-Automata

1. -Automata Ekaterina Mineev Edited by: Guy Lando

2. Today: 1 Introduction - notation - -Automata overview

3. Today: 1 Introduction - notation - -Automata overview 2 Nondeterministic models - Büchi acceptance - Muller acceptance - Rabin acceptance - Streett acceptance - parity condition

4. Today(cont.): 2.1 Equivalency of nondeterministic models

5. Today(cont.): 2.1 Equivalency of nondeterministic models 3 Deterministic models - Büchi condition - equivalency of deterministic* models

6. Today(cont.): 2.1 Equivalency of nondeterministic models 3 Deterministic models - Büchi condition - equivalency of deterministic* models 4 Some lower bound for transformations

7. Today(cont.): 2.1 Equivalency of nondeterministic models 3 Deterministic models - Büchi condition - equivalency of deterministic* models 4 Some lower bound for transformations 5 Weak acceptance conditions - Staiger-Wagner acceptance

8. Today(cont.): 2.1 Equivalency of nondeterministic models 3 Deterministic models - Büchi condition - equivalency of deterministic* models 4 Some lower bound for transformations 5 Weak acceptance conditions - Staiger-Wagner acceptance 6 Conclusion

9. Notation

10. Notation •  := {0, 1, 2, 3, …}

11. Notation •  := {0, 1, 2, 3, …} •  - finite alphabet

12. Notation •  := {0, 1, 2, 3, …} •  - finite alphabet • * - set of finite words over 

13. Notation •  := {0, 1, 2, 3, …} •  - finite alphabet • * - set of finite words over  •  - set of infinite words (-words) over 

14. Notation •  := {0, 1, 2, 3, …} •  - finite alphabet • * - set of finite words over  •  - set of infinite words (-words) over  • u, v, w – finite words

15. Notation •  := {0, 1, 2, 3, …} •  - finite alphabet • * - set of finite words over  •  - set of infinite words (-words) over  • u, v, w – finite words • , ,  - infinite words

16. Notation •  := {0, 1, 2, 3, …} •  - finite alphabet • * - set of finite words over  •  - set of infinite words (-words) over  • a, b, c – symbols in the alphabet • u, v, w – finite words • , ,  - infinite words •  = (0)(1)(2)…with (i)

17. Notation •  := {0, 1, 2, 3, …} •  - finite alphabet • * - set of finite words over  •  - set of infinite words (-words) over  • a, b, c – symbols in the alphabet • u, v, w – finite words • , ,  - infinite words •  = (0)(1)(2)…with (i) • ,  - runs of automata

18. Notation •  := {0, 1, 2, 3, …} •  - finite alphabet • * - set of finite words over  •  - set of infinite words (-words) over  • a, b, c – symbols in the alphabet • u, v, w – finite words • , ,  - infinite words •  = (0)(1)(2)…with (i) • ,  - runs of automata • -language – set of -words

19. Notation(cont.)

20. Notation(cont.) • ||a– number of occurrences of a in 

21. Notation(cont.) • ||a– number of occurrences of a in  • Occ() := {ai (i)=a}

22. Notation(cont.) • ||a– number of occurrences of a in  • Occ() := {ai (i)=a} • Inf () := {ai j>i (j)=a}

23. Notation(cont.) • ||a– number of occurrences of a in  • Occ() := {ai (i)=a} • Inf () := {ai j>i (j)=a} • 2M– powerset of a set M

24. Notation(cont.) • ||a– number of occurrences of a in  • Occ() := {ai (i)=a} • Inf () := {ai j>i (j)=a} • 2M– powerset of a set M • REG – class of regular languages

25. Notation(cont.) • ||a– number of occurrences of a in  • Occ() := {ai (i)=a} • Inf () := {ai j>i (j)=a} • 2M– powerset of a set M • REG – class of regular languages • L(A) := {*A accepts } - -language recognized by A

26. -Automata -Automaton is (Q, , , qI, Acc)

