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transcript
An application of iterative
pushdown automata to contour
words of balls and truncated balls
in hyperbolic tessellations
Maurice Margenstern
Universite Paul Verlaine − Metz,LITA, EA 3097, and LORIA, CNRS
Journees Montoises 2010September, 9, 2010
Amiens, France
in this talk:
1. iterated pushdown automata
2. tessellations in the hyperbolic plane
3. contour words of balls
4. in 3D
1. iterated pushdown automata
we will see, successively:
1.1 pushdown automata
1.2 iterated pushdown automata
1.1 pushdown automata
it consists of two structures:
a finite automaton
finitely many states, finite alphabet
reading the input word letter after
letter with possible ǫ-transitions
and an additional feature:
a stack:
list of letters, top only accessible
while reading, two possible actions:
push and pop
transitions: fixed by the read letter
and by the top of the stack
1.2 iterated pushdown automata
again a finite automaton,
but with an iterated store:
again stack structure
but the elements are stacks of stacks
below, formal presentation based
on a paper by G. Senizergues
and S. Fratani,
Annals of pure and applied logic,
(2006)
formally define k-iterated store as fol-
lows:
0-pds(Γ) = {ǫ}k+1-pds(Γ) = (Γ[k-pds(Γ)])∗
it-pds = ∪k
k-pds(Γ)
unique representation of a k+1 store
ω as:
ω = A[flag].rest, A ∈ Γ,
with flag k-store, and rest k+1-store
we have: lgth(ω) = lgth(rest)+1
operations on k-iterated stores:
notion of top symbols:
topsym(ǫ) = ǫ,
topsym(A[flag].rest) = A.topsym(flag)
the pop operation:
popj(ǫ) undefined
popj+1(A[f ].r) = A[popj(f)].r,
A ∈ Γ
operations on k-iterated stores:
the push operation:
push1(w)(ǫ) = w, w ∈ Γ∗
pushj(γ)(ǫ) undefined for j > 1
push1(w)(A[f ].r) = w1[f ]..wh[f ].r,
pushj+1(w)(A[f ].r) = A[pushj(w)(f)].r,
w = w1..wh ∈ Γ∗, wi ∈ Γ
operations performed within the top symbols
k-iterated pushdown automata:
operate on a word as a finite
automaton, using a k-store and the
push and pop operations
computation:
there are final states
word accepted if and only if at least
one path of computation leads to a
final state
theorem (G. Senizergues, 2003) −the languages recognized by a k-iterated
pushdown automaton define a stricly
increasing hierarchy with respect to k
key example:
2-pds automata recognize:
{afn | {fn}n∈IN : Fibonacci sequence}
three states: q0, q1 and q2;
input word in {a}∗; Γ = {Z, X1, X2, F};
initial state: q0; initial stack: Z[ǫ];transition function δ:δ(q0, ǫ, Z) = {(q0, push2(F)), (q0, push1(X2))}δ(q0, ǫ, ZF) = {(q0, push2(FF)), (q0, push1(X2))}δ(q0, ǫ, X1F) = (q1, pop2)δ(q0, ǫ, X2F) = (q2, pop2)δ(q0, a, X1) = (q0, pop1)δ(q0, a, X2) = (q0, pop1)δ(q1, ǫ, X1F) = (q0, push1(X1X2))δ(q2, ǫ, X2F) = (q0, push1(X1))δ(q1, ǫ, X1) = (q0, push1(X1X2))δ(q2, ǫ, X2) = (q0, push1(X1))
