An application of iterative pushdown automata to contour words of balls and truncated balls in hyperbolic tessellations Maurice Margenstern Universit´ e Paul Verlaine − Metz, LITA, EA 3097, and LORIA, CNRS Journ´ ees Montoises 2010 September, 9, 2010 Amiens, France
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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