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

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

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 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)

(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

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

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

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).

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

Paquin’s construction 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 Conjugate of a prefix of Tribonacci word

(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

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

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

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

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 Minimization of DFA K-ary alphabet Finite episturmian words Minimization of DMA

Cyclic Moore automaton associated to a circular word aabaaabaac

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.

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 7 2 1 4 2 1 1 1 2 1 1

Theorem: reduction tree and derivation tree are isomorphic!

THANK YOU!