end



Output



Output L=3



program copy

Output L=3



program copy L = mem + 1;

Output L=3



program copy L = mem + 1; print(“program copy”);

Output L=3



program copy L = mem + 1; print(“program copy”);

Output L=4



program copy L = mem + 1; print(“program copy”); print(“ L = mem + 1;”);

Output L=4



program copy L = mem + 1; print(“program copy”); print(“ L = mem + 1;”);

Output L=5



program copy L = mem + 1; print(“program copy”); print(“ L = mem + 1;”); loop until line[L] =“end”

Output L=5



program copy L = mem + 1; print(“program copy”); print(“ L = mem + 1;”); loop until line[L] =“end”

Output L=6



program copy L = mem + 1; print(“program copy”); print(“ L = mem + 1;”); loop until line[L] =“end” {

Output L=6



program copy L = mem + 1; print(“program copy”); print(“ L = mem + 1;”); loop until line[L] =“end” {

Output L=7



program copy L = mem + 1; print(“program copy”); print(“ L = mem + 1;”); loop until line[L] =“end” { print(line[L]);

Output L=7



program copy L = mem + 1; print(“program copy”); print(“ L = mem + 1;”); loop until line[L] =“end” { print(line[L]);

Output L=8



program copy L = mem + 1; print(“program copy”); print(“ L = mem + 1;”); loop until line[L] =“end” { print(line[L]); L = L+1;

Output L=8



program copy L = mem + 1; print(“program copy”); print(“ L = mem + 1;”); loop until line[L] =“end” { print(line[L]); L = L+1;

Output L=9



program copy L = mem + 1; print(“program copy”); print(“ L = mem + 1;”); loop until line[L] =“end” { print(line[L]); L = L+1; }

Output L=9



program copy L = mem + 1; print(“program copy”); print(“ L = mem + 1;”); loop until line[L] =“end” { print(line[L]); L = L+1; }

Output L=10



program copy L = mem + 1; print(“program copy”); print(“ L = mem + 1;”); loop until line[L] =“end” { print(line[L]); L = L+1; } print(“end”);

Output L=10



program copy L = mem + 1; print(“program copy”); print(“ L = mem + 1;”); loop until line[L] =“end” { print(line[L]); L = L+1; } print(“end”);

Output L=11



program copy L = mem + 1; print(“program copy”); print(“ L = mem + 1;”); loop until line[L] =“end” { print(line[L]); L = L+1; } print(“end”); end

Output L=11

Significance of the self-copying program

• The essence of self-copying in this program is to use the same information stored in memory in two ways:

– interpret it as instructions in a computer program




– interpret it as “data” to be used by the instructions in the computer program





• This is also a key mechanism in biological self-reproduction






– DNA = program and data







• This principle was formulated by von Neumann in the 1940s, before the details of biological self-reproduction were well-understood.

Programs and interpreters


• Notice that the self-copying program needs an external interpreter—the part of the computer that carries out the instructions



• Rough analogy to biology:





– RNA, ribosomes, and enzymes = interpreter






• DNA contains instructions for copying itself, and also for building its own interpreter






• DNA contains instructions for copying itself, and also for building its own interpreter

• Von Neumann’s self-reproducing automaton also did this.

What are cellular automaton actually used for? • Different perspectives:

– CAs are models of physical (or biological or social) systems

– CAs are alternative methods for approximating differential equations

– CAs are devices that can simulate standard computers

– CAs are parallel computers that can perform image processing, random number generation, cryptography, etc.

– CAs are a framework for implementing molecular scale computation

– CAs are a framework for exploring how “collective computation” might take place in natural systems (and that might be imitated in novel human-made computational systems)

Dynamics and Computation in Cellular Automata

http://math.hws.edu/xJava/CA/EdgeOfChaosCA1.html

http://math.hws.edu/xJava/CA/EdgeOfChaosCA1.html

1. Fixed point

2. Periodic

3. Chaotic

4. “Complex”

– long transients

– universal computation?

