CD5560 FABER Formal Languages, Automata and Models of Computation Lecture 3 Mälardalen University

1. CD5560 FABER Formal Languages, Automata and Models of Computation Lecture 3 Mälardalen University 2010

2. Content • Finite Automata, FA • Deterministic Finite Automata, DFA • Nondeterministic Automata NFA • NFA  DFA Equivalence

3. Finite Automata FA (Finite State Machines)

4. There is no formal general definition for "automaton". Instead, there are various kinds of automata, each with it's own formal definition. Generally, an automaton • has some form of input • has some form of output • has internal states, • may or may not have some form of storage • is hard-wired rather than programmable

5. Finite Automaton Input String Output Finite Automaton String

6. Input String Output “Accept” or “Reject” Finite Automaton Finite Accepter

7. FA as Directed Graph • Nodes = States • Edges = Transitions • An edge with several symbols is a short-hand for several edges:

8. Deterministic Finite Automata DFA

9. DFA • Deterministic • there is no element of choice • Finite • only a finite number of states and arcs • Acceptors • produce only a yes/no answer

10. Alphabet = Transition Graph abba -Finite Acceptor initial state final state “accept” transition state

11. Formally • For a DFA • Language accepted by : alphabet transition function initial state final states

12. Language rejected by Observation • Language accepted by

13. Regular Languages A language is regular if there is a DFA such that All regular languages form a language family

14. : set of states : input alphabet : transition function : initial state : set of final states Formal definitions • Deterministic Finite Accepter (DFA)

15. Input Aplhabet

16. Set of States

17. Initial State

18. Set of Final States

19. Transition Function

23. Transition Function

24. Extended Transition Function

28. Observation: There is a walk from to with label

29. Recursive Definition

30. ( ) d = * q , ab 0 ( ) d d = * ( q , a ), b 0 ( ( ( ) ) ) d d d l = * q , , a , b 0 ( ( ) ) d d = q , a , b 0 ( ) q d = q , b 2 1

31. Languages Accepted by DFAs • Take DFA • Definition: • The language contains • all input strings accepted by • = { strings that drive to a final state}

32. Alphabet = Example accept

33. Alphabet = Another Example accept accept accept

34. Formally • For a DFA • Language accepted by : alphabet transition function initial state final states

35. Language rejected by Observation • Language accepted by

36. Alphabet = More Examples accept trap state

37. Alphabet = = { all strings with prefix } accept

38. = { all strings without substring } Alphabet =

39. A language is regular if there is a DFA such that Regular Languages All regular languages form a language family

40. Alphabet = Example The language is regular

41. Nondeterministic Automata NFA

42. NFA • Nondeterministic • there is an element of choice: in a given state NFA can act on a given string in different ways. Several start/final states are allowed. -transitions are allowed. • Finite • only a finite number of states and arcs • Acceptors • produce only a yes/no answer

43. Nondeterministic Finite Accepter (NFA) Alphabet = Two choices

44. First Choice

45. First Choice

46. First Choice

47. First Choice “accept”

48. Second Choice

49. Second Choice

50. Second Choice No transition: the automaton hangs