Combinatorics on words
Binary alphabet
Finite Sturmian words
Theory of Automata
Minimization of DFA
K-ary alphabet
Finite episturmian words
Minimization of DMA

Infinite words – binary alphabet {a,b}
n+1 factors of lenght n for each n 0;
one right special factor for each length n;
(factor that appears followed by two different letters resp.)
Example: Fibonacci word
abaababaabaababaababaab…
Given (p,q) coprime, the Christoffel word having p occurrences of a's and q occurrences
of b's is obtained by considering the path under the segment in the lattice NxN, from
the point (0,0) to the point (p,q) and by coding by ‘a’ a horizontal step and by ‘b’ a
vertical step.
Example: (5,3)
aabaabab
(5,3)
infinite


Exactly n+1 factors of lenght n for each
n  0;
One right special factor for each length
Example: Fibonacci word
abaababaabaababaababaab…
finite
(w) - Christoffel classes – circular
Sturmian words


Exactly n+1 circular factors of
lenght n for each nw-1;
One right circular special factor
for each length n  w-2
Example: finite Fibonacci word
abaababaabaababaababaab
a
b
a
b
a
a
b
a
b a b
a b a a
a
a
b
b
a
b
a
a

Are closed under reversal and have at most one
right special factor of each length.
Example: Tribonacci word over {a,b,c}
abacabaabacaba…

Are closed under reversal and have at most one
right special factor of each length.
Example: Tribonacci word over {a,b,c}
abacabaabacaba…
epichristoffel classes
or
circular episturmian words
A finite word is an epichristoffel word if it is the image of a letter by an episturmian
morphism and if it is the smallest word of its conjugacy class (epichristoffel class).
(6, 3, 1) →(2, 3, 1)→(2, 0, 1) →(1, 0, 1) →(0, 0, 1).
[Paquin ’09: On a generalization of Christoffel words: epichristoffel words]
There exists an epichristoffel class having letter frequencies (p,q,r) if and only if
iterating the described process we obtain a triple with all 0’s and a 1.
Unique up to
changes of letters
a
b
a
a
(6, 3, 1) →(2, 3, 1) →(2, 0, 1) →(1, 0, 1) →(0, 0, 1).
Episturmian morphism:
ψa(a) = a;
ψa(x) = ax, if x ∈ A \ {a};
ψabaa(c) = ψaba(ac) = ψab(aac) = ψa(bababc) = abaabaabac
Directive sequence Δ
infinite

At most one right special
factor for each length
Example: Tribonacci
word
abacabaabacaba…
finite
(w) - epichristoffel classes - circular
episturmian words
One right circular special
factor for each length n  !!!
 …how many h-special?!

Example: abaabaabac
prefix of a conjugate of
Tribonacci word
c
a
b
a
a
b
a
a
a
b
a
b
a
a
(5, 3) →(2, 3) →(2, 1) →(1, 1) →(0, 1).
Episturmian morphism:
ψa(a) = a;
ψa(x) = ax, if x ∈ A \ {a};
ψabaa(b) = ψaba(ab) = ψab(aab) = ψa(babab) = abaabaab
a
a
b
a
(7, 2, 1) →(4, 2, 1) →(1, 2, 1) →(1, 0, 1) →(0, 0, 1).
ψaaba(c) = aabaaabaac
Δ=aaba
(abaabac)
(aabaaabaac)
Δ=aaba
Epichristoffel classes
(ab)
(a)
Δi the prefix of Δ up to the first occurrence of ai in Δ
Each letter ai induces a factorization in a set of factors
Xai={ψΔi aj (ai), for each j}
Xa= {a, ba, ca} then (aabaaabaac)
Xb= {aab, aaab, aacaab} then (aaabaacaab)
Xc={aabaabaac, …, … } then (aabaaabaac)
by coding…
up to changes of letters
Theorem: Each epichristoffel
class determines a reduction
tree, unique up to changes
of letters
Combinatorics on words
Binary alphabet
Finite Sturmian words
Theory of Automata
Minimization of DFA
K-ary alphabet
Finite episturmian words
Minimization of DMA
aabaaabaac
Minimization by a variant
of Hopcroft’s algorithm
Theorem: If the cyclic automaton is associated to an epichristoffel class the
algorithm has a unique execution.
(aabaaabaac)
(7, 2, 1) →(4, 2, 1) →(1, 2, 1) →(1, 0, 1) →(0, 0, 1)
10
7
2
1
4
2
1
1
1
2
1
1
1
1
1
1
THANK YOU!