Context-Free Languages

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Context-Free Languages

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Regular Languages
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Context-Free Languages
Regular Languages
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Context-Free Languages
Context-Free Grammars
Pushdown Automata
stack
automaton
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Context-Free Grammars

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Example

A context-free grammar
A derivation
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A context-free grammar
Another derivation
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(((( ))))
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Example
A context-free grammar
A derivation
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A context-free grammar
Another derivation
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Example
A context-free grammar
A derivation
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A context-free grammar
A derivation
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() ((( ))) (( ))
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Definition Context-Free Grammars
Grammar
Variables
Terminal symbols
Start variables
Productions of the form
is string of variables and terminals
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Definition Context-Free Languages

• A language is context-free
• if and only if
• there is a grammar with

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Derivation Order

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Derivation Trees

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Derivation Tree
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Derivation Tree
yield
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Partial Derivation Trees

Partial derivation tree
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Partial derivation tree
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sentential form

Partial derivation tree
yield
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Sometimes, derivation order doesnt matter
Leftmost

Rightmost
Same derivation tree
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Ambiguity

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leftmost derivation
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leftmost derivation
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The grammar
is ambiguous
string
has two derivation trees
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The grammar
is ambiguous
string
has two leftmost derivations
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Definition
A context-free grammar is ambiguous if
some string has two or
more derivation trees
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In other words
A context-free grammar is ambiguous if
some string has two or
more leftmost derivations
(or rightmost)
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Why do we care about ambiguity?
take
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Correct result
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• Ambiguity is bad for programming languages

• We want to remove ambiguity

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We fix the ambiguous grammar
New non-ambiguous grammar
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Unique derivation tree
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The grammar
is non-ambiguous
Every string has a unique
derivation tree
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Inherent Ambiguity

• Some context free languages
• have only ambiguous grammars

Example
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The string
has two derivation trees