Quantifier Extensions of Multidimensional Sofic Shifts

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by  Ilkka Törmä

PROCEEDINGSOFTHE
AMERICANMATHEMATICALSOCIETY
Volume00,Number0,Pages000–000
S0002-9939(XX)0000-0
QUANTIFIER EXTENSIONS OF MULTIDIMENSIONAL SOFIC
SHIFTS
4 ILKKATO¨RMA¨
1
0
2 (Communicated byNimishShah)

l
u
Abstract. Wedefineapairofsimplecombinatorialoperations onsubshifts,
J
called existential and universal extensions, and study their basic properties.

3 Weprovethattheexistentialextensionofasoficshiftbyanothersoficshiftis
2 alwayssofic,andthesameholdsfortheuniversalextensioninonedimension.
However, we also show by a construction that universal extensions of two-

] dimensional soficshifts maynotbesofic, evenifthe subshiftweextend byis
S
verysimple.
D
.
h
t
a
m
1. Introduction
[

Subshifts of finite type and sofic shifts are, in some sense, the simplest objects
2
v studied in symbolic dynamics. Subshifts of finite type are symbolic systems whose
4 structureisdefinedbylocalinteractions,andsoficshiftsareobtainedfromthemby
9 forgettingsome of their structure. One-dimensionalSFTs andsofic shifts arequite
2
well-behavedandtheirtheoryiswellunderstood,althoughsomefundamentalopen
2
problems still remain. For a good overview of the theory, see [6]. In particular,
.
1 since one-dimensional sofic shifts correspond to regular languages, most natural
0
(and some unnatural) operations respect the property of being sofic.
4
The higher-dimensional case is much more complex, and the language of a two-
1
: dimensionalSFTcanalreadybeuncomputable. Infact,eventheemptinessproblem
v
oftwo-dimensionalSFTsisundecidable[2]. Theclassoftwo-dimensionalsoficshifts
i
X is badly understood, in the sense that there are only a couple ofgeneralconditions
r for showingthat a particular subshift is not sofic,and not many morespecific ones
a
[3, 4, 5, 10]. In this article, we study a pair of particular operations on subshifts,
calledthequantifierextensions,which‘extend’agivensubshiftbyanothersubshift.
They both respect the property of being a sofic shift in one dimension, and are
inspired by the concept of multi-choice shift spaces, as defined in [7] and studied
further in [8]. The main theorem shows that one of the operations does not take
sofic shifts to sofic shifts in the two-dimensional setting, even when extending by
very simple subshifts. This also solves an open problem presented in [7].
ReceivedbytheeditorsJuly24,2014.
2010 Mathematics Subject Classification. Primary37B50.
ResearchsupportedbytheAcademyofFinlandGrant131558.
(cid:13)cXXXXAmericanMathematicalSociety
1
2 ILKKATO¨RMA¨
2. Definitions
Let S be a finite set, called the alphabet, and fix a dimension d 1. The set of
≥
finite words over S is denoted S∗, and the length of a word w S∗ by w. A d-
dimensional pattern over S is a function P : D S, where D ∈Zd is th|e d|omain
→ ⊂
of P, denoted D(P). The set of finite d-dimensional patterns over S is denoted
(S). A full pattern x : Zd S is called a configuration. A pattern P : D S
d
Poccurs inanotherpatternQ:→E S,denotedP ⊏Q,if~v+w~ E andQ =→P
~v+w~ ~v
hold for some w~ Zd and all ~v →D. ∈
∈ ∈
Each set of finite patterns F (S) defines a subshift as the set of configura-
d
tions X = x SZd P F :⊂PP⊏ x . We say F is a set of forbidden patterns
{ ∈ | ∀ ∈ 6 }
for X. If F is finite, X is a subshift of finite type, SFT for short, and if every
patternofF hasdomain ~0,~e forsomei [1,d],where ~e ,...,~e isthenatural
i 1 d
generating set of Zd, X i{s a ti}ling system.∈The set P { (S) x} X : P ⊏ x
d
is called the language of X, and is denoted (X).{Fo∈r PD Z|d∃, w∈e also denot}e
(X) = (X) SD, and if d = 1 and n BN, we denote ⊂ (X) = (X) Sn.
D n
AB subshiftBX ∩SZd is strongly irreducible ∈(with constant MB> 0) if fBor any∩two
domains D,D′⊂ Zd such that min ~n m~ ~n D,m~ D′ M where
⊂ {k − k | ∈ ∈ } ≥ k·k
denotes the maximum norm, and any two patterns p D(X) and p′ D′(X),
∈ B ∈ B
there exists x X with xD =p and xD′ =p′.
