Normal Forms for Context-free Grammars

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Normal FormsforContext-free Grammars

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Chomsky Normal Form
All productions have form
and
variable
variable
terminal
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Examples
Chomsky Normal Form
Not 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 variable
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Introduce intermediate variable
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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
First remove Nullable variables Unit
productions
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For every symbol
Add production
In productions replace with
New variable
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Replace any production
with
New intermediate variables
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Theorem
For any context-free grammar there is an
equivalent grammar in Chomsky Normal Form
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Observations

• Chomsky normal forms are good
• for parsing and proving theorems

• It is very easy to find the Chomsky normal
• form of any context-free grammar

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Greinbach Normal Form
All productions have form
symbol
variables
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Examples
Greinbach Normal Form
Not 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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Observations

• Greinbach normal forms are very good
• for parsing

• It is hard to find the Greinbach normal
• form of any context-free grammar

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An ApplicationofChomsky Normal Forms
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The CYK Membership Algorithm
Input

• Grammar in Chomsky Normal Form

• String

Output
find if
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The Algorithm
Input example

• 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
Observation
The CYK algorithm can be easily converted to a
parser
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Pushdown AutomataPDAs

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Pushdown Automaton -- PDA
Input String
Stack
States
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Initial Stack Symbol
Stack
Stack
bottom
special symbol
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The States
Pop symbol
Input symbol
Push symbol
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input
stack
top
Replace
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input
stack
top
Push
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input
stack
top
Pop
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input
stack
top
No Change
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Non-Determinism
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NPDA Non-Deterministic PDA
Example
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Execution Example
Time 0
Input
Stack
current state
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Time 1
Input
Stack
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Time 2
Input
Stack
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Time 3
Input
Stack
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Time 4
Input
Stack
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Time 5
Input
Stack
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Time 6
Input
Stack
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Time 7
Input
Stack
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Time 8
Input
Stack
accept
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A string is accepted if

• All the input is consumed
• The last state is a final state

We do not care about the stack contents
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The input string is accepted by the NPDA
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In general,
is the language accepted by the NPDA
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Another NPDA example
NPDA
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Execution Example
Time 0
Input
Stack
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Time 1
Input
Stack
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Time 2
Input
Stack
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Time 3
Input
Stack
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Time 4
Input
Stack
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Time 5
Input
Stack
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Time 6
Input
Stack
accept