1. Moore automata and epichristoffel words
G. Castiglione and M. Sciortino
University of Palermo
ICTCS 2012, Varese sept 18-21

2. Outline Combinatorics on words Theory of Automata Binary alphabet
Finite Sturmian words
Minimization of DFA
K-ary alphabet
Finite episturmian words
Minimization of DMA

3. Sturmian words 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…

4. Christoffel word Example: (5,3) aabaabab
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)
Conjugate of standard words (particular prefixes of Sturmian words)

5. (w) - Christoffel classes – circular Sturmian words
The finite version
infinite
finite
(w) - Christoffel classes – circular Sturmian words
Exactly n+1 factors of lenght n for each n  0;
One right special factor for each length
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: Fibonacci word
abaababaabaababaababaab…
Example: finite Fibonacci word
abaababaabaababaababaab a b

6. K-ary alphabet, Episturmian words
Are closed under reversal and have at most one right special factor of each length.
Example: Tribonacci word over {a,b,c}
abacabaabacaba…
3-special factor

7. K-ary alphabet, episturmian words
Are closed under reversal and have at most one right special factor of each length.
Example: Tribonacci word over {a,b,c}
abacabaabacaba…
2-special factor

8. epichristoffel classes circular episturmian words
The finite case
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).

9. Unique up to changes of letters
Epichristoffel class
(6, 3, 1)
→(2, 3, 1)
→(2, 0, 1) →(1, 0, 1) →(0, 0, 1).
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.
[Paquin ’09: On a generalization of Christoffel words: epichristoffel words]
Unique up to changes of letters

10. Paquin’s constructionb 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
Conjugate of a prefix of Tribonacci word

11. (w) - epichristoffel classes - circular episturmian words
The finite version
infinite
finite
(w) - epichristoffel classes - circular episturmian words
At most one right special factor for each length
One right circular special factor for each length n  !!!
…how many h-special?!
Example: Tribonacci word
abacabaabacaba…
Example: abaabaabac
prefix of a conjugate of
Tribonacci word a b c

12. Paquin’s construction (binary case)
(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
Conjugate of a prefix of Fibonacci word

13. A factorization of epichristoffel classesb a
(7, 2, 1) →(4, 2, 1) →(1, 2, 1) →(1, 0, 1) →(0, 0, 1).
ψaaba(c) = aabaaabaac
Δ=aaba

14. A factorization of epichristoffel class
(abaabac)
(ab)
(a)
Epichristoffel classes
(aabaaabaac)
Δ=aaba
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

15. Reduction tree
Theorem: Each epichristoffel class determines a reduction tree, unique up to changes of letters

16. Outline Combinatorics on words Theory of Automata Binary alphabet
Finite Sturmian words
Minimization of DFA
K-ary alphabet
Finite episturmian words
Minimization of DMA

17. Cyclic Moore automaton associated to a circular word
aabaaabaac

18. Derivation tree
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.

19. Derivation tree (aabaaabaac) 10 7 2 1 4 2 1 1 1 2 1 1
(7, 2, 1) →(4, 2, 1) →(1, 2, 1) →(1, 0, 1) →(0, 0, 1) 10

20. Theorem: reduction tree and derivation tree are isomorphic!

21. THANK YOU!