1 Pushdown Automata PDAs

2 Pushdown Automaton -- PDA Input String Stack States

3 Initial Stack Symbol Stack bottom special symbol

4 The States Input symbol Pop symbol Push symbol

5 top input stack

6 top input stack

7 top input stack

8 top input stack

9 Non-Determinism

10 NPDA: Non-Deterministic PDA Example:

11 Execution Example: Input current state Time 0 Stack

12 Input Time 1 Stack

13 Input Stack Time 2

14 Input Stack Time 3

15 Input Stack Time 4

16 Input Stack Time 5

17 Input Stack Time 6

18 Input Stack Time 7

19 Input Time 8 accept Stack

20 A string is accepted if: All the input is consumed The last state is a final state We do not care about the stack contents

21 The input string is accepted by the NPDA:

22 is the language accepted by the NPDA: In general,

23 Another NPDA example NPDA

24

25 Execution Example: Input Time 0 Stack

26 Input Time 1 Stack

27 Input Time 2 Stack

28 Input Time 3 Stack

29 Input Time 4 Stack

30 Input Time 5 Stack

31 Input Time 6 Stack accept

32 Pushing Strings Input symbol Pop symbol Push string

33 top input stack pushed string Example:

34 Another NPDA example NPDA

35

36 Execution: Time 0 Input current state Stack

37 Time 1 Input Stack

38 Time 3 Input Stack

39 Time 4 Input Stack

40 Time 5 Input Stack

41 Time 6 Input Stack

42 Time 7 Input Stack

43 Time 8 Input Stack accept

44 Formalities for NPDAs

45 Transition function:

46 Transition function:

47 Formal Definition Non-Deterministic Pushdown Automaton NPDA States Input alphabet Stack alphabet Transition function Final states Stack start symbol

48 Instantaneous Description Current state Remaining input Current stack contents

49 Input Stack Time 4: Example:Instantaneous Description

50 Input Stack Time 5: Example:Instantaneous Description

51 We write: (Time 4)(Time 5)

52 A computation:

53 For convenience we write:

54 Formal Definition Language of NPDA : Start state Final state

55 Example: NPDA : thus

56 NPDA : thus

57 NPDA : Therefore:

58 Converting Context-Free Grammars to NPDAs

59 An example grammar: What is the equivalent NPDA?

60 Grammar: NPDA:

61 The NPDA simulates leftmost derivations of the grammar L(grammar) = L(NPDA)

62 Grammar: A leftmost derivation:

63 NPDA execution: Input Stack Time 0 current state

64 Input Stack Time 1

65 Input Stack Time 2

66 Input Stack Time 3

67 Input Stack Time 4

68 Input Stack Time 5

69 Input Stack Time 6

70 Input Stack Time 7

71 Input Stack Time 8

72 Input Stack Time 9

73 Input Stack Time 10 accept

74 In general: Given any grammar We can construct a NPDA With

75 Constructing NPDA from grammar : For any production For any terminal

76 Grammar generates string if and only if NPDA accepts

77 Therefore: For any context-free language there is an NPDA that accepts the same language