Notes on local determinism

Days 12, 13 and 14 of Comp Sci 480

Motivation

• When does a tile set work correctly?

• Can you prove this?

• It would be nice to have a set of sufficient conditions, say X, Y, Z, etc., such that, if your tile set satisfied conditions X, Y, Z, etc., then the tile set is correct

Correctness

• What does it mean for a tile set to be “correct”?

• The tile set should uniquely produce something– The sequence of tile additions might be

different each time (or there might be a large finite number of such sequences)

– But the end result is always the same

First attempt

• What if a tile set is very strongly deterministic, in the following sense…– sequence of tile additions is unique

• Such a tile set always does the same thing• It definitely uniquely produces some

assembly• “Unique assembly sequence” is a very

strong sufficient condition• Perhaps we can relax this condition…

Second attempt

• What if a tile set has the property that every tile binds “deterministically”

• E.g., if tile A binds to tile B (on some side) and at some location, then no other tile type can bind to B on the same side

• Does this tile set always do the same thing?

• No!

54

Counter example

S 1

A

B

C

DC DD

3

43

2

A

B

A

B

Every tile binds “deterministically”, but the tile system does not uniquely produce the loop.

21

21

21

3

2

3

2

A

B

A

B

A

B

A

B

B

C

B

C

3

4

3

4

DC

??

54

54

DC

Temperature τ = 1

Local determinism

• A special property of a particular assembly sequence (i.e., sequence of tile additions) for a tile set

• Key fact: if a tile set has a locally deterministic assembly sequence, then EVERY assembly sequence for that same tile set is locally deterministic– Proven by Soloveichik and Winfree (2004)

• Local determinism of an assembly sequence (for an assembly) guarantees the unique production of that assembly

Some notation

• From here on out, assume a = (a0,a1,…) is an assembly sequence that leads to a final (not necessarily terminal) assembly A.

• Sides of tiles are denoted as d, with possible values North, South, East and West.

• For a tile type t, – strt(d) is the strength of the glue on side d of tile type t

– labelt(d) is the label of the glue on side d of tile type t

Input, output and terminal sides

• Let (x,y) be an arbitrary point at which a tile is placed in assembly A.

• INa(x,y) = { the set of sides, on which the tile placed at position (x,y), in a, initially binds }– This is the set of input sides for the tile A(x,y) in assembly sequence a.

• OUTa(x,y) = { the set of sides that other tiles use to initially bind to the tile placed at position (x,y) in a }– This is the set of “propagating output” sides for the tile A(x,y) in

assembly sequence a.

• TERMa(x,y) = { North, South, East, West } - INa(x,y) - OUTa(x,y)– Sides of the tile A(x,y) that are neither input nor output sides.

A

B

X

IN/OUT/TERM Example #1

S 1

A

Temperature τ = 2

B

C

A

B

X

X2

D

X C

D

1

2

Note that, IN, OUT and TERM sides are properties of an assembly sequence – NOT the tiles themselves…

A

B

X

B

C C

D

1

2

X2

D

X

X2

D

X C

D

B

C 1

2

IN = { North, South, West }OUT = { }TERM = { East }

IN/OUT/TERM Example #2

S 1

A

Temperature τ = 2

B

C

A

B

X

X2

D

X C

D

1

2

A

B

X

B

C C

D

1

2

X2

D

X

A

B

X

X2

D

X C

D

B

C 1

2

1

2

IN = { South, West }OUT = { North }TERM = { East }

Local determinism definition

• Let a = (a0, a1, …) be an assembly sequence in a tile set T, which leads to a final structure called A,

• For every point (x,y) that receives a tile in A, let t = A(x,y)

• We say that a is locally deterministic if the following conditions are true:– For all d ϵ INa(x,y), Σd strt(d) = τ

• When a new tile binds, it binds “just barely”

– If t’ ϵ T, with t ≠ t’, then for all d ϵ INa(x,y) U TERMa(x,y) with labelt’(d) = labelt(d), Σd strt’(d) < τ

• Nothing can grow from non-propagating sides except “as desired”.

– The final assembly A is terminal

• We say that a tile set T is locally deterministic if there exists a locally deterministic assembly sequence in it.

Local determinism example

S 1

A

Temperature τ = 2

B

C

A

B

X2

D

X C

D

1

2

A

B

B

C

B

C1

2

1

2

C

D

C

D

X2

D

X

X2

D

X

A

B

Local determinism example

S 1

A

Temperature τ = 2

B

C

A

B

X2

D

X C

D

1

2

A

B

B

C

1

2

C

D

X2

D

X

Final assembly is terminal

Every tile binds (initially) with exactly strength 2

Need to verify deterministic binding via IN and TERM sides…

…All good!

