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Moore automata and e pichristoffel words

Moore automata and e pichristoffel words. G. Castiglione and M. Sciortino University of Palermo. ICTCS 2012, Varese sept 18-21. Outline. Combinatorics on words. Theory of Automata. Binary alphabet. Minimization of DFA. Finite Sturmian words. K- ary alphabet.

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Moore automata and e pichristoffel words

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

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

  3. Sturmianwords • Infinite words–binaryalphabet {a,b} n+1 factors of lenght n for each n 0; one right special factor for eachlengthn; (factorthatappearsfollowed by twodifferentlettersresp.) Example: Fibonacci word abaababaabaababaababaab…

  4. Christoffel word Given (p,q) coprime, the Christoffelword 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. The finite version infinite finite (w) -Christoffelclasses–circularSturmianwords • Exactly n+1 factors of lenght n for each n  0; • One right special factor for eachlength • Exactly n+1 circularfactorsof lenght n for eachnw-1; • One right circularspecial factor for eachlength n  w-2 Example: Fibonacci word abaababaabaababaababaab… Example: finite Fibonacci word abaababaabaababaababaab

  6. K-aryalphabet, Episturmianwords • 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-aryalphabet, episturmianwords • 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. The finite case epichristoffelclasses or circularepisturmianwords A finite word is an epichristoffel word if it is the image of a letter by an episturmianmorphism and if it is the smallest word of its conjugacyclass(epichristoffel class).

  9. Epichristoffelclass (6, 3, 1) →(2, 3, 1) →(2, 0, 1) →(1, 0, 1) →(0, 0, 1). [Paquin’09: On a generalization of Christoffel words: epichristoffelwords] There exists an epichristoffelclass 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

  10. Paquin’sconstruction b a a a (6, 3, 1) →(2, 3, 1) →(2, 0, 1) →(1, 0, 1) →(0, 0, 1). Episturmianmorphism: 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 sequenceΔ

  11. The finite version infinite finite (w) -epichristoffelclasses-circularepisturmianwords • At mostone right special factor for eachlength • One right circularspecial factor for eachlength n  !!! • …howmany h-special?! Example:Tribonacci word abacabaabacaba… Example:abaabaabac prefix of a conjugate of Tribonacci word

  12. Paquin’sconstruction (binary case) a a a b (5, 3) →(2, 3) →(2, 1) →(1, 1) →(0, 1). Episturmianmorphism: 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 classes a b a 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) Epichristoffel classes (aabaaabaac) (ab) Δ=aaba (a) Δithe prefix of Δ up to the first occurrence of aiin Δ Each letter ai induces a factorization in a set of factors Xai={ψΔiaj(ai), for eachj} 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 Minimization of DFA Finite Sturmian words K-ary alphabet Finite episturmian words Minimization of DMA

  17. Cyclic Moore automatonassociated to a circular word aabaaabaac

  18. Derivationtree Minimization by a variant of Hopcroft’salgorithm Theorem: If the cyclic automaton is associated to an epichristoffel class the algorithm has a unique execution.

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

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

  21. THANK YOU!

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