Grammars

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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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• 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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Examples

Grammar
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Another Example

• Grammar G

Derivations
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Language of a Grammar

• For a grammar
• with start variable

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Example

• For grammar G

Since
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A Convenient Notation

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Linear Grammars

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A Linear Grammar

• Grammars with
• at most one variable on 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 the 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

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

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Observation

• Regular grammars generate
• regular languages
• Examples

Grammar G
Grammar G
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Theorem

• A Language is regular
• if and only if
• there is a grammar such that