A function∈φ: X Y between two subshifts X SZd and Y TZd is called a
block map if there ex→ists a local function Φ : (X)⊂ T, where⊂D Zd is finite,
D
B → ⊂
such that φ(x) = Φ(x ) for all x X and ~v Zd. If D = ~0 , then φ is a
~v D+~v
∈ ∈ { }
symbol map. The image of an SFT under a block map is called a sofic shift. All
sofic shifts also occur as images of tiling systems under symbol maps.
Werecallthe arithmetical hierarchy, whichclassifiescertainsubsetsofNaccord-
ingtotheircomplexity. Afirst-orderarithmeticalformulawithboundedquantifiers
is classified as Π0 and Σ0. If a formula φ with free variable k is Π0, then k : φ
0 0 n ∃
is Σ0 . Similarly, if φ is Σ0, then k : φ is Π0 . Every first-order arithmetical
formnu+la1 is equivalent to a Πn0 or Σ0∀one, for sonm+1e n N. If φ has a unique free
variable k, then the set N n= k n N φ(k) is give∈n the same classification as
φ
φ. Note that N is Π0 if and o{nly∈if N| N }is Σ0. A set N N is C-hard for
a class C, if forφeverynM C there is a\ coφmputanble function⊂f : N N with
∈ →
M = f−1(N), and C-complete if also N C. The class Σ0 (Π0) contains exactly
∈ 1 1
the (co-)recursively enumerable sets. When classifying subsets of other countable
sets than N, for example (S), we implicitly choose a computable bijection to N.
d
P
A subshift is given the same classification as its language.
3. The Quantifier Extensions
We begin by defining our objects of interest, the quantifier extension subshifts.
Definition 1. Let X,Y SZd be d-dimensional subshifts, let / S be a new
symbol and denote Sˆ = S⊂ . For patterns P (Sˆ) and Q⋄ ∈SD(P), denote
d
∪{⋄} ∈ P ∈
by P(Q) SD(P) the pattern with
∈
P(Q) = Q~v, if P~v =⋄
~v (cid:26) P~v, otherwise.
We define two quantifier extension subshifts for X and Y, the universal extension
A(X,Y) SˆZd and the existential extension E(X,Y) SˆZd, with the respective
⊂ ⊂
QUANTIFIER EXTENSIONS OF MULTIDIMENSIONAL SOFIC SHIFTS 3
sets of forbidden patterns
P (S) Q (Y):P(Q) / (X) and
d D(P)
{ ∈P |∃ ∈B ∈B }
P (S) Q (Y):P(Q) / (X)
d D(P)
{ ∈P |∀ ∈B ∈B }
The symbol represents a ‘hole’ in a configuration x SˆZd that can be filled
⋄ ∈
with the contents of another configurationy Y. The extensions quantify over all
∈
suchy todecidewhetherxisvalid. Ourperspectiveinthispaperistostudy which
properties the extensions respect. We mainly focus on the universal extension, as
the question of soficness is particularly interesting for it. First, we show that the
universal extension respects the property of being an SFT in the following sense.
Proposition 1. Let X SZd be an SFT and Y SZd any subshift. Then the
⊂ ⊂
universal extension A(X,Y) is an SFT.
Proof. Let F (S) be a finite set of forbidden patterns for X. We claim that
d
⊂ P
Fˆ = P (Sˆ) Q (Y) : P(Q) F is a set of forbidden patterns for
d D(P)
{ ∈ P | ∃ ∈ B ∈ }
A(X,Y). Let x SˆZd be arbitrary. If x A(X,Y), it is clear that no patterns of
∈ ∈
Fˆ occur in x. Conversely, if x / A(X,Y), then there exists a configuration y Y
∈ ∈
such that x(y) / X, or in other words, there exists D Z2 such that x(y) F.
∈ ⊂ D ∈
This implies that the pattern P =x is in Fˆ, since the pattern Q =y (Y)
D D D
∈B
satisfies P(Q) F. Since Fˆ is finite, A(X,Y) is an SFT, as claimed. (cid:3)
∈
Example 1. The above result does not hold for the existential extension, even if
Y = 0,1 Z and X 0,1 Z is mixing. Namely, let X be defined by the single
{ } ⊂ { }
forbidden pattern 0100, and consider the configurations x = ∞01(0 )∞ and x′ =
⋄
∞(0 )0∞. We havex E(X,Y)by substituting 1 toevery , andx′ E(X,Y)by
⋄ ∈ ⋄ ∈
substituting 0. It is easy to see that these are the only possible substitutions, and
thus ∞01(0 )n0∞ / E(X,Y) for all n 1. Thus E(X,Y) is not an SFT.
⋄ ∈ ≥
For soficness, we can say the following.
