Embedding computations in tilings
(a perspective of the course)
Andrei Romashchenko
30 May 2016
1/8
What is a tile?
2/8
What is a tile?
In this mini-course:
Color: an element of a finite set C = {·, ·, ·, ·, ·, ·, ·}
2/8
What is a tile?
In this mini-course:
Color: an element of a finite set C = {·, ·, ·, ·, ·, ·, ·}
Wang Tile: a unit square with colored sides.
2/8
What is a tile?
In this mini-course:
Color: an element of a finite set C = {·, ·, ·, ·, ·, ·, ·}
Wang Tile: a unit square with colored sides.
i.e, an element of C 4 , e.g.,
2/8
What is a tile?
In this mini-course:
Color: an element of a finite set C = {·, ·, ·, ·, ·, ·, ·}
Wang Tile: a unit square with colored sides.
i.e, an element of C 4 , e.g.,
Tile set: a set τ ⊂ C 4
2/8
What is a tile?
In this mini-course:
Color: an element of a finite set C = {·, ·, ·, ·, ·, ·, ·}
Wang Tile: a unit square with colored sides.
i.e, an element of C 4 , e.g.,
Tile set: a set τ ⊂ C 4
Tiling: a mapping f : Z2 → τ
that respects the matching rules
2/8
A shift of finite type (SFT):
3/8
A shift of finite type (SFT):
I
a finite set of letters τ
3/8
A shift of finite type (SFT):
I
a finite set of letters τ
I
a finite set of forbidden (finite) patterns F
I
SFT: the set of all configurations f : Z2 → τ that does not contain
forbidden patterns
3/8
A shift of finite type (SFT):
I
a finite set of letters τ
I
a finite set of forbidden (finite) patterns F
I
SFT: the set of all configurations f : Z2 → τ that does not contain
forbidden patterns
Remark: for every set of Wang tiles τ the set of all τ -tilings is an SFT
3/8
τ -tiling:
a mapping f : Z2 → τ that respects the local rules.
4/8
τ -tiling:
a mapping f : Z2 → τ that respects the local rules.
T ∈ Z2 is a period if f (x + T ) = f (x) for all x.
4/8
super-classic facts:
5/8
super-classic facts:
I
SFT ∼ tilings
5/8
super-classic facts:
I
SFT ∼ tilings
I
if you can tile arbitrarily large square, than you can tile the infinite
plane (compacteness)
5/8
super-classic facts:
I
SFT ∼ tilings
I
if you can tile arbitrarily large square, than you can tile the infinite
plane (compacteness)
I
if there exists a τ -tiling with one period T , then there exists another
tiling with two non collinear periods T1 , T2
5/8
super-classic facts:
I
SFT ∼ tilings
I
if you can tile arbitrarily large square, than you can tile the infinite
plane (compacteness)
I
if there exists a τ -tiling with one period T , then there exists another
tiling with two non collinear periods T1 , T2
I
there exist tile sets τ s.t. all τ -tilings are aperiodic
5/8
super-classic facts:
I
SFT ∼ tilings
I
if you can tile arbitrarily large square, than you can tile the infinite
plane (compacteness)
I
if there exists a τ -tiling with one period T , then there exists another
tiling with two non collinear periods T1 , T2
I
there exist tile sets τ s.t. all τ -tilings are aperiodic
I
there exists a tile set τ s.t. all τ -tilings are non-computable
5/8
super-classic facts:
I
SFT ∼ tilings
I
if you can tile arbitrarily large square, than you can tile the infinite
plane (compacteness)
I
if there exists a τ -tiling with one period T , then there exists another
tiling with two non collinear periods T1 , T2
I
there exist tile sets τ s.t. all τ -tilings are aperiodic
I
there exists a tile set τ s.t. all τ -tilings are non-computable
I
given a tile set τ we cannot algorithmically decide whether there
exists a τ -tiling of Z2
5/8
super-classic facts:
I
SFT ∼ tilings
I
if you can tile arbitrarily large square, than you can tile the infinite
plane (compacteness)
I
if there exists a τ -tiling with one period T , then there exists another
tiling with two non collinear periods T1 , T2
I
there exist tile sets τ s.t. all τ -tilings are aperiodic
I
there exists a tile set τ s.t. all τ -tilings are non-computable
I
given a tile set τ we cannot algorithmically decide whether there
exists a τ -tiling of Z2
other super-classic facts:
5/8
super-classic facts:
I
SFT ∼ tilings
I
if you can tile arbitrarily large square, than you can tile the infinite
plane (compacteness)
I
if there exists a τ -tiling with one period T , then there exists another
tiling with two non collinear periods T1 , T2
I
there exist tile sets τ s.t. all τ -tilings are aperiodic
I
there exists a tile set τ s.t. all τ -tilings are non-computable
I
given a tile set τ we cannot algorithmically decide whether there
exists a τ -tiling of Z2
other super-classic facts:
I
in any reasonable programming language you can write a program π
that prints its own text
5/8
super-classic facts:
I
SFT ∼ tilings
I
if you can tile arbitrarily large square, than you can tile the infinite
plane (compacteness)
I
if there exists a τ -tiling with one period T , then there exists another
tiling with two non collinear periods T1 , T2
I
