How to obtain a recursively enumerable subshift
with a bidimensional SFT
(ANR Subtile meeting, LAMFA Amiens)
N. Aubrun and M. Sablik
IGM, University Paris-Est and LATP, University of Provence
October 22, 2009
N. Aubrun and M. Sablik ()
How to obtain a RE subshift ?
October 22, 2009
1 / 26
Outline
1
Subshifts and tilings
Denitions
Four classes of operations on subshifts
2
The main result
A simulation result
Sketch of the proof
3
Computation zones for Turing machines
Adding Turing machine calculations in computation zones
4
Turing machines calculations
The machines MF and MSearch
5
The nal construction
N. Aubrun and M. Sablik ()
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Subshifts and classes of subshifts
Zd , A nite alphabet
F a set of d -dimensional nite patterns
conguration c ∈ AZ
subshift dened by the set of forbidden patterns F :
d
XF = {c ∈ AZd /∀p ∈ F , p does not appear in c } ⊆ AZd
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Subshifts and classes of subshifts
Zd , A nite alphabet
F a set of d -dimensional nite patterns
conguration c ∈ AZ
subshift dened by the set of forbidden patterns F :
d
XF = {c ∈ AZd /∀p ∈ F , p does not appear in c } ⊆ AZd
Σ fullshift ⇔ Σ =
XF , F = ∅
Σ subshift of nite type (SFT) ⇔ ∃F nite set of patterns, Σ =
XF
tilings ≈ 2-dimensional SFT
Σ recursively enumerable (RE) subshift ⇔ ∃F RE set of patterns, Σ = XF .
N. Aubrun and M. Sablik ()
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An example of 2-dimensional SFT
F=
N. Aubrun and M. Sablik ()
ß
™
,
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An example of 2-dimensional SFT
F=
ß
™
,
⇓
..
.
..
.
XF :
... , ...
...
..
.
N. Aubrun and M. Sablik ()
...
..
.
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Operations on subshifts, simulation
S = set of all subshifts (all dimensions and alphabets)
Operation on subshifts:
I
I
op : S → S
or op : S × S → S
Closure of a class of subshifts U under a set of operations Op :
C lOp (U) = smallest set stable by Op which contains U.
T simulates T0 by Op if T0 ∈ C lOp (T) (denoted by T0 ≤Op T).
C lOp (T) = {T0 : T0 ≤Op T}.
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Product operation P
Denition
If Σ = XF ⊆ AZ and Σ0 = XF 0 ⊆ BZ are two subshifts with the same
dimension:
d
d
d
Σ × Σ0 = XF ×F 0 ⊆ (A × B)Z .
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Product operation P
Denition
If Σ = XF ⊆ AZ and Σ0 = XF 0 ⊆ BZ are two subshifts with the same
dimension:
d
d
d
Σ × Σ0 = XF ×F 0 ⊆ (A × B)Z .
Σ = XF ⊆ {a, b , c }Z
2
F=
ß
x
a
a x
,
y
b
Σ0 = XF 0 ⊆
F0 =
b x
,
y
c
,
ß
N. Aubrun and M. Sablik ()
Z2
™
,






™ 


c





y



















Σ × Σ0 ⊆ {a, b , c , a, b , c }Z
2
⇒
F × F0 =
ß
x
a
How to obtain a RE subshift ?
a x
,
y
a
a x
,
y
a
a
,...
y
October 22, 2009
™
5 / 26
Product operation P
Denition
If Σ = XF ⊆ AZ and Σ0 = XF 0 ⊆ BZ are two subshifts with the same
dimension:
d
d
d
Σ × Σ0 = XF ×F 0 ⊆ (A × B)Z .
Σ = XF ⊆ {a, b , c }Z
2
F=
ß
x
a
a x
,
y
b
Σ0 = XF 0 ⊆
F0 =
b x
,
y
c
,
ß
Remark :
N. Aubrun and M. Sablik ()
Z2
™
,






