Chomsky and Greibach Normal Forms

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Chomsky and Greibach Normal Forms

Chomsky and Greibach Normal Forms– p.1/24

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Simplifying a CFG

It is often convenient to simplify CFG One of the simplest and most useful simplified forms of

CFG is called the Chomsky normal form

Another normal form usually used in algebraic specifications is Greibach normal form

Note the difference between grammar cleaning and simplification

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Simplifying a CFG

It is often convenient to simplify CFG

One of the simplest and most useful simplified forms of CFG is called the Chomsky normal form

Another normal form usually used in algebraic specifications is Greibach normal form

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Simplifying a CFG

It is often convenient to simplify CFG

One of the simplest and most useful simplified forms of CFG is called the Chomsky normal form

Another normal form usually used in algebraic specifications is Greibach normal form

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Simplifying a CFG

It is often convenient to simplify CFG

One of the simplest and most useful simplified forms of CFG is called the Chomsky normal form

Another normal form usually used in algebraic specifications is Greibach normal form

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Note

Normal forms are useful when more advanced topics in

computation theory are approached, as we shall see further

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Definition

A context-free grammar is in Chomsky normal form if every rule is of the form:

where

is a terminal,

are nonterminals, and

may not be the start variable (the axiom)

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Note

The rule

, where is the start variable, is not ex-

cluded from a CFG in Chomsky normal form.

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Theorem 2.9

Any context-free language is generated by a context-free grammar in Chomsky normal form.

Proof idea:

Show that any CFG can be converted into a CFG in Chomsky normal form

Conversion procedure has several stages where the rules that violate Chomsky normal form conditions are replaced with

equivalent rules that satisfy these conditions

Order of transformations: (1) add a new start variable, (2)

eliminate all -rules, (3) eliminate unit-rules, (4) convert other rules Check that the obtained CFG defines the same language

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Theorem 2.9

Any context-free language is generated by a context-free grammar in Chomsky normal form.

Proof idea:

Show that any CFG can be converted into a CFG in Chomsky normal form

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Theorem 2.9

Any context-free language is generated by a context-free grammar in Chomsky normal form.

Proof idea:

Show that any CFG can be converted into a CFG

in Chomsky normal form

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Theorem 2.9

Any context-free language is generated by a context-free grammar in Chomsky normal form.

Proof idea:

equivalent rules that satisfy these conditions Order of transformations: (1) add a new start variable, (2)

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Theorem 2.9

Any context-free language is generated by a context-free grammar in Chomsky normal form.

Proof idea:

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Theorem 2.9

Any context-free language is generated by a context-free grammar in Chomsky normal form.

Proof idea:

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Proof

Let

be the original CFG.

Step 1: add a new start symbol to , and the rule to

Note: this change guarantees that the start symbol of does not occur on the of any rule

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Proof

Let

be the original CFG.

Step 1: add a new start symbol to , and the rule to Note: this change guarantees that the start symbol of does not occur

on the of any rule

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Proof

Let

be the original CFG.

Step 1: add a new start symbol to , and the rule to Note: this change guarantees that the start symbol of

does not occur on the of any rule

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Step 2: eliminate -rules

Repeat

1. Eliminate the rule from where is not the start symbol 2. For each occurrence of on the of a rule, add a new rule to

with that occurrence of deleted

Example: replace by

; replace by

3. Replace the rule , (if it is present) by

unless the rule has been previously eliminated

until all

rules are eliminated

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Step 3: remove unit rules

Repeat

1. Remove a unit rule

2. For each rule , add the rule to , unless

was a unit rule previously removed

until all unit rules are eliminated

Note:

is a string of variables and terminals

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Convert all remaining rules

Repeat

1. Replace a rule , , where each , , is a variable or a terminal, by:

, , , where , , , are new variables

2. If replace any terminal with a new variable and add the rule

until no rules of the form

with

remain

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Convert all remaining rules

Repeat

until no rules of the form

with

remain

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Convert all remaining rules

Repeat

until no rules of the form

with

remain

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Example CFG conversion

Consider the grammar whose rules are:

Notation: symbols removed are green and those added are red.