Proposition 2. Let X,Y SZd be sofic shifts. Then the existential extension
⊂
E(X,Y) is sofic. If d=1, then the universal extension A(X,Y) is also sofic.
Proof. Let φ : X′ X and ψ : Y′ Y be surjective block maps, where X′ and
Y′ are SFTs, and d→enote Z = #, →Zd. Then E(X,Y) is obtained from the SFT
{ ⋄}
(x,y,z) ~v Zd :z = = x =y X′ Y′ Z,
~v ~v ~v
{ |∀ ∈ ⋄ ⇒ }⊂ × ×
by applying the block map ξ defined by
, if z =
ξ(x,y,z)~0 =(cid:26) ⋄φ(x)~0, oth~0erwi⋄se.
In the case d = 1, a subshift is sofic if and only if its language is regular, if
and only if it can be defined by a regular forbidden set [6, Chapter 3]. The sets
L= (S) (X) and (Y) are thus regular. Now, A(X,Y) is defined by the set
1
P \B B
w Sˆn v (Y):w(v) L
n
{ ∈ |∃ ∈B ∈ }
n[∈N
of forbidden words, which is clearly regular. Thus the extension is sofic. (cid:3)
In higher dimensions, automata theory is replaced by computability theory. In
particular,languagesofmultidimensionalsoficshiftsareco-recursivelyenumerable.
4 ILKKATO¨RMA¨
Lemma 1. The language of every multidimensional sofic shift is Π0.
1
Zd
Proof. Let X S be a sofic shift, let F (R) be a finite set of forbidden
d
patternsdefinin⊂ganSFTY RZd,andletφ⊂:YP X beasurjectivesymbolmap.
For a hypercube pattern P ⊂S[0,n−1]d, we have P→ (X) if and only if
∈ ∈B
∀m∈N:∃Q∈R[−m,n+m−1]d :φ(Q)[0,n−1]d =P ∧(∀T ∈F :T 6⊏Q).
This formula is Π0 by form, and so the language of X is Π0. (cid:3)
1 1
The latter statement of Proposition 2 translates to the following result.
Lemma 2. Let X,Y SZd be Π0 subshifts. Then the universal extension A(X,Y)
⊂ 1
is Π0. If Y is also Σ0, then A(X,Y) is Π0.
2 1 1
Proof. Let P Sˆn×n be arbitrary. We have P / (A(X,Y)) if and only if
∈ ∈B
Q Sn×n :Q (Y) P(Q) / (X).
∃ ∈ ∈B ∧ ∈B
Since the languages of X and Y are Π0, the proposition Q (Y) is Π0, while
1 ∈ B 1
P(Q) / (X)isΣ0. Thusthecomplementof (A(X,Y))isaΣ0language,implying
that ∈(AB(X,Y)) 1is Π0. The latter claim folloBws similarly. 2 (cid:3)
B 2
This bound is sharp, and we use the following powerful result to prove it.
Theorem 1 ([4, 1]). Let X SZ be a Π0 subshift. Then the vertically periodic
two-dimensional subshift y ∈SZ2 x X1: (i,j) Z2 :y =x is sofic.
(i,j) i
{ ∈ |∃ ∈ ∀ ∈ }
Proposition 3. There exist countable sofic shifts X,Y SZ2 such that the lan-
⊂
guage of the universal extension A(X,Y) is Π0-complete.
2
Proof. BecauseofLemma2andTheorem1,itsufficestoconstructone-dimensional
countable Π0 subshifts X,Y SZ such that A(X,Y) is Π0-hard.
1 ⊂ 2
Define S = 0,...,6 , and define X by the set of forbidden patterns
{ }
46 ij i,j S,i>j 01a2b+13c4 a,b,c N,a=c .
{ }∪{ | ∈ }∪{ | ∈ }
Sincethissetisrecursive,X isaΠ0 subshift,andeasilyseentobecountable. Next,
1
let Φ be an arithmetical formula with bounded quantifiers such that the set
N = k N m N: n N:Φ(k,m,n)
{ ∈ |∀ ∈ ∃ ∈ }
is Π0-hard. Define Y by the set
2
0,1,46 ij i,j S,i>j 23a4b5 a,b N, n N: Φ(a,b,n)
{ }∪{ | ∈ }∪{ | ∈ ∃ ∈ ¬ }
of forbidden patterns. Since this set is Σ0 by form, Y is a countable Π0 subshift.
1 1
Define the function w : N L = 01k2 k N by w(k) = 01k2 . We note
→ { ⋄ | ∈ } ⋄
that for all s S, there exists t (Y) such that ts is forbidden in X, which
1
∈ ∈ B
impliesthat sisforbiddeninA(X,Y). Thusw(k)occursinA(X,Y)ifandonlyif
⋄
the infinite tail w(k) ∞ does. By the definition of X and Y, this is the case if and
onlyif23k4m5 / (Y⋄)forallm N. Butthis isequivalenttok N,whichmeans
that N =w−1(∈LB (A(X,Y))),∈and thus the languageof A(X,Y∈) is Π0-hard. (cid:3)
∩B 2
As a corollary of Lemma 1 and the above proposition, we obtain the following
counterpart of Proposition 2.
