Single Final State for NFAs

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Single Final State for NFAs
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• Any NFA can be converted
• to an equivalent NFA
• with a single final state

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Example
NFA
Equivalent NFA
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In General
NFA
Equivalent NFA
Single final state
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Extreme Case

NFA without final state
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Properties of Regular Languages

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For regular languages and we will
prove that
Star
Reversal
Complement
Intersection
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We say Regular languages are closed under
Star
Reversal
Complement
Intersection
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(No Transcript)
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Example
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Union

• NFA for

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Example

NFA for
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Concatenation

• NFA for

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Example

• NFA for

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Star Operation

• NFA for

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Example

• NFA for

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Reverse
NFA for
1. Reverse all transitions
2. Make initial state final state and vice
versa
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Example
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Complement
1. Take the DFA that accepts
2. Make final states non-final, and
vice-versa
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Example
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Intersection
DeMorgans Law
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Example
regular
regular
regular
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Regular Expressions

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Regular Expressions

• Regular expressions
• describe regular languages
• Example
• describes the language

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Recursive Definition
Primitive regular expressions
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Examples
A regular expression
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Languages of Regular Expressions

• language of regular expression
• Example

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Definition

• For primitive regular expressions

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Definition (continued)

• For regular expressions and

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Example

• Regular expression

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Example

• Regular expression

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Example

• Regular expression

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Example

• Regular expression

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Example

• Regular expression

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Equivalent Regular Expressions

• Definition
• Regular expressions and
• are equivalent if

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Example

all strings without two consecutive 0
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Regular ExpressionsandRegular Languages

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Theorem
Languages Generated by Regular Expressions
Regular Languages
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Theorem - Part 1
Languages Generated by Regular Expressions
Regular Languages
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Theorem - Part 2
Languages Generated by Regular Expressions
Regular Languages
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Proof - Part 1
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Induction Basis

• Primitive Regular Expressions

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Inductive Hypothesis

• Assume
• for regular expressions and
• that
• and are regular languages

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Inductive Step

• We will prove

Are regular Languages
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• By definition of regular expressions

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By inductive hypothesis we know and
are regular languages
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• Therefore

Are regular languages
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• And trivially

is a regular language
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Proof Part 2
2. For any regular language there is
a regular expression with
Proof by construction of regular expression
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• Since is regular take the
• NFA that accepts it

Single final state
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• From construct the equivalent
• Generalized Transition Graph
• in which transition labels are regular
  expressions

Example
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• Another Example

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• Reducing the states

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• Resulting Regular Expression

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In General

• Removing states

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• The final transition graph

The resulting regular expression
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Theorem

• If for some DFA A
• Then there is a regular expression R such that

Proof (by induction on the size of R) Let As
states be 1, 2, , n for some integer
n Construct a collection of regular expressions
that describe progressively broader sets of paths
in DFA A
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• Let be the regex that represent the set of
  strings w such that w is the label of a path from
  state i to j in A, and that path has no
  intermediate state whos label is greater than k.
• Note beginning end points, i.e. i j are not
  intermediate so they can be greater than k

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Inductive Definition

• Basis k 0
• Since all states are gt 1, restriction on the
  paths is that the path must have no intermediate
  states at all.
• If i ? j An arc from i to j
• Depending of the DFA A may be f, a or
• (If there are m symbols that label arcs from i to
  j)

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Inductive Definition

• If i j all loops from i to itself
• legal paths of length 0
• (If there are m symbols that label arcs from i to
  i)

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Inductive Definition

• Induction
• Suppose that there is a path from i to j that
  goes through no state with label higher than k.
  Consider two cases
• path does not go thru state k at all.

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• path goes thru state k at least once

- More than once
- once