Moore automata and epichristoffel words

G. Castiglione and M. SciortinoUniversity of Palermo

ICTCS 2012, Varese sept 18-21

Outline

Combinatorics on words Theory of Automata

Binary alphabet

Finite Sturmian words

K-ary alphabet

Finite episturmian words Minimization of DMA

Minimization of DFA

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…

Christoffel wordGiven (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

Conjugate of standard words (particular prefixes of Sturmian words)

(5,3)

a b aa

bab

aa

baaba

ba

abab

aa b

The finite version

infinite finite (w) - Christoffel classes – circular

Sturmian words

Example: Fibonacci wordabaababaabaababaababaab…

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 nw-1;

One right circular special factor for each length n w-2

Example: finite Fibonacci wordabaababaabaababaababaab

Example: Tribonacci word over {a,b,c} abacabaabacaba…

K-ary alphabet, Episturmian words

Are closed under reversal and have at most one right special factor of each length.

3-special factor

K-ary alphabet, episturmian words

Example: Tribonacci word over {a,b,c}abacabaabacaba…

2-special factor

Are closed under reversal and have at most one right special factor of each length.

The finite caseepichristoffel 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).

Epichristoffel class

(6, 3, 1)→(2, 3, 1)→(2, 0, 1) →(1, 0, 1) →(0, 0, 1).

Unique up to changes of letters

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]

Paquin’s construction

(6, 3, 1) →(2, 3, 1) →(2, 0, 1) →(1, 0, 1) →(0, 0, 1).a b a a

Episturmian morphism:

ψa(a) = a;ψa(x) = ax, if x A \ {a};∈

ψabaa(c) = ψaba(ac) = ψab(aac) = ψa(bababc) = abaabaabac

Conjugate of a prefix of Tribonacci word

Directive sequence Δ

ab

a

a

ba

a

b

a

cExample: abaabaabacprefix of a conjugate of Tribonacci word

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 wordabacabaabacaba…

Paquin’s construction (binary case)

(5, 3) →(2, 3) →(2, 1) →(1, 1) →(0, 1).a b a a

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

(7, 2, 1) →(4, 2, 1) →(1, 2, 1) →(1, 0, 1) →(0, 0, 1).a a b a

ψaaba(c) = aabaaabaac

A factorization of epichristoffel classes

Δ=aaba

A factorization of epichristoffel class

Each letter ai induces a factorization in a set of factors Xai={ψΔi aj

(ai), for each j}

(aabaaabaac)

Δ=aaba

Δi the prefix of Δ up to the first occurrence of ai in Δ

Xb= {aab, aaab, aacaab} then (aaabaacaab)

Xa= {a, ba, ca} then (aabaaabaac)

Xc={aabaabaac, …, … } then (aabaaabaac)

(abaabac)

(ab)

(a)

Epichristoffel classes

by coding…up to changes of letters

Reduction tree

Theorem: Each epichristoffel class determines a reduction tree, unique up to changes

of letters

Outline

Combinatorics on words Theory of Automata

Binary alphabet

Finite Sturmian words

K-ary alphabet

Finite episturmian words Minimization of DMA

Minimization of DFA

Cyclic Moore automaton associated to a circular word

aabaaabaac

Derivation treeMinimization by a variant of Hopcroft’s algorithm

Theorem: If the cyclic automaton is associated to an epichristoffel class the algorithm has a unique execution.

Derivation tree

(7, 2, 1) →(4, 2, 1) →(1, 2, 1) →(1, 0, 1) →(0, 0, 1)

10

7 2 1

4 2 1 1 1

1 1

1 1

2 1 1

(aabaaabaac)

Theorem: reduction tree and derivation tree are isomorphic!

THANK YOU!