Corollary 1. There exist countable sofic shifts X,Y SZ2 such that the universal
⊂
extension A(X,Y) is not sofic.
QUANTIFIER EXTENSIONS OF MULTIDIMENSIONAL SOFIC SHIFTS 5
While this resultisinterestinginitself, theproofisnotverysatisfying,sincethe
subshift Y that we extend by is computationally complex, and we use the simpler
structure of X only to check a universally quantified property of (Y). However,
B
if we restrict Y to be a recursive subshift (both Π0 and Σ0), Lemma 2 shows that
1 1
the language of A(X,Y) is Π0, so a recursion theoretic proof will no longer work.
1
In the next section, we find pairs of computationally simple sofic shifts X and
Y such that A(X,Y) is not sofic. The following special case was presented as an
open problem by professor Brian Marcus at the 2013 PIMS/EQINOCS Automata
Theory and Symbolic Dynamics Workshop: Is the extension A(X, 0,1 Z2) a sofic
shiftforeverysoficX 0,1 Z2? In[7],itwasaskedwhetherthem{ulti-}choice shift
⊂{ }
space (see the cited article for the definition) associated to a two-dimensionalsofic
shift is necessarilysofic,andthe aboveproblemis a restatementofthis questionin
the binary case. Theorem 2 in particular shows that the answer is negative.
4. Main Theorem
In this section, we classify those two-dimensional subshifts that only yield sofic
extensions. Before stating the result, we give the following definition.
Definition 2. A sofic shift X SZd is countably covered if it is the image of a
⊂
countable SFT via a block map.
This notion is not standard in the literature. Of course, all countably covered
sofic shifts are countable. Now, our main result is the following.
Theorem 2. Let Y SZ2 be a subshift. The following are equivalent:
⊂
(1) Y is finite.
(2) A(X,Y) is sofic for all sofic shifts X RZ2 over all alphabets R.
(3) A(X,Y) is sofic for all countably cove⊂red sofic shifts X SZ2.
⊂
Wecontinuewithaseriesoflemmas. Mostofthemarenotneededinthespecial
caseofbinaryfullshifts,butweusethembecauseTheorem2ismuchmoregeneral.
Lemma 3. Let X SZ2 be a countably covered sofic shift, and let Y (S R)Z2
be an SFT. If Y (⊂X RZ2) is countable, then it is a countably covere⊂d sofi×c shift.
∩ ×
Proof. Let X = f(Z), where Z TZ2 is a countable SFT and f : Z X a
block map with neighborhood N⊂ Z2. Let F TD and G (S R→)D be
sets of forbidden patterns for Z an⊂d Y, where D⊂ Z2 is finite⊂. The×n the set
⊂
F RD (P,Q) (T R)D+N (f(P),Q ) G defines the SFT Y′ =
D
(z×,z′) ∪Z{ RZ2 (∈f(z),z×′) Y , w|hichis cou|ntab∈le sin}ce Y is. The image ofY′
{under th∈e bl×ock ma|p (z,z′) ∈(f(}z),z′) is exactly Y, and the claim follows. (cid:3)
7→
The following is one of the few known methods for showing a multidimensional
subshiftto benonsofic. Aproofofithasappearedatleastin[5],butthetechnique
is much older, presumably originatingfromthe theory of picture languages. In the
proof, the notation ∂D for a domain D Z2 stands for the set
⊂
(a,b) Z2 (a,b) / D, (a+1,b),(a 1,b),(a,b+1),(a,b 1) D = Z2.
{ ∈ | ∈ { − − }∩ 6 ∅}⊂
Lemma 4. Let X SZ2 be a sofic shift. Then there exists C >0 such that for all
⊂
n 1 and all sets Λ X of size at least Cn+1, there exist x = y Λ such that
c(x≥,y,n) X, where⊂c(x,y,n) SZ2 is defined by c(x,y,n) 6 =∈x and
[0,n−1]2 [0,n−1]2
∈ ∈
c(x,y,n)Z2\[0,n−1]2 =yZ2\[0,n−1]2.