27. -Automata -Automaton is (Q, , , qI, Acc) • Q – finite set of states

28. -Automata -Automaton is (Q, , , qI, Acc) • Q – finite set of states •  - finite alphabet

29. -Automata -Automaton is (Q, , , qI, Acc) • Q – finite set of states •  - finite alphabet •  : Q  2Q/Q – state transition function

30. -Automata -Automaton is (Q, , , qI, Acc) • Q – finite set of states •  - finite alphabet •  : Q  2Q/Q – state transition function • qIQ – initial state

31. -Automata -Automaton is (Q, , , qI, Acc) • Q – finite set of states •  - finite alphabet •  : Q  2Q/Q – state transition function • qIQ – initial state • Acc – acceptance component

32. -Automata -Automaton is (Q, , , qI, Acc) • Q – finite set of states •  - finite alphabet •  : Q  2Q/Q – state transition function • qIQ – initial state • Acc – acceptance component can be given in different way!!!

33. -Automata – (notes 1-3) -Automaton is (Q, , , qI, Acc) • Q – finite set of states •  - finite alphabet •  : Q  2Q/Q – state transition function • qIQ– initial state • Acc– acceptance component can be given in different way!!! |A| = |Q| - size of automaton Acc size sometimes used too in complexity estimations

34. Büchi acceptance

35. Büchi acceptance -Automaton (Q, , , qI, FQ) is Büchi if Acc is Büchi acceptance:

36. Büchi acceptance – (notes 4) -Automaton (Q, , , qI, FQ) is Büchi if Acc is Büchi acceptance: A word * is accepted by A iff there exists a run  of A on  satisfying the condition: Inf()F  

37. Example 1 L := {{a, b}|  ends with aor with (ab)}

38. Büchi acceptance(cont.) – (notes 5) • is accepted by A iff some run of A on  visit some final state qF infinitely often, i.e. W(q0, q)W(q, q)

39. Büchi acceptance(cont.) – (notes 5-8**) • is accepted by A iff some run of A on  visit some final state qF infinitely often, i.e. W(q0, q)W(q, q) The Büchi recognizable -languages are the -languages of the form: L=ki=1 UiVi with k and Ui , Vi  REG for i=1, 2, 3, …

40. Büchi acceptance(cont.) The family of -languages is also called the -Kleene closure of the class of regular languages denoted -KC(REG)

41. Muller acceptance

42. Muller acceptance -Automaton (Q, , , qI, F 2Q) is Muller if Acc is Muller acceptance:

43. Muller acceptance – (notes 9) -Automaton (Q, , , qI, F 2Q) is Muller if Acc is Muller acceptance: A word * is accepted by A iff there exists a run  of A on  satisfying the condition: Inf()F

44. Example 2 L := {{a, b}|  ends with aor with (ab)} F = { {qa}, {qa,qb} }

45. Büchi and Muller automata Nondeterministic Büchi automata and nondeterministic Muller automata are equivalent in expressive power

46. Büchi and Muller automata Nondeterministic Büchi automata and nondeterministic Muller automata are equivalent in expressive power One direction is simple: F := { KQ | KF  }

47. Büchi and Muller automata – (notes 10-12**) Nondeterministic Büchi automata and nondeterministic Muller automata are equivalent in expressive power One direction is simple: F := { KQ | KF  } Second is complex and multiples states number exponentially

48. Rabin acceptance

49. Rabin acceptance -Automaton (Q, , , qI, ),  = {(E1, F1),…,(Ek, Fk)}with Ei, Fi  Q is Rabin if Acc is Rabin acceptance:

50. Rabin acceptance – (notes 13) -Automaton (Q, , , qI, ),  = {(E1, F1),…,(Ek, Fk)}with Ei, Fi  Q is Rabin if Acc is Rabin acceptance: A word * is accepted by A iff there exists a run  of A on  satisfying the condition: (E,F) . (Inf()E = )  (Inf()F  )