™ 


c





y



















Σ × Σ0 ⊆ {a, b , c , a, b , c }Z
2
⇒
F × F0 =
ß
C lP (SFT ) = SFT
x
a
How to obtain a RE subshift ?
a x
,
y
a
a x
,
y
a
a
,...
y
October 22, 2009
™
5 / 26
Finite type operation FT
Denition
If Σ = XF ⊆ AZ is a subshift and F 0 a nite set of patterns :
d
φFT (F 0 , Σ) = XF ∪F 0 .
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Finite type operation FT
Denition
If Σ = XF ⊆ AZ is a subshift and F 0 a nite set of patterns :
d
φFT (F 0 , Σ) = XF ∪F 0 .
Remark :
N. Aubrun and M. Sablik ()
C lFT (FS) = SFT
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Factor operation F
Denition
If Σ ⊆ AZ is a subshift and Π : AZ → BZ a morphism (continue and
d
σ ◦ Π = Π ◦ σ ), Π(Σ) ⊆ B Z is a factor of Σ.
d
N. Aubrun and M. Sablik ()
d
d
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Factor operation F
Denition
If Σ ⊆ AZ is a subshift and Π : AZ → BZ a morphism (continue and
d
σ ◦ Π = Π ◦ σ ), Π(Σ) ⊆ B Z is a factor of Σ.
d
d
d
Example :
A = {0, 1, 2} and Σ ⊆ AZ
Σ = X{00,22,01,12} and Π(0) = Π(2) = 0,Π(1) = 1
⇒ Π(Σ) = {x ∈ {0, 1, 2}Z / blocks of consecutive 0 are of even length}
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Factor operation F
Denition
If Σ ⊆ AZ is a subshift and Π : AZ → BZ a morphism (continue and
d
σ ◦ Π = Π ◦ σ ), Π(Σ) ⊆ B Z is a factor of Σ.
d
d
d
Example :
A = {0, 1, 2} and Σ ⊆ AZ
Σ = X{00,22,01,12} and Π(0) = Π(2) = 0,Π(1) = 1
⇒ Π(Σ) = {x ∈ {0, 1, 2}Z / blocks of consecutive 0 are of even length}
Remark : SFT ( C lF (SFT ). C lF (SFT ) is the class of soc subshifts. In
dimension 1, soc subshifts are XL where L is regular.
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Subaction operation SA
Denition
G a subgroupe of Zd generated u1, u2, . . . , ud (d ≤ d ). If Σ ⊆ AZd is a subshift:
n
d
φSA (G, Σ) = y ∈ AZ : ∃x ∈ Σ tel que ∀i1 , . . . , id ∈ Zd ,
0
0
0
0
0
yi1 ,...,id 0 = xi1 u1 +···+id 0 ud 0 .
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Example of sub-action
F=
ß
™
,
,
,
,...,
G = {(i , i ) ∈ Z2, i ∈ Z}
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Example of sub-action
F=
ß
™
,
,
,
,...,
G = {(i , i ) ∈ Z2, i ∈ Z}
G
φSA ( , XF ) = X¶
N. Aubrun and M. Sablik ()
,
How to obtain a RE subshift ?
i
,i =0...2
©
October 22, 2009
9 / 26
Some simulation results
SFT ( C lSA (SFT )
C lSA (SFT ) =??
C lSA (RE ) = RE
C lF ,SA (SFT ) = RE (Hochman)
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Outline
1
Subshifts and tilings
Denitions
Four classes of operations on subshifts
2
The main result
A simulation result
Sketch of the proof
3
Computation zones for Turing machines
Adding Turing machine calculations in computation zones
4
Turing machines calculations
The machines MF and MSearch
5
The nal construction
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The main result
Theorem (Hochman)
Any RE subshift of dimension d can be obtain with factor and subaction
operations from an SFT of dimension d + 2.
C lF ,SA (SFT ∩ Sd +2 ) ∩ S≤d = RE ∩ S≤d
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The main result
Theorem (Hochman)
Any RE subshift of dimension d can be obtain with factor and subaction
operations from an SFT of dimension d + 2.
C lF ,SA (SFT ∩ Sd +2 ) ∩ S≤d = RE ∩ S≤d
Theorem (A&S)
Any RE subshift of dimension d can be obtain with factor and subaction
operations from an SFT of dimension d+1.
C lF ,SA (SFT ∩ Sd +1 ) ∩ S≤d = RE ∩ S≤d
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Some elements of the proof
We want to simulate a RE subshift Σ ⊆ AΣ Z with a 2-dimensional SFT
(particular case d = 1, but easy to generalize).
We need two Turing machines:
MF : enumerates forbidden patterns for Σ and chacks the validity of Σ
MSearch : explore the bits of AΣ Z
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Coding a Turing machine with a SFT
Denition: Turing machine
A Turing machine is M = (Q , Γ, ], q0 , δ, QF ) where:
Q is a nite set of states; q0 ∈ Q is the initial state;
Γ is a nite alphabet;
]∈
/ Γ is the blank symbol
δ : Q × Γ → Q × Γ × {←, · , →} is the transition function;
F ⊂ QF is the set of nal states.
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Coding a Turing machine with a SFT
Denition: Turing machine
