Post on 02-Jan-2016
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!