6 ILKKATO¨RMA¨
Proof. SinceX issofic,thereexistatilingsystemY TZ2 andasurjectivesymbol
⊂
map φ : Y X. We claim that C = T 4 suffices. Let n 1, and for all x X,
→ | | ≥ ∈
chooseapreimagex˜ φ−1(x). Letx,y X. Ifwehavex˜ =y˜ ,then
∂[0,n−1]2 ∂[0,n−1]2
∈ ∈
the configuration c(x˜,y˜,n) TZ2 is in Y, so φ(c(x˜,y˜,n)) = c(x,y,n) X. Since
the number of x˜ for∈x X is at most Cn = T 4n, the claim fol∈lows. (cid:3)
∂[0,n−1]2
∈ | |
The proof of Theorem 2 requires the use of a computational device, and the
following nonstandard version of the classical counter machine suits our needs.
Definition 3. A counter machine with string input (CMS for short) is a tuple
M = (k,k′,S,Σ,δ,q ,q ,q ), where k N is the number of counters, k′ < k
0 a r
∈
the number of output counters, Σ is a finite state set, S the finite input alphabet,
q ,q ,q Σ the initial, accepting and rejecting states and
0 a r
∈
δ (Σ [1,k] Z,P Σ) (Σ [1,k] 1,0,1 Σ) (Σ S¯ Σ)
⊂ × ×{ }× ∪ × ×{− }× ∪ × ×
the transition relation, where S¯=S # for a new symbol # / S. An instanta-
neousdescription (ID)ofM isanelem∪e{nt}(q,w,n ,...,n )ofΣ ∈S∗ Nk,withthe
1 k
× ×
interpretationthatM isinstateqwithinputwordwandcountervaluesn ,...,n .
1 k
The CMS M operates in possibly nondeterministic steps as follows. If we have
(p,i,Z,q) δ ((p,i,P,q) δ), then from any ID (p,w,n ,...,n ) with n = 0
1 k i
∈ ∈
(n > 0, respectively), M may move to the ID (q,w,n ,...,n ). If (p,i,r,q) δ
i 0 k
∈
with r 1,0,1 , then from any ID (p,w,n ,...,n ,...,n ), M may move to
1 i k
∈ {− }
(q,w,n ,...,n +r,...,n ). Finally, if (p,s,q) δ for s S ((p,#,q) δ), then
1 i k
∈ ∈ ∈
then from any ID (p,w,n ,...,n ) with w =s (w n , respectively), M may
1 k nk | | ≤ k
moveto(q,w,n ,...,n ). We assumethatM neverdecrementsacounterbelow0.
1 k
The CMS is initialized from an ID (q ,w,0,...,0) for w S∗, and halts when
0
it reaches the states qa or qr. The tuple (n1,...,nk′) Nk′ ∈of the first k′ counter
∈
values in the accepting state q is the output of M on w, and is denoted M(w).
a
Note that n is the value of the first counter, not the greatest counter value. If M
1
reaches the rejecting state q or never halts, no output is generated.
r
In one step, a CMS may increment or decrement one of its counters, check
whether a counter is zero, or check the input symbol under the last counter, if one
exists. When it halts in the accepting state, the counter values of the final ID are
considered as outputs. Thus a CMS can be interpreted as a partial function from
S∗ to Nk′. The classical reference for counter machines is [9], although CMSs are
not defined there. Conventional counter machines are computationally universal,
and it is not hard to see that the same holds for these devices.
Lemma5. Everyrecursivepartial functionf :S∗ Nk′ can berealizedas aCMS.
→
We alsoneedthe followingresultfromsymbolicdynamics. Itisslightlystronger
than the version in [6], but the missing details can be extracted from its proof.
Lemma 6 (Marker Lemma). Let S be a finite alphabet. For all n N there exists
a block map f :SZ 0,1 Z, called the n-marker map, such that:∈
n
→{ }
(1) The radius of f is at most r =n|S|2n+1.
n n
(2) For k <n 1,the word 10k1 does not occur in any configuration of f (SZ).
n
(3) If x SZ−is such that f (x) = 02n−1, then x is periodic
n [−n+1,n−1] [−n,n]
∈
with period less than n.
(4) The function (n,w) F (w), where w Srn and F is the local function
n n
7→ ∈
of f , is recursive.
n
QUANTIFIER EXTENSIONS OF MULTIDIMENSIONAL SOFIC SHIFTS 7
Proof. Lemma 10.1.8 of [6] states the existence of a clopen set C SZ such that
⊂
the shifted sets σi(C) are disjoint for all i [0,n 1], and if σi(x) / C for all
∈ − ∈
i [ n+1,n 1], then x is periodic with period less than n. We define
[−n,n]
f∈(x)−= 1 if an−d only if σi(x) C, and then f : SZ 0,1 Z is a block map
n i n
satisfying(2)and(3). Theother∈claimsfollowfromtheco→nst{ruct}ionofC in[6]. (cid:3)
Finally,wehaveageneralconstructionofgrid-likecountablycoveredsoficshifts.