A Turing machine is M = (Q , Γ, ], q0 , δ, QF ) where:
Q is a nite set of states; q0 ∈ Q is the initial state;
Γ is a nite alphabet;
]∈
/ Γ is the blank symbol
δ : Q × Γ → Q × Γ × {←, · , →} is the transition function;
F ⊂ QF is the set of nal states.
The rule δ(q1 , x ) = (q2 , y , ←) will be coded by:
(q2 , z )
z
⇒ subshift of nite type
N. Aubrun and M. Sablik ()
y
z0
(q1 , x ) z 0
TM for a Turing machine M
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Coding a Turing machine with a SFT
Denition: Turing machine
A Turing machine is M = (Q , Γ, ], q0 , δ, QF ) where:
Q is a nite set of states; q0 ∈ Q is the initial state;
Γ is a nite alphabet;
]∈
/ Γ is the blank symbol
δ : Q × Γ → Q × Γ × {←, · , →} is the transition function;
F ⊂ QF is the set of nal states.
The rule δ(q1 , x ) = (q2 , y , ←) will be coded by:
(q2 , z )
z
y
z0
(q1 , x ) z 0
TM for a Turing machine M
Problem : beginning of the calculation, multiple heads, . . .
⇒ subshift of nite type
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Dierent levels of the construction
We construct a 2-dimensional soc subshift:
level 1: some x ∈ AΣZ (a potential x ∈ Σ),
level 2: computation zones, used by machines MF and MSearch,
level 3: Turing machines MF , and communication with MSearch,
level 4: Turing machines MSearch and internal communication between them.
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Level 1: a potential x ∈ Σ
fullshift AΣ Z
nite type condition y(i ,j ) = y(i ,j +1)
same conguration x ∈ AΣ Z on every row
2
rst level T1 = φFT
Åß
a
, a 6= b ∈ AΣ
b
™
,
AΣ Z
2
ã
⇒ We want to exclude forbidden patterns of Σ.
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Outline
1
Subshifts and tilings
Denitions
Four classes of operations on subshifts
2
The main result
A simulation result
Sketch of the proof
3
Computation zones for Turing machines
Adding Turing machine calculations in computation zones
4
Turing machines calculations
The machines MF and MSearch
5
The nal construction
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Initialization of a TM calculation
If a particular tile appears once, it can be used to initialize a calculation:
But it is impossible to force the presence of one copy of a particular tile with a
subshift (compacity).
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Substitutive soc subshifts
a substitution is a function s : A → AUk where Uk = [1; k ] × [1; k ]
a s-pattern is a pattern obtained by iteration of the substitution s on a letter
the subshift generated by a substitution s is the set of congurations x such
that any pattern that appears in x also appears in a s -pattern
Theorem (Mozes)
If a substitution satises some good properties, then the subshift it generates is a
soc subshift.
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The substitution ˜s
Mozes theorem ⇒ soc subshift TSquares
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The substitutive subshift TSquares
subshift TSquares :
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The substitutive subshift TSquares
subshift TSquares :
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The substitutive subshift TSquares
subshift TSquares :
: computation tile (Turing machine calculation)
: obstruction tile (communication inside a same zone)
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Computation and obstruction tiles
a
a
a
a
a
a
a
(q0 , ])
a
a
a
(q0 , ])
(q1 , ])
]
(q0 , ])
(q0 , ])
(q1 , ])
]
(q0 , ])
N. Aubrun and M. Sablik ()
a
a
a
a
a
b
b
b
b
(q1 , ])
(q1 , ])
(q1 , ])
]
k
k
k
(q2 , ])
]
]
]
]
k
a
(q0 , ])
(q2 , ])
]
a
(q0 , ])
]
How to obtain a RE subshift ?
k
(q1 , ])
]
(q2 , ])
]
(q1 , ])
]
]
(q3 , ])
(q3 , ])
(q3 , ])
]
]
]
]
]
October 22, 2009
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The computation zones: stripes
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Using computation zones (stripes)
condition Clock : a clock is increased at each calculation step, and clock≥ 0
condition Init : the entry word where clock= 0 is the empty word, and the
tape is initially lled with ] symbol;
condition Head : the initial state q0 appears where clock= 0;
