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					            Single Final State for NFA




Fall 2004              COMP 335          1
Any NFA can be converted

to an equivalent NFA

with a single final state



Fall 2004             COMP 335   2
            Example
                            NFA




                        Equivalent NFA




Fall 2004    COMP 335                    3
                       In General
            NFA




            Equivalent NFA


                                        Single
                                        final state
Fall 2004                    COMP 335             4
                  Extreme Case

       NFA without final state




                                   Add a final state
                                   Without transitions



Fall 2004               COMP 335                    5
              Properties of
            Regular Languages




Fall 2004          COMP 335     6
For regular languages                 and
we will prove that:

                 Union:

      Concatenation:

                  Star:                     Are regular
                                            Languages
               Reversal:

            Complement:
        Intersection:
Fall 2004                  COMP 335                       7
We say: Regular languages are closed under


                 Union:

      Concatenation:

                  Star:

               Reversal:

            Complement:
        Intersection:
Fall 2004                  COMP 335          8
Regular language             Regular language




            NFA                     NFA




 Single final state              Single final state
Fall 2004             COMP 335                        9
            Example




Fall 2004    COMP 335   10
            Union
NFA for




Fall 2004   COMP 335   11
            Example
NFA for




Fall 2004    COMP 335   12
            Concatenation

NFA for




Fall 2004       COMP 335    13
            Example

NFA for




Fall 2004    COMP 335   14
            Star Operation
NFA for




Fall 2004        COMP 335    15
            Example

NFA for




Fall 2004    COMP 335   16
                           Reverse

                                        NFA for




            1. Reverse all transitions

            2. Make initial state final state
               and vice versa
Fall 2004                    COMP 335             17
            Example




Fall 2004    COMP 335   18
                  Complement




    1. Take the DFA that accepts

    2. Make final states non-final,
       and vice-versa
Fall 2004              COMP 335       19
            Example




Fall 2004    COMP 335   20
                Intersection
    DeMorgan’s Law:

                                 regular

                                 regular

                                 regular

                                 regular

                                 regular
Fall 2004             COMP 335             21
                Example


            regular


            regular          regular




Fall 2004         COMP 335             22
            Regular Expressions




Fall 2004           COMP 335      23
            Regular Expressions
Regular expressions
describe regular languages



Example:

            describes the language



Fall 2004            COMP 335        24
            Recursive Definition
Primitive regular expressions:

Given regular expressions        and




                    Are regular expressions



Fall 2004            COMP 335                 25
                  Examples


  A regular expression:




 Not a regular expression:




Fall 2004            COMP 335   26
            Languages of Regular Expressions

            : language of regular expression



Example:




Fall 2004                   COMP 335           27
                 Definition

For primitive regular expressions :




Fall 2004            COMP 335         28
            Definition (continued)

For regular expressions         and




Fall 2004            COMP 335         29
                 Example
Regular expression:




Fall 2004             COMP 335   30
                 Example

Regular expression




Fall 2004            COMP 335   31
                 Example

Regular expression




Fall 2004            COMP 335   32
                   Example

Regular expression



            = {all strings with at least
                two consecutive 0}




Fall 2004            COMP 335              33
                  Example

Regular expression



            = { all strings without
                 two consecutive 0 }




Fall 2004            COMP 335          34
            Equivalent Regular Expressions

Definition:

      Regular expressions            and


      are equivalent if




Fall 2004                 COMP 335           35
                          Example
            = { all strings without
                 two consecutive 0 }




                                             and
                                        are equivalent
                                        Reg. expressions
Fall 2004                    COMP 335                      36
            Regular Expressions
                    and
             Regular Languages




Fall 2004           COMP 335      37
                   Theorem

    Languages
                                 Regular
    Generated by
                                 Languages
    Regular Expressions




Fall 2004             COMP 335               38
                    Theorem - Part 1

    Languages
                                          Regular
    Generated by
                                          Languages
    Regular Expressions


   1.       For any regular expression
            the language         is regular


Fall 2004                  COMP 335                   39
                     Theorem - Part 2

    Languages
                                              Regular
    Generated by
                                              Languages
    Regular Expressions


 2.         For any regular language     , there is
            a regular expression       with


Fall 2004                   COMP 335                      40
                   Proof - Part 1

1.     For any regular expression
        the language        is regular



        Proof by induction on the size of




Fall 2004                COMP 335           41
              Induction Basis
Primitive Regular Expressions:
    NFAs




                                 regular
                                 languages




Fall 2004           COMP 335            42
            Inductive Hypothesis

Assume for regular expressions     and
that      and        are regular languages




Fall 2004           COMP 335                 43
                 Inductive Step
We will prove:




                                 are regular
                                 Languages.




Fall 2004             COMP 335                 44
  By definition of regular expressions:




Fall 2004             COMP 335            45
  By inductive hypothesis we know:
         and        are regular languages



  We also know:
  Regular languages are closed under:
            Union
            Concatenation
            Star

Fall 2004               COMP 335            46
 Therefore:




                         Are regular
                         languages




Fall 2004     COMP 335                 47
And trivially:


                 is a regular language




Fall 2004        COMP 335                48
                   Proof – Part 2

2.     For any regular language       there is
       a regular expression        with




  Proof by construction of regular expression




Fall 2004               COMP 335                 49
Since        is regular, take an
NFA         that accepts it




              Single final state
Fall 2004               COMP 335   50
From , construct an equivalent
Generalized Transition Graph in which
transition labels are regular expressions



Example:




Fall 2004               COMP 335            51
Another Example:




Fall 2004          COMP 335   52
Reducing the states:




Fall 2004              COMP 335   53
Resulting Regular Expression:




Fall 2004           COMP 335    54
                   In General
Removing states:




Fall 2004             COMP 335   55
The final transition graph:




The resulting regular expression:




Fall 2004            COMP 335       56

				
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