there exist tile sets τ s.t. all τ -tilings are aperiodic
I
there exists a tile set τ s.t. all τ -tilings are non-computable
I
given a tile set τ we cannot algorithmically decide whether there
exists a τ -tiling of Z2
other super-classic facts:
I
in any reasonable programming language you can write a program π
that prints its own text
I
in any reasonable programming language you may assume that your
program has an access to its own text
5/8
super-classic facts:
I
SFT ∼ tilings
I
if you can tile arbitrarily large square, than you can tile the infinite
plane (compacteness)
I
if there exists a τ -tiling with one period T , then there exists another
tiling with two non collinear periods T1 , T2
I
there exist tile sets τ s.t. all τ -tilings are aperiodic
I
there exists a tile set τ s.t. all τ -tilings are non-computable
I
given a tile set τ we cannot algorithmically decide whether there
exists a τ -tiling of Z2
other super-classic facts:
I
in any reasonable programming language you can write a program π
that prints its own text
I
in any reasonable programming language you may assume that your
program has an access to its own text
I
any effective (polynomial time) real-life algorithm can be performed
by a Turing machine in polynomial time
5/8
This mini-course:
6/8
This mini-course:
Two techniques
6/8
This mini-course:
Two techniques of embedding a computation in a tiling
I
from self-referential programs to self-similar tilings
6/8
This mini-course:
Two techniques of embedding a computation in a tiling
I
from self-referential programs to self-similar tilings
[goes back to J. von Neumann]
6/8
This mini-course:
Two techniques of embedding a computation in a tiling
I
from self-referential programs to self-similar tilings
[goes back to J. von Neumann]
I
from arithmetic in Sturmian numeration system to tilings
6/8
This mini-course:
Two techniques of embedding a computation in a tiling
I
from self-referential programs to self-similar tilings
[goes back to J. von Neumann]
I
from arithmetic in Sturmian numeration system to tilings
[J. Kari]
6/8
This mini-course:
Two techniques of embedding a computation in a tiling
I
from self-referential programs to self-similar tilings
[goes back to J. von Neumann]
I
from arithmetic in Sturmian numeration system to tilings
[J. Kari]
Very standard application:
I
a construction of an aperiodic tile set
Less standard application:
I
aperiodicity + quasiperiodicity
6/8
This mini-course:
Two techniques of embedding a computation in a tiling
I
from self-referential programs to self-similar tilings
[goes back to J. von Neumann]
I
from arithmetic in Sturmian numeration system to tilings
[J. Kari]
Very standard application:
I
a construction of an aperiodic tile set
Less standard application:
I
aperiodicity + quasiperiodicity (and even minimality)
6/8
Possible topics of this min-course
7/8
Possible topics of this min-course
Some applications of the self-simulating tilings:
7/8
Possible topics of this min-course
Some applications of the self-simulating tilings:
I
the tiling problem is undecidable [Berger 1966]
I
a tile set with only non computable tilings [Hanf & Myers 1974]
I
a tile set with highly aperiodic tilings [?]
I
robust (error-correcting) tilings [?]
7/8
Possible topics of this min-course
Some applications of the self-simulating tilings:
I
the tiling problem is undecidable [Berger 1966]
I
a tile set with only non computable tilings [Hanf & Myers 1974]
I
a tile set with highly aperiodic tilings [?]
I
robust (error-correcting) tilings [?]
I
an effective shift is isomorphic to a subaction of a sofic shift
[Hochman 2009, Aubrun & Sablik 2013]
I
a minimal effective shift can be simulated by a minimal SFT [?]
7/8
Possible topics of this min-course
Some applications of the self-simulating tilings:
I
the tiling problem is undecidable [Berger 1966]
I
a tile set with only non computable tilings [Hanf & Myers 1974]
I
a tile set with highly aperiodic tilings [?]
I
robust (error-correcting) tilings [?]
I
an effective shift is isomorphic to a subaction of a sofic shift
[Hochman 2009, Aubrun & Sablik 2013]
I
a minimal effective shift can be simulated by a minimal SFT [?]
Another remarkable result:
I
Kari’s technique gives non self-similar tilings [T. Monteil]
7/8
Possible topics of this min-course
Some applications of the self-simulating tilings:
I
the tiling problem is undecidable [Berger 1966]
I
a tile set with only non computable tilings [Hanf & Myers 1974]
I
a tile set with highly aperiodic tilings [?]
I
robust (error-correcting) tilings [?]
I
an effective shift is isomorphic to a subaction of a sofic shift
[Hochman 2009, Aubrun & Sablik 2013]
I
a minimal effective shift can be simulated by a minimal SFT [?]
Another remarkable result:
I
Kari’s technique gives non self-similar tilings [T. Monteil]
7/8
Come to the lectures!
8/8