1 Converting NPDAs to Context-Free Grammars

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

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

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

5 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

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

7 Example of a NPDA in correct form:

8 The Grammar Construction In grammar variables have form Terminals are input symbols states stack symbol

9 For each transition We add production For all states

10 For each transition We add production For all states in the NPDA

11 Start Variable: Stack bottom symbol Start state final state

12 Example: Grammar production:

13 Example: Grammar productions:

14 Example: Grammar production:

15 Resulting Grammar:

16

17 Derivation of string

18 In general, in grammar if and only if is accepted by the NPDA

19 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

20 Deterministic PDAs (DPDAs)

21 DPDAs Allowed Transitions:

22 Not allowed: Not allowed even when is

23 Allowed:

24 Not allowed: Not allowed even when is

25 DPDA example

26 The language is deterministic context-free

27 In general: A language is deterministic context-free if there is some NPDA that accepts it

28 Example of Non-DPDA

29 Not allowed transitions for DPDAs

30 NPDAs have more power than DPDAs

31 We will show: There is which is not a context-free language deterministic context-free (accepted by a NPDA) (not accepted by a DPDA)

32 The language is:

33 The language is context-free Context-free grammar for : (there is an NPDA that accepts )

34 is not deterministic context-free Theorem: The language (there is no DPDA that accepts )

35 Proof: Assume for contradiction that is deterministic context free Therefore: There exists a DPDA that accepts

36 The DPDA with accepts

37 The language is not context-free (we will prove it at the next class) A fact:

38 Another fact: The language is not context-free

39 We will construct a NPDA that accepts

40 First, we modify : Modified Replace with

41 The NPDA that accepts Modified Original

42 is accepted by a NPDA Therefore: is context-free Contradiction! ( is not context-free)

43 Therefore: There is no DPDA that accepts End of Proof