???

??? ???

???

???

Local determinism example?

S 1

A

Temperature τ = 2

B

C

X2

D

X C

D

1

2

B

C

B

C1

2

1

2

C

D

C

D

X2

D

X

X2

D

X

A

B

A

B

X

A

B

A

B

X

A

B

A

B

X

Final assembly is terminal

Every tile binds deterministically via IN and TERM sides

This tile type initially binds with strength 3 > 2…

… NOT locally deterministic

3

2

Local determinism example?

S 1

A

B

C

3

43

2It’s clear that this tile set produces one and only assembly – is it locally deterministic?

21

21

3

2

A

B

B

B

A

B

B

B

B

C

3

4Temperature τ = 1

??4

C

??4

C 21

3

4

??4

C

B

C

A

B

B

B

Is this tile set locally deterministic?

Temperature τ = 1

NO!

Some tiles will bind with > strength 1

Local determinism theorem

Theorem (Soloveichik and Winfree, 2004). If there exists a locally deterministic assembly sequence a of T, then T uniquely produces an assembly.

Just exhibit one locally deterministic assembly sequence of T.

Reminder (key fact): if one assembly sequence is locally deterministic, then they ALL are (for a given tile set).

Not locally deterministic

• Prove or disprove: if a tile set is not locally deterministic, then it does not uniquely produce an assembly

• Disprove: the previous “loop” example is a good example of a tile set that is NOT locally deterministic but uniquely builds an assembly

Example

• Prove/disprove: if one assembly sequence of a tile set is NOT locally deterministic, then the tile set is not locally deterministic

• Prove: – Proof by contradiction, so assume that there is a tile

set that is locally deterministic but has one assembly sequence that is not locally deterministic

– If the tile set is locally deterministic, then we know every assembly sequence is locally deterministic

– This contradicts our assumption that one assembly sequence is not locally deterministic

– Done

Example

• If a tile set has exactly two assembly sequences and one is locally deterministic, prove that the other is locally deterministic as well.

• DO NOT use that “key fact”, which says that if one assembly sequence is locally deterministic, then they all are.

• This is a good (fun) homework problem!

Example• Recall: an assembly sequence a is locally deterministic if the following

conditions are true:– For all d ϵ INa(x,y), Σd strt(d) = τ– If t’ ϵ T, with t ≠ t’, then for all d ϵ INa(x,y) U TERMa(x,y) with labelt’(d) =

labelt(d), Σd strt(d) < τ– The final assembly A is terminal

• What if we change the second condition to this:– If t’ ϵ T, with t ≠ t’, then for all d ϵ INa(x,y) with labelt’(d) = labelt(d), Σd

strt(d) < τ• Deleted the “U TERMa(x,y)” part

– Call this modified version of local determinism LD1 (the original is LD0)

– Does the local determinism theorem still hold for LD1? That is, if a tile set has a locally deterministic assembly sequence, for LD1, does it uniquely produce an assembly? Prove or disprove!

– DISPROVE!!

Disproof of previous claim

S 1

A

Temperature τ = 2

B

C

Y2

X

C

D

1

2

B

C

B

C1

2

1

2

C

D

C

D

A

B

A

B

X

A

B

A

B

X

X2

D

X2

D

X2

D

A

B

A

B

X

Disproof of previous claim

S 1

A

Temperature τ = 2

B

C

Y2

X

C

D

1

2

B

C

1

2

A

B

A

B

XRemove X and its immediate OUT-neighbors (there aren’t any)

No other tile type can bind via just IN sides of X…

…can another tile type bind in place of X in some other assembly sequence?

YES!

X2

D

X2

D

A

B

A

B

X

X2

D

Y2

X

Y2

X

C

D

“LD1 does not imply LD0”

Local determinism summary

• All (most?) of our tile sets we have seen thus far are locally deterministic

• Local determinism: sufficient conditions for unique assembly– NOT necessary conditions, i.e., a tile set can

uniquely produce an assembly and not be locally deterministic

The power of non-determinism in self-assembly…

Finite shapes

• Does it help to purposely use non-determinism in the design of tile sets?

• That is, is there a shape, say X, that can be uniquely assembled with c tile types non-deterministically…– but any deterministic tile set that uniquely

builds the shape X must use > c unique tile types???

• Answer: YES!– Bryans, et. al., prove this in 2013

Infinite shapes

• What about infinite shapes?– Here, tile complexity doesn’t matter (can’t be

a function of the size of the shape)

• Is there an infinite shape X that can be uniquely assembled using a non-deterministic tile set…– but not by any deterministic tile set???

• Answer: YES!– Bryans, et. al., proved this in 2013