CDT314 FABER Formal Languages, Automata and Models of Computation Lecture 3

1. CDT314 FABER Formal Languages, Automata and Models of Computation Lecture 3 School of Innovation, Design and Engineering Mälardalen University 2012 1

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

3. Finite Automata FA (Finite State Machines) Based on C Busch, RPI, Models of Computation

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

12. Set of States

13. Input Alphabet

14. Initial State

15. Set of Final States

16. Transition Function

20. Transition Function

21. Extended Transition Function

24. Observation: There is a walk from to with the label

25. Recursive Definition

26. ( ) 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

27. String Acceptance • Definition: • A string w is accepted by DFA M if w drives M to a final state from the initial state. • Formally: M accepts w iff

28. abbbaa is NOT accepted

29. abba is accepted

30. Language Accepted by DFA • Take a DFA • Definition: • The language contains • all input strings accepted by • = strings that drive to a final state

31. Alphabet = Example accept

32. Alphabet = Another Example accept accept accept

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

34. Language rejected by Observation • Language accepted by

35. Alphabet = More Examples accept trap state

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

37. = { all strings without substring } Alphabet =

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

39. Alphabet = Example The language is regular

40. Nondeterministic Finite Automata NFA

41. 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

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

43. First Choice

44. First Choice

45. First Choice

46. First Choice “accept”

47. Second Choice

48. Second Choice

49. Second Choice No transition: the automaton hangs

50. Second Choice “reject”