indeed,it is enough to prove the following
lemma for any nonnegative k and m:
(q0, afkam, X2[Fk].ω) ⇒∗
δ (q0, am, ω)
(q0, afk+1am, X1[Fk].ω) ⇒∗
δ (q0, am, ω)
the case k = 0 is obviousassuming the above for k, and any m,
consider (q0, afk+2am, X1[Fk+1].ω)
then
(q0, afk+2am, X1[Fk+1].ω)
⊢ (q1, afk+2am, X1[Fk].ω)
δ(q0, ǫ, X1F) = (q1, pop2)
⊢ (q0, afk+2am, X1[Fk].X2[F
k].ω)δ(q1, ǫ, X1F) = (q0, push1(X1X2))
⊢ (q0, afkam, X2[Fk].ω) ⇒∗
δ (q0, am, ω)as fk+2 = fk+1 + fkand induction hypothesis
2. tessellations
in the hyperbolic plane
we shall see:
2.1 Poincare’s disc model
2.2 the pentagrid and the heptagrid
2.1 Poincare’s disc model
the disc
2.1 Poincare’s disc model
a point A
A
2.1 Poincare’s disc model
a point A
a line ℓ A
l
2.1 Poincare’s disc model
a secant
through A
which cuts ℓ
A
l
s
2.1 Poincare’s disc model
a parallel p
to ℓ
through A
A
l
sp
P
2.1 Poincare’s disc model
another
parallel q
to ℓ
through A
A
l
sp
q
PQ
2.1 Poincare’s disc model
a non secant
line m to ℓ
through A
A
p
PQ
l
q
m
s
2.1 Poincare’s disc model
the common
perpendicular
to ℓ and to m
A
p
PQ
l
q
m
s
h
PQ
a few useful properties
the sum of angles in a triangle:
always less than π
non-secant lines :
always a unique common perpen-
dicular
no similarity in the Euclidean mean-
ing
2.2 the pentagrid and the heptagrid
from Poincare’s theorem:
infinitely many tilings of thehyperbolic plane by replicating atriangle in its side and, recursively,
the images in their sides
angles of the triangle, necessarily
2π
p,2π
q,2π
r,
with p, q, r positive integers
satisfying:1
p+
1
q+
1
r< 1
from now on focus on
the pentagrid and the heptagrid:
{5,4} {7,3}
navigation in {5,4} and {7,3}
in the pentagrid
a quarter,
navigation in {5,4} and {7,3}
in the pentagrid
a quarter, and the underlying tree
navigation in {5,4} and {7,3}
in the heptagrid
a sector,
navigation in {5,4} and {7,3}
in the heptagrid
a sector, and the underlying tree
navigation in {5,4} and {7,3}
in both cases:
a central tile and α sectors, α ∈ {5,7}
the common tree for {5,4} and {7,3}
in both cases, the Fibonacci tree:
10
100
101
1000
100
1010
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1000
1001
1010
10101
0 1 0 0 1
10000
1000
100
10
0
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001
010
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1010
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101001
101010
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0
0
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000
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10000
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1000001
1000000
1
1
2 3 4
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13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
11 1
1
1 1
1
1
1
1 1
0 1
1 1 1
11 1
1
the common tree for {5,4} and {7,3}
why Fibonacci tree?
