1 … NPDAs continued

2 Pushing Strings Input symbol Pop symbol Push string

3 top input stack Push pushed string Example:

4 Another NPDA example NPDA

5 Time 0 Input current state Stack Execution Example:

6 Time 1 Input Stack

7 Time 3 Input Stack

8 Time 4 Input Stack

9 Time 5 Input Stack

10 Time 6 Input Stack

11 Time 7 Input Stack

12 Time 8 Input Stack accept

13 Formalities for NPDAs

14 Transition function:

15 Transition function:

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

17 Instantaneous Description Current state Remaining input Current stack contents

18 Input Stack Time 4: Example:Instantaneous Description

19 Input Stack Time 5: Example:Instantaneous Description

20 We write: Time 4 Time 5

21 A computation:

22 For convenience we write:

23 Formal Definition Language of NPDA : Initial state Final state

24 Example: NPDA :

25 NPDA :

26 NPDA : Therefore:

27 NPDAs Accept Context-Free Languages

28 Context-Free Languages (Grammars) Languages Accepted by NPDAs Theorem:

29 Context-Free Languages (Grammars) Languages Accepted by NPDAs Proof - Step 1: Convert any context-free grammar to a NPDA with:

30 Context-Free Languages (Grammars) Languages Accepted by NPDAs Proof - Step 2: Convert any NPDA to a context-free grammar with:

31 Converting Context-Free Grammars to NPDAs

32 An example grammar: What is the equivalent NPDA?

33 Grammar: NPDA:

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

35 Grammar: A leftmost derivation:

36 NPDA execution: Input Stack Time 0 current state

37 Input Stack Time 1

38 Input Stack Time 2

39 Input Stack Time 3

40 Input Stack Time 4

41 Input Stack Time 5

42 Input Stack Time 6

43 Input Stack Time 7

44 Input Stack Time 8

45 Input Stack Time 9

46 Input Stack Time 10 accept

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

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

49 Grammar generates string if and only if NPDA accepts

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

51 Converting NPDAs to Context-Free Grammars

52 For any NPDA we will construct a context-free grammar with

53 Intuition:The grammar simulates the machine A derivation in Grammar : Current configuration in NPDA

54 in NPDA Input processedStack contents terminalsvariables A derivation in Grammar :

55 Some Necessary Modifications First, we modify the NPDA: It has a single final state It empties the stack when it accepts the input Original NPDA Empty Stack

56 Second, we modify the NPDA transitions: all transitions will have form or

57 Example of a NPDA in correct form:

58 The Grammar Construction In grammar : Terminals: Input symbols of NPDA states Stack symbol Variables:

59 For each transition We add production

60 For each transition We add production For all states

61 Start Variable: Stack bottom symbol Start state final state

62 Example: Grammar production:

63 Example: Grammar productions:

64 Example: Grammar production:

65 Resulting Grammar:

66

67 Derivation of string

68 In general, in Grammar: if and only if is accepted by the NPDA

69 Explanation: By construction of Grammar: if and only if in the NPDA going from to the stack doesn’t change below and is removed from stack