Reverse of a Regular Language

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Reverse of a Regular Language

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Theorem
The reverse of a regular language is a
regular language
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Proof
Since is regular, there is NFA that
accepts
Example
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Invert Transitions
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Make old initial state a final state
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Add a new initial state
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Resulting machine accepts
is regular
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Grammars

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Grammars

• Grammars express languages
• Example the English language

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• A derivation of the boy walks

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• A derivation of a dog runs

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• Language of the grammar

L a boy runs, a boy walks,
the boy runs, the boy walks,
a dog runs, a dog walks, the
dog runs, the dog walks
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Notation

Variable or Non-terminal
Terminal
Production rule
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Another Example

• Grammar
• Derivation of sentence

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• Grammar
• Derivation of sentence

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• Other derivations

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• Language of the grammar

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More Notation

• Grammar

Set of variables
Set of terminal symbols
Start variable
Set of Production rules
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Example

• Grammar

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More Notation

• Sentential Form
• A sentence that contains
• variables and terminals
• Example

Sentential Forms
sentence
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• We write
• Instead of

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• In general we write
• If

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• By default

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Example

Grammar
Derivations
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Example
Grammar
Derivations
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Another Grammar Example

• Grammar

Derivations
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More Derivations

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Language of a Grammar

• For a grammar
• with start variable

String of terminals
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Example

• For grammar

Since
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A Convenient Notation

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Linear Grammars

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Linear Grammars

• Grammars with
• at most one variable at the right side
• of a production
• Examples

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A Non-Linear Grammar

Grammar
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Another Linear Grammar

• Grammar

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Right-Linear Grammars

• All productions have form
• Example

or
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Left-Linear Grammars

• All productions have form
• Example

or
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Regular Grammars

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Regular Grammars

• A regular grammar is any
• right-linear or left-linear grammar
• Examples

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Observation

• Regular grammars generate regular languages
• Examples

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Regular Grammars GenerateRegular Languages

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Theorem
Languages Generated by Regular Grammars
Regular Languages
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Theorem - Part 1
Languages Generated by Regular Grammars
Regular Languages
Any regular grammar generates a regular language
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Theorem - Part 2
Languages Generated by Regular Grammars
Regular Languages
Any regular language is generated by a regular
grammar
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Proof Part 1
Languages Generated by Regular Grammars
Regular Languages
The language generated by any
regular grammar is regular
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The case of Right-Linear Grammars

• Let be a right-linear grammar
• We will prove is regular
• Proof idea We will construct NFA
• with

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• Grammar is right-linear

Example
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• Construct NFA such that
• every state is a grammar variable

special final state
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• Add edges for each production

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NFA
Grammar

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In General

• A right-linear grammar
• has variables
• and productions

or
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• We construct the NFA such that
• each variable corresponds to a node

special final state
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• For each production
• we add transitions and intermediate nodes

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• For each production
• we add transitions and intermediate nodes

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• Resulting NFA looks like this

It holds that
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The case of Left-Linear Grammars

• Let be a left-linear grammar
• We will prove is regular
• Proof idea
• We will construct a right-linear
• grammar with

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Since is left-linear grammar the
productions look like
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• Construct right-linear grammar

In
In
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Construct right-linear grammar
In
In
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• It is easy to see that
• Since is right-linear, we have

Regular Language
Regular Language
Regular Language
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Proof - Part 2
Languages Generated by Regular Grammars
Regular Languages
Any regular language is generated by
some regular grammar
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Any regular language is generated by
some regular grammar
Proof idea
Let be the NFA with .
Construct from
a regular grammar such that
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• Since is regular
• there is an NFA such that

Example
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• Convert to a right-linear grammar

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In General
For any transition
Add production
variable
terminal
variable
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For any final state
Add production
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• Since is right-linear grammar
• is also a regular grammar
• with