Context-Free%20Languages

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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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A context-free grammar G
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(((( ))))
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A context-free grammar G
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A context-free grammar G
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() ((( ))) (( ))
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Definition of Context-Free Grammars

Any grammar G(V,T,S,P)
V variables T terminal symbols S start
variable
P production rules of the form
string of variables and terminals
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Definition of Context-Free Languages

• A language L is context-free
• if and only if
• there is a grammar G with L L(G)

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Derivation Order

Leftmost derivation
Rightmost derivation
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Derivation Trees

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Derivation Tree
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sentence
Derivation Tree
yield
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Partial Derivation Trees

Partial derivation tree
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sentential form

Partial derivation tree
yield
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Often, derivation order doesnt matter

Same Derivation Tree
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Ambiguity

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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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In general
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
rightmost
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Why do we care about ambiguity?
take
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The right solution
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Therefore
Ambiguity is bad for programming languages
We want to remove the ambiguity
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We fix the ambiguous grammar
New grammar
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Unique derivation tree
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The grammar
Is not ambiguous
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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