Lemma 7. For all m,n N , there exists a countably covered sofic shift
∈ ∪{∞}
X =X over the alphabet # 0,1 2 such that:
G(m,n)
{ }∪{ }
For all x X, the set D(x) = ~v Z2 x = # is a (possibly infinite)
~v
• ∈ { ∈ | 6 }
rectangle.
For all k 2, 1 a m+k and 1 b n+k , there is a configuration
• ≥ ≤ ≤ k ≤ ≤ k
x = xk,a,b X such tha(cid:0)t D((cid:1)x) = [0,ak] [(cid:0)0,bk(cid:1)], and for all ~v = (i,j)
∈ × ∈
D(x) we have π (x ) = 1 (π (x ) = 1) if and only if i 0modk (j
1 ~v 2 ~v
≡ ≡
0modk, respectively), where π ,π : 0,1 2 0,1 are the projections to
1 2
{ } →{ }
the first and second components.
Ifm= (n= ), theneveryhorizontal (vertical) lineof1’sin thesecond
• ∞ ∞
(first) layer of X is infinite to the right (upwards), and otherwise, every
configuration of X contains only finitely many vertical (horizontal) lines of
1’s in its first (second) layer.
Z2
Proof. We construct a countable SFT Y A and a symbol map π : A 0,1
⊂ →{ }
suchthatπ(Y)=X. ThealphabetAisthesetoftilesinFigure1,wherethelabels
C rangeover[0,ℓ]ifℓ N,and ifℓ= . Notethatsometilesareforbiddenif
ℓ
∈ {∞} ∞
morn isinfinite. Every2 2patternwhere the linesorcolorsofsometiles donot
×
match (including the diagonal lines) is forbidden in Y. Then the regions colored
by L, R, B, and T in a configuration y Y, if nonempty, form a left half plane,
∈
a right half plane, a downwardinfinite rectangle,and an upwardinfinite rectangle,
respectively. The rectangular area not contained in them is called the grid of y.
It is divided into rectangles by the grid lines (the thick lines in Figure 1), which
stretchfromoneendofthegridtotheother. Theserectanglesareactuallysquares,
allofthesamesize,becauseofthediagonallines. Wealsoforbidevery2 2pattern
×
containing such a square, so that every square contains at least one interior tile,
shown in the fifth column of the figure. If m (n) is infinite, then so is the width
(height) of every grid, as it cannot have a right (top, respectively) border.
The labels C of the interior tiles have the following rules if m = . Inside
m
6 ∞
a square of the grid, they must be horizontally constant and downward increasing
(all patterns cd for c = d and c for c > d are forbidden). On the border of two
6 d
horizontally adjacent squares, all 3 2 patterns except
×
−+− , −+ − , e | e , e | e , e |e+1 , m|c , e | e+1 , m | c
e | e e | e+1 f |f f |f+1 m|e+1 m|c −+ − − +−
h i h i h i h i h i h i h i h i
forc,d [0,m]ande,f [0,m 1]areforbidden(thesymbols , and+represent
∈ ∈ − − |
horizontal, vertical and crossing grid lines, including T-junctions). Denote by Vk
m
the set of length-k downward increasing column vectors over [0,m]. The above
rules imply that for all horizontally adjacent grid squares with k k interior tiles,
×
the column vector formed by the bottom-right labels of the interior tiles of the
right square is the lexicographical successor of that of the left square with respect
to the set Vk. Thus the width of any grid containing such a square is at most
m
8 ILKKATO¨RMA¨
Vk = m+k squares. To ensure countability, we also require that the labels next
| m| k
to the le(cid:0)ft bo(cid:1)rder of a grid are all 0. We introduce analogous rules for the top-left
labels,buttransposed,sothatthe heightofthegridisatmost n+k squares. This
k
concludes the definition of Y. We set π(t) = # if and only if t(cid:0)he t(cid:1)ile t contains a
6
grayregion,andthenπ (π(t))=1(π (π(t))=1)ifandonlyiftcontainsavertical
1 2
(horizontal) grid line. The three conditions for X follow easily.
We show that Y is countable, so let y Y. If a finite grid square occurs
∈
in y, then there are countably many choices for the position of the grid, which
uniquely determines its contents (because ofthe restrictionson the columnvectors
introduced above) and the rest of y. If y contains no grid tiles, then it consists of
the L, R, B and T-tiles, for which we have countably many choices. In the case of
infinite squares,since the labels of the interior tiles are decreasing in one direction
and constant in the other, our choices are again restricted to a countable set. (cid:3)
T T T Cm T
L T L R T R
C
n
n= m=
6 ∞ 6 ∞
C
m
L L R R T
C
n
C
m
L L R R B
B B B Cn B
Figure 1. The alphabet of the grid SFT Y in the proof of Lemma 7.