After first step of transformation we get:

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Example CFG conversion

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Example CFG conversion

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Removing rules

Removing :

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Removing rules

Removing :

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Removing rules

Removing :

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Removing unit rule

Removing :

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Removing unit rule

Removing :

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Removing unit rule

Removing :

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More unit rules

Removing :

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More unit rules

Removing :

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More unit rules

Removing :

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Converting remaining rules

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Converting remaining rules

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Converting remaining rules

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Note

The conversion procedure produces several variables

along with several rules

. Since all these represent the same rule, we may simplify the result using a single variable

and a single rule

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Note

The conversion procedure produces several variables

along with several rules

. Since all these represent the same rule, we may simplify the result using a single variable

and a single rule

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Note

The conversion procedure produces several variables

along with several rules

. Since all these represent the same rule, we may simplify the result using a single variable

and a single rule

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Greibach Normal Form

A context-free grammar

is in

Greibach normal form if each rule

has the property:

,

and

.

Note: Greibach normal form provides a justifica- tion of operator prefix-notation usually employed in algebra.

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Greibach Normal Form

A context-free grammar

is in

Greibach normal form if each rule

has the property:

,

and

.

Note: Greibach normal form provides a justifica-

tion of operator prefix-notation usually employed

in algebra.

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Greibach Normal Form

A context-free grammar

is in

Greibach normal form if each rule

has the property:

,

and

.

Note: Greibach normal form provides a justifica- tion of operator prefix-notation usually employed in algebra.

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Greibach Theorem

Every CFL where

can be generated by a CFG in Greibach normal form.

Proof idea: Let

be a CFG generating . Assume that is in Chomsky normal form

Let

be an ordering of nonterminals.

Construct the Greibach normal form from Chomsky normal form

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Greibach Theorem

Every CFL where

can be generated by a CFG in Greibach normal form.

Proof idea: Let

Let

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Greibach Theorem

Every CFL where

can be generated by a CFG in Greibach normal form.

Proof idea: Let

Let

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Construction

1. Modify the rules in so that if then

2. Starting with and proceeding to this is done as follows:

(a) Assume that productions have been modified so that for

, only if

(b) If is a production with , generate a new set of productions substituting for the the rhs of each

production

(c) Repeating (b) at most times we obtain rules of the form

, (d) Replace rules

by removing left-recursive rules

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Construction

, only if

production

, (d) Replace rules

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Construction

, only if

production

, (d) Replace rules

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Removing left-recursion

Left-recursion can be eliminated by the following scheme:

If

are all

left recursive rules, and

are all remaining -rules then chose a new nonterminal, say

Add the new -rules

, Replace the -rules by

,

This construction preserve the language .

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Removing left-recursion

Left-recursion can be eliminated by the following scheme:

If

are all

Add the new -rules

,

This construction preserve the language .

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Removing left-recursion

Left-recursion can be eliminated by the following scheme:

If

are all

Add the new -rules

,

This construction preserve the language .

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21-1

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More on Greibach NF

See Introduction to Automata Theory, Languages,

and Computation, J.E, Hopcroft and J.D Ullman,

Addison-Wesley 1979, p. 94–96

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Example

Convert the CFG

where

into Greibach normal form.

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Example

Convert the CFG

where

into Greibach normal form.

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Example

Convert the CFG

where

into Greibach normal form.

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Solution

1. Step 1: ordering the rules: (Only rules violate ordering conditions, hence only rules need to be changed).

Following the procedure we replace

rules by:

2. Eliminating left-recursion we get:

,

3. All rules start with a terminal. We use them to replace

. This introduces the rules

4. Use production to make them start with a terminal

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Solution

rules by:

,

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Solution

rules by:

,