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

1000

1000

1001

1010

10101

0 1 0 0 1

10000

1000

100

10

0

10

001

010

01

00

1010

0

01

10

101001

101010

10

0

0

10

0

00

10

00

0

10

000

10

000

100

0

0

100

00

100

000

1000

0

1000

00

10000

0

1000001

1000000

1

1

2 3 4

5 6 7 8 9 10 11 12

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

1

2

3

4

56

7

89

10 11

0

1

23

4

5

6

78

910

11

12

13

1415

1617

18

19

projection of the regular rectangular

dodecahedron on the plane of a face

navigation in the dodecagrid

1

2

3

4

56

7

89

10 11

1

2

3

4

56

7

89

10 11

1

2

3

4

56

7

89

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!