condition Stop : when a side of a stripe is reached by the head of the
machine, the calculation stops and the tape content is just copied out (clock
is no longer increased);
condition Final : when a nal state is reached, the tape content is just
copied out for next steps of calculation (clock is no longer increased);
condition Transfer : when the head reaches an obstruction zone, the
information is transferred (horizontally or vertically) to the next computation
tile.
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Using computation zones (stripes)
condition Clock : a clock is increased at each calculation step, and clock≥ 0
condition Init : the entry word where clock= 0 is the empty word, and the
tape is initially lled with ] symbol;
condition Head : the initial state q0 appears where clock= 0;
condition Stop : when a side of a stripe is reached by the head of the
machine, the calculation stops and the tape content is just copied out (clock
is no longer increased);
condition Final : when a nal state is reached, the tape content is just
copied out for next steps of calculation (clock is no longer increased);
condition Transfer : when the head reaches an obstruction zone, the
information is transferred (horizontally or vertically) to the next computation
tile.
Dene the SFT TM , zones for a Turing machine M:
TM,zones = φFT ({Clock, Head, Init, Stop, Final, Transfer}, φP (TSquares, TM))
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Outline
1
Subshifts and tilings
Denitions
Four classes of operations on subshifts
2
The main result
A simulation result
Sketch of the proof
3
Computation zones for Turing machines
Adding Turing machine calculations in computation zones
4
Turing machines calculations
The machines MF and MSearch
5
The nal construction
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Level 3: Enumeration of forbidden patterns
MF generates forbidden patterns for Σ
MF is given a responsibility zone, and ensure no forbidden pattern appear in
it
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Level 3: Enumeration of forbidden patterns
MF generates forbidden patterns for Σ
MF is given a responsibility zone, and ensure no forbidden pattern appear in
it
each time a forbidden pattern is produced, MF checks it does not appear:
Responsability zone of MF
z
}|
a0
a1
f0
f1
f0
a2
f2
f1
f0
...
{
...
...
...
aN
...
f2
f1
...
f2
...
MF asks MSearch for ak (gives the adress k and waits for the answer)
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Responsibility zone of a machine MF
Required: responsability zones ⇒ the entire conguration is scanned
Responsibility zones must overlap:
. . . x−2 x−1 x0 ∈ L(Σ) and x0 x1 x2 · · · ∈ L(Σ) does not imply
. . . x−2 x−1 x0 x1 x2 · · · ∈ L(Σ)
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Level 4: Scan of the entire responsibility zone
MF calls for MSearch to acceed its responsibility zone
MSearch of level n needs the help of MSearch of level n − 1
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Level 4: Scan of the entire responsibility zone
MF calls for MSearch to acceed its responsibility zone
MSearch of level n needs the help of MSearch of level n − 1
MSearch shares its calculation time:
I
answer to
I
help the higher level
MF
N. Aubrun and M. Sablik ()
(with the help of
MSearch
of lower levels)
MSearch
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Outline
1
Subshifts and tilings
Denitions
Four classes of operations on subshifts
2
The main result
A simulation result
Sketch of the proof
3
Computation zones for Turing machines
Adding Turing machine calculations in computation zones
4
Turing machines calculations
The machines MF and MSearch
5
The nal construction
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The nal construction
condition Request : MF asks MSearch for a letter in its responsibility zone
and waits for the answer
condition Communication : internal communication between the MSearch
machines of dierent levels
Tnal = φFT Request, Communication, φP TMF ,zones, TM
zones
Search ,
To obtain Σ :
F operation to only keep letters from AΣ
SA operation to project on the horizontal direction
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