Single Final State for NFAs and DFAs

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Single Final State for NFAs and DFAs
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Observation

• Any Finite Automaton (NFA or DFA)
• can be converted to an equivalent NFA
• with a single final state

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Example
NFA
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In General
NFA
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Extreme Case

NFA without final state
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Some Properties of Regular Languages

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Properties
For regular languages and we will
prove that
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We Say
Regular languages are closed under
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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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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 with at least 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
• 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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• Obtaining the final regular expression