Wolfram’s classes for elementary CAs

Rule:

ECA 110 is a universal computer(Matthew Cook, 2002)

Wolfram’s numbering of ECA:

0 1 1 0 1 1 1 0 = 110 in binary

Outline of A New Kind of Science (Wolfram,2001)(from MM review, Science, 2002)

• Simple programs can produce complex, and random-looking behavior

– Complex and random-looking behavior in nature comes from simple programs.

• Natural systems can be modeled using cellular-automata-like architectures

• Cellular automata are a framework for understanding nature

• Principle of computational equivalence

Principle of Computational Equivalence

1. The ability to support universal computation is very common in nature.

2. Universal computation is an upper limit on the sophistication of computations in nature.

3. Computing processes in nature are almost always equivalent in sophistication.

How can we describe information processing in complex systems?

majority on

A cellular automaton evolved by the genetic algorithm

majority off

Stephen Wolfram's last problem from ``Twenty problems in the theory of cellular automata'' (Wolfram, 1985):

20. What higher-level descriptions of information processing in cellular automata can be given?

"It seems likely that a radically new approach is needed"

What is needed?

1. How can we characterize patterns as computations?

2. How can we design computations in the language of patterns?


Components of computation in spatially extended systems:

– Storage of information

– Transfer of information

– Integration of information from different spatial locations


Components of computation in spatially extended systems: – Storage of information – Transfer of information– Integration of information from different spatial locations

First step: What structures in the observed patterns implement these components?

Second step: How do these structures implement the computation?

– Transfer of information: moving particles

From http://www.stephenwolfram.com/publications/articles/ca/86-caappendix/16/text.html

– Transfer of information: moving particles

– Integration of information from different spatial locations: particle collisions

From http://www.stephenwolfram.com/publications/articles/ca/86-caappendix/16/text.html

How to automatically identify information-carrying structures in spatio-temporal patterns?

Three proposals:

• Filter by regular languages (Crutchfield and Hanson, 1993; Crutchfield et al., 2002)

• Filter by local statistical complexity (Shalizi et al., 2006)

• Filter by local information measures (Lizier et al., 2007)

How to automatically identify information-carrying structures in spatio-temporal patterns?

Filter by regular languages (Crutchfield and Hanson, 1993; Crutchfield et al., 2002)

Regular language: Simple-to-describe periodic pattern

Examples:

Regular domains:

(0)*, (1)* , (01)*

Particles(spatially localized, temporally periodicboundaries or“defects” betweenregular domains)

Regular domainsfiltered out

CA for performing density classification

Regular domains:

(0001)* , (1110)*

Rule 54Regular domains

filtered out

Filter by local statistical complexity (Shalizi et al., 2006)

• Local statistical complexity of site i:

– amount of information from past needed for optimal prediction of future in the vicinity of site i




site i at time t


light cone of pastinfluence on site i

site i at time t





site i at time t



light cone of futureinfluence of site i



site i at time t



light cone of futureinfluence of site i

How well does pastlight cone predict futurelight cone?

Example: Rule 110

Filtering by localstatistical complexity

Original Filtered

Example: Rule 110

Filtering by localstatistical complexity

Note: This filter requires no prior determination of “regular domains”.But more computationally expensive than filtering by regular domains.

Original Filtered

Filter by local information measures(Lizier et al., 2006)


Degree of information storage at site i:

Mutual information between state at site i at t and states of site i at k previous time steps.

Degree of information transfer from site i-j to site i:

Average information in the state of the source (site i-j) about next state of destination (site i)

Degree of information modification at site i:

Degree to which sum of information storage and information transfer do not predict state at site i.




site i at time t








site i at time t

site i k time steps in past








site i at time t

Degree of information storage: how predictable is current state from past states?






Rule 54 Positive informationstorage

From Lizier et al., 2007

Degree of information storage at each site

Positive: information has been stored at this site





Amount of information in the state of the source (site i-j) about next state of destination (site i)










site i at time t

site i j at time t1

(j radius of neighborhood)








site i at time t

site i j at time t1

Degree of information transfer: how predictable is state at site i from previous state at site i j ?

