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Algorithms for hard problems Automata and tree automata

Algorithms for hard problems Automata and tree automata. Juris Viksna, 2013. Finite deterministic automata. initial state. accepting state. transition. state. [Adapted from P.Drineas]. Finite deterministic automata. Finite Automaton (FA). : set of states. : input alphabet.

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Algorithms for hard problems Automata and tree automata

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  1. Algorithms for hard problems Automata and tree automata Juris Viksna, 2013

  2. Finite deterministic automata initial state accepting state transition state [Adapted from P.Drineas]

  3. Finite deterministic automata Finite Automaton (FA) : set of states : input alphabet : transition function d: Q×S  Q : initial state : set of accepting states L(M) = set of all words accepted by M [Adapted from P.Drineas]

  4. Finite non-deterministic automata A word is accepted by NFA, if there exists an accepting path from the initial state to a final state [Adapted from P.Drineas]

  5. Finite non-deterministic automata Set of states, i.e. Input aplhabet, i.e. Transition function d: Q×(S)  P(Q) Initial state Accepting states L(M) = set of all words accepted by M [Adapted from P.Drineas]

  6. Some basic results • the class of languages accepted by NFAs with -transitions is the same as the class of languages accepted by NFAs without -transitions • the class of languages accepted by NFAs is the same as the class of languages accepted by DFAs 0,1 0,1 q3 1 q4 q1 q2 1 0,e Nondeterministic finite automaton M [Adapted from S.Yukita]

  7. Some basic results 0 q010 q110 q000 q100 0 0 1 0 1 1 0 0 1 0 1 q011 q111 q001 q101 1 1 Deterministic finite automaton equivalent to M 1 [Adapted from S.Yukita]

  8. Some basic results Nondeterministic finite automaton M [Adapted from R.Downey, M.Fellows]

  9. Some basic results Corresponding deterministic finite automaton M [Adapted from R.Downey, M.Fellows]

  10. Some basic results NDF with  transitions [Adapted from R.Downey, M.Fellows]

  11. Some basic results Corresponding NDF without  transitions [Adapted from R.Downey, M.Fellows]

  12. Regular expressions [Adapted from R.Downey, M.Fellows]

  13. Regular languages [Adapted from R.Downey, M.Fellows]

  14. Regular languages = languages accepted by DFA/NFA [Adapted from R.Downey, M.Fellows]

  15. Regular languages = languages accepted by DFA/NFA [Adapted from R.Downey, M.Fellows]

  16. Regular languages = languages accepted by DFA/NFA [Adapted from R.Downey, M.Fellows]

  17. Regular languages = languages accepted by DFA/NFA [Adapted from R.Downey, M.Fellows]

  18. Congruences [Adapted from R.Downey, M.Fellows]

  19. Myhill-Nerode theorem [Adapted from R.Downey, M.Fellows]

  20. Myhill-Nerode theorem [Adapted from R.Downey, M.Fellows]

  21. Myhill-Nerode theorem [Adapted from R.Downey, M.Fellows]

  22. Myhill-Nerode theorem [Adapted from R.Downey, M.Fellows]

  23. Myhill’s congruence [Adapted from R.Downey, M.Fellows]

  24. Pumping Lemma [Adapted from R.Downey, M.Fellows]

  25. Myhill’s congruence [Adapted from R.Downey, M.Fellows]

  26. Construction of automata [Adapted from R.Downey, M.Fellows]

  27. Construction of automata [Adapted from R.Downey, M.Fellows]

  28. Construction of automata [Adapted from R.Downey, M.Fellows]

  29. State minimization [Adapted from R.Downey, M.Fellows]

  30. State minimization [Adapted from R.Downey, M.Fellows]

  31. State minimization - example [Adapted from R.Downey, M.Fellows]

  32. Regular grammars A right regular grammar is a formal grammar (N, Σ, P, S) such that all the production rules in P are of one of the following forms: A → a - where A is a non-terminal in N and a is a terminal in Σ A → aB - where A and B are in N and a is in Σ A → ε - where A is in N and ε denotes the empty string, i.e. the string of length 0. In a left regular grammar all rules obey the forms: A → a - where A is a non-terminal in N and a is a terminal in Σ A → Ba - where A and B are in N and a is in Σ A → ε - where A is in N and ε is the empty string. Both right and left grammars generate regular languages

  33. Automata and parameterized algorithms [Adapted from J.Flum,M.Grohe]

  34. Tree automata [Adapted from R.Downey, M.Fellows]

  35. Tree automata [Adapted from R.Downey, M.Fellows]

  36. Tree automata [Adapted from R.Downey, M.Fellows]

  37. Tree automata [Adapted from R.Downey, M.Fellows]

  38. Tree automata [Adapted from R.Downey, M.Fellows]

  39. Tree automata [Adapted from R.Downey, M.Fellows]

  40. Tree grammars [Adapted from R.Downey, M.Fellows]

  41. Tree grammars [Adapted from R.Downey, M.Fellows]

  42. Tree grammars - example [Adapted from R.Downey, M.Fellows]

  43. Normalized tree grammars [Adapted from R.Downey, M.Fellows]

  44. Normalized tree grammars [Adapted from R.Downey, M.Fellows]

  45. Kleene’s theorem for trees [Adapted from R.Downey, M.Fellows]

  46. Kleene’s theorem for trees [Adapted from R.Downey, M.Fellows]

  47. Kleene’s theorem for trees [Adapted from R.Downey, M.Fellows]

  48. Regular tree expressions [Adapted from R.Downey, M.Fellows]

  49. Regular tree expressions [Adapted from R.Downey, M.Fellows]

  50. Regular tree expressions [Adapted from R.Downey, M.Fellows]

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