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					                NPDA’s Accept
            Context-Free Languages




Fall 2004            COMP 335        1
 Theorem:

            Context-Free               Languages
              Languages               Accepted by
             (Grammars)                 NPDAs




Fall 2004                  COMP 335                 2
 Proof - Step 1:

            Context-Free               Languages
              Languages               Accepted by
             (Grammars)                 NPDAs


    Convert any context-free grammar
    to a NPDA      with:


Fall 2004                  COMP 335                 3
 Proof - Step 2:

            Context-Free                  Languages
              Languages                  Accepted by
             (Grammars)                    NPDAs


    Convert any NPDA                  to a context-free
    grammar      with:


Fall 2004                  COMP 335                       4
                Proof - step 1

                 Converting
            Context-Free Grammars
                      to
                    NPDAs




Fall 2004           COMP 335        5
We will convert any context-free grammar

to an NPDA automaton


Such that:



            Simulates leftmost derivations of



Fall 2004                COMP 335               6
                    Leftmost derivation


                          Input               Stack
                      processed               contents
                                   leftmost variable

            Simulation of derivation              Stack

            Input




Fall 2004               COMP 335                          7
                    Leftmost derivation

                                   string of terminals




            Simulation of derivation           Stack

            Input



               end of input is reached
Fall 2004               COMP 335                         8
An example grammar:




 What is an equivalent NPDA?




Fall 2004          COMP 335    9
Grammar:




                       NPDA:




Fall 2004   COMP 335           10
Grammar:




A leftmost derivation:




Fall 2004           COMP 335   11
Derivation:

Input



  Time 0
                         Stack




Fall 2004     COMP 335           12
Derivation:

Input



  Time 0
                         Stack




Fall 2004     COMP 335           13
Derivation:

Input



  Time 1
                         Stack




Fall 2004     COMP 335           14
Derivation:

Input



  Time 2
                         Stack




Fall 2004     COMP 335           15
Derivation:

Input



  Time 3
                         Stack




Fall 2004     COMP 335           16
Derivation:

Input



  Time 4
                         Stack




Fall 2004     COMP 335           17
Derivation:

Input



  Time 5
                         Stack




Fall 2004     COMP 335           18
Derivation:

Input



  Time 6
                         Stack




Fall 2004     COMP 335           19
Derivation:

Input



  Time 7
                         Stack




Fall 2004     COMP 335           20
Derivation:

Input



  Time 8
                         Stack




Fall 2004     COMP 335           21
Derivation:

Input



  Time 9
                         Stack




                         accept


Fall 2004     COMP 335           22
 In general:

            Given any grammar

            We can construct a NPDA

            With




Fall 2004                 COMP 335    23
Constructing NPDA          from grammar     :



                                  For any terminal

  For any production




Fall 2004              COMP 335                 24
      Grammar           generates string

            if and only if

        NPDA             accepts




Fall 2004                    COMP 335      25
Therefore:


            For any context-free language
            there is a NPDA
            that accepts the same language



        Context-Free                 Languages
          Languages                 Accepted by
         (Grammars)                   NPDAs

Fall 2004                COMP 335                 26
                Proof - step 2

                 Converting
                    NPDAs
                      to
            Context-Free Grammars




Fall 2004           COMP 335        27
    For any NPDA


     we will construct a

     context-free grammar             with




Fall 2004                  COMP 335          28
Intuition: The grammar simulates the
           computation of the machine

A derivation in Grammar                   :

                  terminals        variables




            Input processed              Stack contents

              Current configuration in NPDA

Fall 2004                     COMP 335                    29
              Some Necessary Modifications

  Modify (if necessary) the NPDA so that:

  1) The stack is never empty

  2)           has a single final state and empties
            the stack when it accepts a string

  3) Has transitions in a special form:
       d(qi,a,A)={c1,c2,…,cn}, where ci=(qj,l) or ci=(qj,BC)


Fall 2004                        COMP 335                      30
Example of a NPDA in correct form:




Fall 2004          COMP 335          31
            The Grammar Construction
 In grammar       :
                                    Stack symbol

                      Variables:


                                       states



                      Terminals:
                      Input symbols of NPDA
Fall 2004                COMP 335                  32
    For each transition




