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Simplifications of Context-Free Grammars

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Simplifications of Context-Free Grammars

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Equivalent

grammar

Substitute

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Substitute

Equivalent

grammar

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In general:

Substitute

equivalent

grammar

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Nullable Variable:

Example:

Nullable variable

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Removing

Substitute

After we remove all the

all the nullable variables disappear

(except for the start variable)

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Unit Production:

(a single variable in both sides)

Example:

Unit Productions

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Removal of unit productions:

Substitute

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Unit productions of form

can be removed immediately

Remove

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Substitute

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Remove repeated productions

Final grammar

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Useless Production

Some derivations never terminate...

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Useless Production

Another grammar:

Not reachable from S

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In general:

If there is a derivation

consists of

terminals

Then variable is useful

Otherwise, variable is useless

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Productions

useless

Variables

useless

useless

useless

useless

useless

useless

A production is useless

if any of its variables is useless

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Removing Useless Variables and Productions

Example Grammar:

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First:

find all variables that can produce

strings with only terminals or

(possible useful variables)

Round 1:

(the right hand side of production

that has only terminals)

Round 2:

(the right hand side of a production

has terminals and

variables of previous round)

This process can be generalized

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Then, remove productions that use variables

other than

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Second:

Find all variables

reachable from

Use a Dependency Graph

where nodes are variables

unreachable

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Keep only the variables

reachable from S

Final Grammar

Contains only

useful variables

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Step 1: Remove Nullable Variables

Step 2: Remove Unit-Productions

Step 3: Remove Useless Variables

This sequence guarantees that

unwanted variables and productions

are removed

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Normal FormsforContext-free Grammars

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Each production has form:

or

variable

variable

terminal

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Examples:

Chomsky

Normal Form

Not Chomsky

Normal Form

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Example:

Not Chomsky

Normal Form

We will convert it to Chomsky Normal Form

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Introduce new variables for the terminals:

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Introduce new intermediate variable

to break first production:

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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

(which doesn’t produce )

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

(Useless variables optional)

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Then, for every symbol :

New variable:

Add production

In productions with length at least 2

replace with

Productions of form

do not need to change!

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Replace any production

with

New intermediate variables:

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Observations

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

- It is easy to find the Chomsky normal
- form for any context-free grammar
- (which doesn’t generate )

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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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Observations

- Greinbach normal forms are very good
- for parsing strings (better than Chomsky Normal Forms)

- However, it is difficult to find the
- Greinbach normal of a grammar

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