because
level n in the tree: f2n+1 nodes
two kinds of nodes: Black and White
with the rules:
B → BW , W → BWW
hence the
connection with 2-iterated pds
3. contour words of balls
we shall see:
3.1 balls in {5,4} and {7,3}
3.2 contour words and
their recognition
3.1 balls in {5,4} and in {7,3}
common definition:
path between a tile A and a tile B,
A 6= B:
sequence {Ti}0≤i≤n of tiles with:
Ti and Ti+1 share a side, i < n,
T0 = A, Tn = B
3.1 balls in {5,4} and in {7,3}
let {Ti}0≤i≤n be a path
from A and B
n is the length of the path
now, d(A, B) is the length of a
shortest path between A and B,
A 6= B,
d(A, B) = 0, when A = B
3.1 balls in {5,4} and in {7,3}
a ball of radius n around A is the
set of tiles at a distance from A not
greater than n: denoted by BA(n)
3.2 contour wordsand their recognition
contour of a ball:
the contour of the ball BA(n),denoted by ∂BA(n),is the set of tiles at a distance n from A
exactly
3.2 contour wordsand their recognition
illustration:
pentagrid heptagrid
3.2 contour wordsand their recognition
a contour word of BA(n) is a word
asn where sn is the length of ∂BA(n)
theorem (MM, arXiv 2009)
the language of the contour words of
the balls of the pentagrid can be
recognized by a 2-iterated pushdown
automaton; the same for the
heptagrid
sketch of the proof:
each of the 5 sectors
is spanned by a
Fibonacci tree
the 2-iterated
pushdown automaton
performs a traversal
of the tree
same proof for the heptagrid:
7 sectors here,
each one spanned by the
same Fibonacci tree
the 2-iterated
pushdown automaton
performs a traversal
of the tree
two states: q0 and q1;
input word in {b, w}∗; Γ = {Z, B, W, F};
initial state: q0; initial stack: Z[ǫ];transition function δ, α ∈ {5,7}:δ(q0, ǫ, Z) = {(q0, push2(F)), (q0, push1(W
α))}δ(q0, ǫ, ZF) = {(q0, push2(FF)), (q0, push1(W
α))}δ(q0, ǫ, WF) = (q1, pop2)δ(q0, ǫ, BF) = (q1, pop2)δ(q0, b, B) = (q0, pop1)δ(q0, w, W ) = (q0, pop1)δ(q1, ǫ, WF) = (q0, push1(BWW ))δ(q1, ǫ, BF) = (q0, push1(BW ))δ(q1, ǫ, W ) = (q0, push1(BWW ))δ(q1, ǫ, B) = (q0, push1(BW ))
the theorem can be extended to
contour words of sectors:
pentagrid, see MM arXiv paper
similar extension to
contour words of sectors:
heptagrid, see MM arXiv paper
the theorem can be extended
to balls and sectors
of the tilings {p,4} of IH2,
and the tilings {p+2,3} of IH2,
p ≥ 5 in both cases
to balls
of the tiling {5,3,4} of IH3
and the tiling {5,3,3,4} of IH4
4. in 3D
we shall see:
4.1 {5,3,4}: the dodecagrid
4.2 recognition of contour words
in the dodecagrid
4.1 {5,3,4}: the dodecagrid
first, extend Poincae’s disc model:
the points: inside the unit nD ball
the border, the unit n−1D ball= the points at infinity
hyperplanes:
trace of diametral hyperplanes
or of n−1D balls, orthogonal to the border
k-hyperplanes, 1 ≤ k ≤ n−2:
intersection of k+1-hyperplanes
only 4 tessellations in the hyperbolic
3D space
notation {p, q, r}:
p sides around a face
q faces around a vertex
r polyhedra around an edge
the tessellations:
{3,5,3} {4,3,5} {5,3,4} {5,3,5}
the dodecagrid: {5,3,4}
an important tool: Schlegel diagrams
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89
10 11
0
1
23
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78
910
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1415
1617
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projection of the regular rectangular
dodecahedron on the plane of a face
navigation in the dodecagrid
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2
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4
56
7
89
10 11
1
2
3
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56
7
89
10 11
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2
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10 11
O H T
this gives us a tree in bijection with
the tiling, whose generating rules are:
O → O2H6T , H → OH6T , T → H5T
a sector is the set of tiles generated
by the tree Twe have 3 kinds of γ-sectors:
with γ ∈ O-, H- and T -sectors,
depending on the root of the tree Tγ
a k-truncated γ-sector is generated
by the first k levels of Tγ
a ball Bk is the union of 8 k-truncated
O-sectors around a vertex
we can state:
theorem (MM, arxiv 2009)
there is a 2-iterated pushdown automa-
ton which recognizes the countour
words of ball Bk in the dodecagrid;
for each γ ∈ {O, H, T}, there is
a 2-iterated pushdown automaton
which recognizes the contour word
of any k-truncated γ-sector
Thank you for your attention!