    We add production




Fall 2004               COMP 335   33
             For each transition




            We add productions




            For all possible states
            in the automaton
Fall 2004                   COMP 335   34
                      Stack bottom symbol


    Start Variable:


            Start state              final state




Fall 2004                 COMP 335                 35
  Example:




Grammar production:
Fall 2004         COMP 335   36
  Example:




  Grammar productions:



Fall 2004           COMP 335   37
  Example:




Grammar production:

Fall 2004             COMP 335   38
Resulting Grammar:




Fall 2004            COMP 335   39
Fall 2004   COMP 335   40
            Derivation of string




Fall 2004                  COMP 335   41
            In general:




                          if and only if

            the NPDA goes from        to
            by reading string       and
            the stack doesn’t change below
            and then      is removed from stack
Fall 2004                     COMP 335            42
      Therefore:




                   if and only if


            is accepted by the NPDA




Fall 2004               COMP 335      43
Therefore:

            For any NPDA
            there is a context-free grammar
            that accepts the same language


       Context-Free                   Languages
         Languages                   Accepted by
        (Grammars)                     NPDAs


Fall 2004                 COMP 335                 44
            Deterministic PDA

                 DPDA




Fall 2004          COMP 335     45
            Deterministic PDA: DPDA
Allowed transitions:




               (deterministic choices)
Fall 2004              COMP 335          46
  Allowed transitions:




             (deterministic choices)
Fall 2004                COMP 335      47
 Not allowed:




            (non deterministic choices)
Fall 2004             COMP 335            48
            DPDA example




Fall 2004       COMP 335   49
        The language

            is deterministic context-free




Fall 2004                  COMP 335         50
Definition:
A language      is deterministic context-free
if there exists some DPDA that accepts it




Fall 2004            COMP 335                   51
            Example of Non-DPDA (NPDA)




Fall 2004              COMP 335          52
            Not allowed in DPDAs




Fall 2004                COMP 335   53
                  NPDAs

            Have More Power than

                  DPDAs



Fall 2004           COMP 335       54
        It holds that:

            Deterministic
                                       Context-Free
            Context-Free
                                        Languages
             Languages
                                          NPDAs
               (DPDA)


              Since every DPDA is also a NPDA




Fall 2004                   COMP 335                  55
      We will actually show:

            Deterministic
                                       Context-Free
            Context-Free
                                        Languages
             Languages
                                          (NPDA)
               (DPDA)



            We will show that there exists
            a context-free language which is not
            accepted by any DPDA
Fall 2004                   COMP 335                  56
The language is:




 We will show:

            •   is context-free

            •   is not deterministic context-free


Fall 2004                 COMP 335                  57
Language     is context-free

Context-free grammar for       :




Fall 2004          COMP 335        58
Theorem:

            The language

            is not deterministic context-free



            (there is no DPDA that accepts      )




Fall 2004                  COMP 335                 59
Proof: Assume for contradiction that


             is deterministic context free



 Therefore:

       there is a DPDA         that accepts


Fall 2004                COMP 335             60
DPDA        with


            accepts




                   accepts
Fall 2004              COMP 335   61
DPDA        with



Such a path exists because of the determinism




Fall 2004           COMP 335               62
 Fact 1:        The language
                is not context-free


                  Context-free languages


                    Regular languages



      (we will prove this at a later class using
      pumping lemma for context-free languages)
Fall 2004                 COMP 335                 63
 Fact 2:        The language
                is not context-free




            (we can prove this using pumping lemma
            for context-free languages)
Fall 2004                   COMP 335                 64
  We will construct a NPDA that accepts:




            which is a contradiction!


Fall 2004             COMP 335             65
Modify      Replace
            with




Fall 2004       COMP 335   66
The NPDA that accepts

       Connect final states of
       with final states of




Fall 2004               COMP 335   67
Since                           is accepted by a NPDA

            it is context-free


             Contradiction!



 (since                            is not context-free)



Fall 2004                     COMP 335                    68
Therefore:

             Not deterministic context free



             There is no DPDA that accepts




                      End of Proof
 Fall 2004                COMP 335            69

				
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