We are now ready to prove Theorem 2. In its proof, we will modify the sofic
shifts X by superimposing new symbols on top of their configurations with
G(m,n)
Lemma 3. They provide a rigid geometric structure for the construction.
Proof of Theorem 2. (1 2): LetY SZ2 be finite, anddenote Y = y1,...,yk .
The yi must all be perio⇒dic, and we le⊂t p N be a common horizontal{and vertic}al
∈
periodforallofthem. LetZ beatwo-dimensionalSFTandφ:Z X asurjective
blockmap,anddefinetheSFTZ′ Zk Yk ,# Z2 asfollows.→Aconfiguration
z′ =(z1,...,zk,yn1,...,ynk,t) Z⊂k Y×k ×{,⋄# Z}2 is in Z′ if and only if
∈ × ×{⋄ }
yn1,...,ynk = y1,...,yk , which can be checked by p p patterns,
• {if t = , then}φ(z{i) =yni f}or all i [1,k], and ×
• ~v ⋄ ~v ~v ∈
if t =#, then φ(zi) =φ(zj) for all i,j [1,k].
~v ~v ~v
• ∈
We then define the block map ψ :Z′ Sˆ by
→
, if t = ,
ψ(z′)~v =(cid:26) ⋄φ(z1)~v, oth~verwi⋄se.
It is easily verified that ψ(Z′)=A(X,Y), and thus the extension is sofic.
(2 3): Trivial.
(3⇒1): Let Y SZ2 be infinite. Then for all p N there exists y Y which is
⇒ ⊂ ∈ ∈
not p-periodic either in the horizontal or the vertical direction. If either condition
cannot be satisfied for some p, then the other can be satisfied for all p. Thus we
may assume that Y has no common horizontal period. We also have S 2.
| |≥
QUANTIFIER EXTENSIONS OF MULTIDIMENSIONAL SOFIC SHIFTS 9
Our goal is now to construct a countably covered sofic shift X SZ2 such that
⊂
A(X,Y) is not sofic. We proceed by constructing a countably coveredsofic shift Z
and a block map φ : Z SZ2 whose image we define as X. After this, it will be
→
easy to show that the extension is not sofic, using Lemma 4. The subshift Z is a
subset of Z Z Z , where Z is the input layer, Z the computation layer, and
1 2 3 1 2
× ×
Z the output layer. We define the layers sequentially:
3
(1) Define Z , a countably covered sofic shift.
1
(2) Define the SFT Z , and a countable sofic shift Z′ Z Z .
2 1 2
⊂ ×
(3) Define Z , a countably covered sofic shift, and the sofic shift Z Z′ Z .
3 3
⊂ ×
We begin with Z , which is defined by superimposing a label fromS 0,1 on
1
×{ }
each vertical column of the grid shift X given by Lemma 7. More formally,
G(1,∞)
Z = Z′ (X ( (cid:3) S 0,1 )Z2), where Z′ is the SFT with forbidden
pa1tterns1(∩t,ℓ)Gw(h1,e∞re)×exa{ctly}∪oneo×f{t }(1,0),(1,1) an1dℓ=(cid:3)holds,andall1 2
∈{ } 6 ×
patterns (t1,ℓ1) whereℓ andℓ differfrom(cid:3)andeachother. Thesymbol(cid:3)means
(t2,ℓ2) 1 2
‘no label’. As all configurations of X contain only finitely many columns of
G(1,∞)
1’s, Z is countable, and thus a countably covered sofic shift by Lemma 3.
1
We moveonto the computationlayerZ ,whosedesignis verysimilarto thatof
2
Construction1in[11]. Let / S 0,1 beanewsymbol,andletM =(k,k′,(S
∗∈ ×{ } ×
0,1 ) ,Σ,δ,q ,q ,q ) be a CMS whose functionality we define later; for now,
0 a r
{ } ∪{∗}
weonly require thatits input alphabetis S˜=S 0,1 . The alphabetofZ
2
×{ }∪{∗}
isS = L,R,H ( 0,1 Γ) 0,1 k ,whereΓ=δ (S˜ # ) , , .
2
{ }∪ { }∪ ×{ } × ∪{ } ×{⇐ ← →}
First,every2 2(cid:0)patternthatcontains(cid:1)L, RorH,butisnotofoneofthe forms
×
LL LR L a La a R LH LH H H
LL LR LR L b LR LH L a H H
RR a R a b H R H H H H H H
RR RR c R R R H R a R a b
wherea,b,c ( 0,1 Γ) 0,1 k,isforbidden. Ifaconfigurationz Z contains
2
∈ { }∪ ×{ } ∈
only letters of L,R,H , then it consists of two half planes, or only contains one
{ }
letter. TheL-regionofz,if nonempty,is alefthalfplane, whilethe R-regionis the
intersection of right and southeast half planes, which may be the whole of Z2 or
empty. The H-region, if nonempty, is the intersection of an upper half plane and
the region without L-tiles or R-tiles. The rest of z is called the computation cone.