Rule 54 Positive t (i, j, n+1)

Degree of information transferat each site (for j = 1)

Positive: information has been transferred from site i j to site i








site i at time t









site i at time tsite i j at time t1











Degree of information modification: how unpredictable is state at site i from storage and transfer?










Degree of information modification: how unpredictable is state at site i from storage and transfer?

“local separable information”:positive: no info modificationnegative: info modification

Local information modification

Rule 54 Negative s(i, n) Negative s (i, n) plotted on top of information transferNegative: State at site i is not well

predicted by information storage or transfer.

majority on

A cellular automaton evolved by the genetic algorithm (Performance 80%)

majority off

ParticlesRegular domains:

(0)*, (1)* , (01)*

From Crutchfield et al., 2001

laws of “particle physics”

Hordijk, Crutchfield, and Mitchell, 1996: Models of CA computation in terms of particle kinematics

(Note “condensation phase”)

Particle model of CA computation

CAs evolvedfor density classification

CAs evolvedfor synchronization

generation 17 generation 18

How can we design computations in the language of patterns?

“Programming” the CA in the language of the high-level structures, and compiling the program to the level of the CA rule

Open problem!

Amorphous Computing

Abelson, Sussman et al., MIT

Amorphous Computing(Abelson et al., 2000; 2007)

• Main ideas:– Produce vast quantities of tiny, unreliable computing elements

(“particles”) at very low cost

– Give them limited wireless communication abilities so each particle can communicate with nearby particles

– Spray paint them onto a surface. Spatial arrangement, and thus communication, will be irregular. Processing and communication will be asynchronous.

– Have them self-organize into a reliable network that does something useful

Some possible applications

• Smart buildings that sense usage and adjust to save energy

• Smart bridges that monitor traffic load and structural integrity

• Smart arrays of microphones for opimizing acoustics

• Self-assembly of nano-machines

One example:Origami-based self-assembly

(R. Nagpal et al.)

• Set-up: – “Spray paint” thousands of MEMS “agents” on a 2D square

foldable material.

– Agents have no knowledge of global position or interconnect topology.

– All agents have identical program.

– Agents communicate locally (out to distance r)

– Agents run asynchronously

– Agents will collectively “fold” material to desired shape.

• High-level language: Origami Shape Language

– Six paper-folding axioms that can be used to construct a large class of flat folded shapes.

• Low-level language primitives:

– gradients

– neighborhood query

– cell-to-cell contact

– polarity inversion

– flexible folding

Folding a cup

Origami Shape Language

Primitives: points pi, lines Li, regions Ri

Axioms (Huzita):

1. Given two points p1 and p2, fold a line through them.

2. Given two points p1 and p2, fold p1 onto p2 (creates crease

that bisects the line p1p2 at right angles

3. Given two lines L1 and L2, fold L1 onto L2 (constructs the crease that bisects the angle between L1 and L2)

4. Given p1 and L1, fold L1 onto itself through p1 (constructs a crease through p1 perpendicular to L1).

5. Given p1 and p2 and L1, make a fold that places p1 on L1 and passes through p2 (constructs the tanget to the parabola (p1 L1) through p2)

6. Given p1 and p2 and lines L1 and L2, make a fold that places p1 on L1 and p2 on L2 (constructs the tanget to two parabolas)

Primitive operations we’ll need:

(create-region p1 L1)

(within-region r1 op1...): restricts operations to give region

(execute-fold L1 type landmark-point)

Fold types: Valley (apical), Mountain (basal) apical: puts apical surface on inside basal: puts basal surface on inside

Landmark-point: defines new “apical” and “basal” surfaces after fold by defining side of fold that will reverse polarity

(intersect L1 L1): returns a point that is the intersection of the two lines

Corner points: c1-c4Edge lines: e12-e41

Low-level agent operations(Nagpal, 2001)

How to implement OSL in low-level cell programsExample

I will put a link to all my slides on the CSSS wiki (under “Readings” “Melanie Mitchell”).

Thanks for listening!

cellular automata and amorphous computing melanie mitchell portland state university and santa fe...

Documents

print program

turing machine slide

computer program

halting problem slide

programming language

game of life slide

melanie mitchell slide

selfreproducing machine