Simplifications of Context-Free Grammars

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Simplifications of Context-Free Grammars

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A Substitution Rule
equivalent grammar
substitute B
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In general
substitute B
equivalent grammar
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Useless Productions
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never terminates...
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Useless Production
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Another grammar
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Not reachable from S
Useless Production
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In general
If
Then
is useful
is useless
Otherwise
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Removing Useless Productions
Example Grammar
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First
find all variables that produce strings with only
terminals
Round 1
Round 2
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Keep only the variables that produce terminal
symbols
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Second
Find all variables reachable from S
Dependency Graph
not reachable
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Keep only the variables reachable from S
Final Grammar
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Nullable Variables
Nullable Variable
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Removing Nullable Variables
Example Grammar
Nullable variable
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Substitute
Final Grammar
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Unit-Productions
Unit Production
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Removing Unit Productions
Observation
Is removed immediately
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Example Grammar
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substitute
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(No Transcript)
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substitute
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Remove repeated productions
Final grammar
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Removing All

• 1. Remove Nullable Variables
• 2. Remove Unit-Productions
• 3. Remove Useless Variables

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Normal FormsforContext-free Grammars

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Chomsky Normal Form
All productions have form
OR
variable
variable
terminal
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Examples
NOT
Chomsky Normal Form
Chomsky Normal Form
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Convertion to Chomsky Normal Form

• Example

NOT
Chomsky Normal Form
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Introduce variables for terminals
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Introduce intermediate variables
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Final grammar in Chomsky Normal Form
Initial grammar
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In general
From any context-free grammar Not in
Chomsky Normal Form
We can obtain An equivalent grammar
in Chomsky Normal Form
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The Procedure
Remove nullable variables unit productions
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For every symbol
Add new variable
Add production
In productions, replace with
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Replace any production
with
New variables
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Theorem
For any context-free grammar there is an
equivalent grammar in Chomsky Normal Form
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Greinbach Normal Form
All rules have form
variables
symbol
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Examples
NOT
Greinbach Normal Form
Greinbach Normal Form
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Conversion to Greinbach Normal Form
Greinbach Normal Form
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Theorem
For any context-free grammar
there is an equivalent grammar in Greinbach
Normal Form
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An ApplicationofChomsky Normal Forms
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The CYK Membership Algorithm
Inputs

• Grammar
• in Chomsky Normal Form

• String w

Output
find if
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Algorithm Input Examples
Grammar
String
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(No Transcript)
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(No Transcript)
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(No Transcript)
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(No Transcript)
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Therefore
Time Complexity
It can be easily converted to a parser