We enforce by 2 1 forbidden patterns that each horizontal row of the compu-
tation cone is of the×form 1ℓγ0m−ℓ−1 k 1ni0m−ni, where γ Γ is called the
zig zag head and m N is the length o×fQthie=r1ow. The product den∈otes a family of
∈
layers,not concatenation. The row correspondsto a computation step ofM where
the k counters have the respective values n ,...,n . The zig zag head remembers
1 k
the transition of the step and the symbol read by the last counter. The length-1
k
row on the bottom of the cone is (t , , ) 0, where t δ is the transition
0 ∗ → × i=1 0 ∈
M takes from its initial ID (q0,w,0,...,0) forQan input word w with w0 = .
∗
A computation of M is simulated on the rows of the cone, with time increasing
upwards. The zig zag head starts from the left border of the cone in some state
(t,s, ) Γ, moving right at speed 2. When it hits the right border, it changes
→ ∈
its state to (t,s, ) and starts moving left at speed 1. If the transition t involves
⇐
updating a counter i, and the head steps on said counter, its value n is updated.
i
The head assumes the state (t,s, ), or (t,r, ) in case of the last counter, where
← ←
r S˜ # isarbitrary. Whentheheadreachestheleftborderinstate(t,s, ),it
∈ ∪{ } ←
choosesanewtransitiontoexecute,basedont,s(interpretedastheletterunderthe
10 ILKKATO¨RMA¨
last counter), and the set of counters with value 0. Finally, the computation cone
ends in a horizontalrow of H-symbols precisely when the final state q is reached.
f
This simulation can be enforced by 2 2 forbidden patterns. The definition of Z
2
×
is complete. See Figure 2 for a visualization of the computation cone.
LLHHHHHHHHHHHHHHHHHH
LLHHHHHHHHHHHHHHHHHH
LL 2 2 2 1 1 1 1
←
LL 2 2 2 1 1 1 1
←
LL 2 2 2 1 1 1 1
←
LL 2 2 2 1 1 1 1
←
LL 2 2 2 1 1 1 1
←
LL 2 2 2 1 1 1 1
← .
.
.
LL 1 1 RR
→
LL 1 1 RRR
→
LL 1 1 RRRR
→
LL 1 1 RRRRR
→
LL 1 1 RRRRRR
→
LL 1 RRRRRRR
→
LL 1 RRRRRRRR
←
LL 1 RRRRRRRRR
←
LL 1 RRRRRRRRRR
←
LL 1 RRRRRRRRRRR
←
LL 1 RRRRRRRRRRRR
←
LL 1 RRRRRRRRRRRRR
→
LL 1 RRRRRRRRRRRRRR
→
LL RRRRRRRRRRRRRRR
→
LL RRRRRRRRRRRRRRRR
←
LL RRRRRRRRRRRRRRRRR
→
LLRRRRRRRRRRRRRRRRRR
LLRRRRRRRRRRRRRRRRRR
Figure 2. The base of a computation cone in a configuration of
Z . The tiles are not in a one-to-one correspondence with the
2
actual letters of S . The limits of the cone are indicated by thick
2
lines, and the head by an arrow that indicates its direction. The
counters aredenoted by 1 and2, andthe counter1 is incremented
in the first two computation steps. The halting step, which lies
much higher, is also shown.
The subshift Z′ Z Z is defined by finitely many additional forbidden
1 2
⊂ ×
patterns. First,thewidth-1bottomrowofthe computationconeinZ canonlybe
2
paired with the bottom left corner of the grid in Z , or the upper right corner of a
1
pattern ab with a=c=(0,0)=b, andvice versa. Second, the symbolof S˜ #
c 6 ∪{ }
the zig zag head chooses when updating the last counter is determined by the
symbolonthe firstlayeratthatcoordinate. Ifthis symbolhaslabel ℓ S 0,1 ,
∈ ×{ }
thentheheadgetsℓ;ifthesymbolis#,theheadgets#;otherwise, . Theupward
∗
infinite grid of Z thus encodes an input string for the simulation of M.
1
Claim 1. If M is deterministic, then Z′ is a countably covered sofic shift. Also,
if a configuration z Z′ contains the bottom row of a grid of width (n 1)n+1
∈ −
whose labels encode a word w (S 0,1 )n, and if M(w n−1 n−1w ) =
0 n−1
∈ ×{ } ∗ ···∗