CBSSS 6/25/02

Computing Beyond Silicon Summer School

Physics becomes thecomputer

Norm Margolus

CBSSS 6/25/02

Physics becomes the computerEmulating Physics

» Finite-state, locality, invertibility,and conservation laws

Physical Worlds» Incorporating comp-universality

at small and large scalesSpatial Computers

» Architectures and algorithms forlarge-scale spatial computations

Nature as Computer» Physical concepts enter CS and

computer concepts enter Physics

0/1

CBSSS 6/25/02

Review: Why emulate physics?

• Comp must adapt tomicroscopic physics

• Comp models may helpus understand nature

• Rich dynamics

• Started with locality(Cellular Automata).

CBSSS 6/25/02

Review: Conway’s “Life”• Captures physical locality and finite-

stateBut,• 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.

CBSSS 6/25/02

Review: CA’s with conservations

1. The data are rearranged withoutany interaction, or

2. The data are partitioned intodisjoint groups of bits that changeas a unit. Data that affect morethan one such group don’t change.

To make reversibility and other conservations manifest, we

employ a multi-step update, in each step of which either

Conservations allow computations to map efficiently onto

microscopic physics, and also allow them to have interesting

macroscopic behavior. Such CA’s have hardly been studied.

b c da c d ab

xx xhg

CBSSS 6/25/02

CBSSS 6/25/02

Physical WorldsSome regular spatial systems:

1. Programmable gate arrays atthe atomic scale

2. Fundamental finite-statemodels of physics

3. Rich “toy universes”

• All of these systems must becomputation universal

CBSSS 6/25/02

Computation UniversalityIf you can build basic logic elements and connect themtogether, then you can construct any logic function -- yoursystem 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

CBSSS 6/25/02

Computation UniversalityIf you can build basic logic elements and connect themtogether, then you can construct any logic function -- yoursystem 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

CBSSS 6/25/02

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 ofpatterns of bits in theLife CA

CBSSS 6/25/02

What’s wrong with Life?• One can build signals,

wires, and logic out ofpatterns of bits in theLife 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.

CBSSS 6/25/02

Billiard Ball Logic

• Simple reversible logic gatescan be universal

• Turn continuous model intodigital at discrete times!

• (A,B)→ AND(A,B) isn’treversible by itself

• Can do better than just throwaway extra outputs

• Need to also show that youcan compose gatesFredkin’s reversible

Billiard Ball Logic Gate

CBSSS 6/25/02

Billiard Ball Logic

Fixed mirrors allow signalsto be routed around.

Mirrors allow signals tocross without interaction.

CBSSS 6/25/02

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.

CBSSS 6/25/02

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.

CBSSS 6/25/02

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).

CBSSS 6/25/02

“Critters” is universal

A BBMCA collision:

Critters “glider” collision:

CBSSS 6/25/02

UCA with momentum conservation

Hard sphere collision

• Hard-sphere collisionconserves momentum

• Can’t make simple CAout of this that does

• Problem: finite impactparameter required

• Suggestion: find a newphysical model!

CBSSS 6/25/02

UCA with momentum conservation

Hard sphere collision Soft sphere collision

CBSSS 6/25/02

UCA with momentum conservation

Can shrink balls to points! Soft sphere collision

CBSSS 6/25/02

UCA with momentum conservation

SSM rule: rotations alsoact like this. All othercases remain unchanged.This is a Lattice Gas:movement and interactionsteps alternate.Can shrink balls to points!

CBSSS 6/25/02

UCA with momentum conservation

SSM rule: rotations alsoact like this. All othercases remain unchanged.This is a Lattice Gas:movement and interactionsteps alternate.Add mirrors at lattice

points to guide balls.

CBSSS 6/25/02

UCA with momentum conservation

Add mirrors at latticepoints to guide balls. SSM rule with mirrors

CBSSS 6/25/02

UCA with momentum conservation

Add mirrors at latticepoints to guide balls.

Mirrors allow signals tocross without interacting.

CBSSS 6/25/02

SSM collisions on other lattices

Triangular lattice 3D Cubic lattice

CBSSS 6/25/02

Getting rid of mirrors

Mirrors allow signals tocross without interacting.

• SSM with mirrors doesnot conservemomentum

• Mirrors must haveinfinite mass

• Want both universalityand mom conservation

• Can do this with just theSSM collision!

CBSSS 6/25/02

Getting rid of mirrors

Mirrors allow signals tocross without interacting.

Adding a rest particleallows signals to cross.

CBSSS 6/25/02

Getting rid of mirrors

Adding a rest particleallows signals to cross.

• The rule is very simplewithout mirrors: just onecollision and it’s inverse.

• All other cases, includingthe rest particle case, gostraight through.

CBSSS 6/25/02

Getting rid of mirrors

Pairing every signal with itscomplement allows constantstreams of 1’s to act like mirrors

• The rule is very simplewithout mirrors: just onecollision and it’s inverse.

• All other cases, includingthe rest particle case, gostraight through.

CBSSS 6/25/02

Getting rid of mirrors• Fredkin Gate,

built in SSM• No mirrors• Constants of 1

act as mirrors• Dual-rail pairs

used as signals• Can show that

1’s can be reusedby buildingBBMCA in SSM

CBSSS 6/25/02

Macroscopic universalityWith exact microscopic control of every bit, the SSM model lets us compute reversibly and withmomentum conservation, but

• an interesting world should have macroscopiccomplexity!

• Relativistic invariance would allow large-scalestructures 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.

CBSSS 6/25/02

Relativistic conservation⇐Non-relativistically,

mass and energy areconserved separately

⇐Simple lattice gassesthat conserve only mand mv are more like relthan non-rel systems!

€

E∑ = ′ E ∑Ei

r v i∑ = ′ E i ′ r v i∑

€

12 miv2∑ = 1

2 ′ m i∑ ′ v 2

mi∑ = ′ m i∑mi

r v i∑ = ′ m i ′ r v i∑

€

(since r p = γmr v = γmc2 × r v /c 2)

Non-relativistic:

Relativistic:

(energy)

(mass)

(mom)

(energy)

(mom)

CBSSS 6/25/02

Relativistic conservation

• We used dual-railsignalling to allowconstant 1’s to act asmirrors

• Dual rail signals don’trotate very easily

• Suggestion: make anLGA in which youdon’t need dual-rail

AA

B B

Dual-rail signals have a defectwhen it comes to allowing rotatedsignals to interact with each other.

CBSSS 6/25/02

Relativistic conservation

The rule we infer from this is:

CBSSS 6/25/02

Can we add macroscopic forces?

3D momentum conserving crystallization.

becomes:

Particles six sites apart alongthe lattice attract each other.

CBSSS 6/25/02

Can we add macroscopic forces?

Crystallization using irreversible forces (Jeff Yepez, AFOSR)

CBSSS 6/25/02

Summary• 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

CBSSS 6/25/02

Physics becomes the computerEmulating Physics

» Finite-state, locality, invertibility,and conservation laws

Physical Worlds» Incorporating comp-universality

at small and large scalesSpatial Computers

» Architectures and algorithms forlarge-scale spatial computations

Nature as Computer» Physical concepts enter CS and

computer concepts enter Physics

0/1

CBSSS 6/25/02

Problem from last lecture: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.

CBSSS 6/25/02

Problem from last lecture: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.

Problem:• Show that the waves

running along theboundary obey the waveequation exactly

Hint:• The wave equation’s

solutions consist of asuperposition of right- andleft-going waves