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Normal Forms for CFGs | Theory of Computation - Computer Science Engineering (CSE) PDF Download

Part III. Normal Forms for CFGs

Standard form or normal forms have been developed for context free grammars. Any CFG can be converted to these forms.
The advantages of representing a CFG in this form are,
1. Complexity of the CFG will be less, and
2. It will be easier to implement the CFG.

Two commonly used normal forms for CFGs are,
1. Chomsky normal form (CNF), and
2. Greibach normal form (GNF).

4 Chomsky Normal Form (CNF)
A context free grammar, G is in Chomsky Normal Form, if every production is of the form,
A → a,
A → BC,
S → ε; for this production, S is the start symbol.

Thus the RHS of a production can be a single terminal, or two non-terminals. Also it can be ε, if LHS is the start symbol.

Example 1:
Following CFG is in Chomsky normal form (CNF):
S→ AB|b|ε,
A→ BC|a,
B→ b,
C→ c
where S is the start symbol.

Example 2:
Following CFG is not in Chomsky normal form (CNF):
S→ ASB|AB|b,
A→ BC|a,
A→ a,
B→ bb
where S is the start symbol.

The above is not in CNF because the productions,
S→ ASB,
B→ bb
do not satisfy the conditions of CNF.

4.1 Conversion to CNF
Every CFG can be reduced to CNF. Following examples show how this is done.

Example 1:
Consider the following CFG,
S→ bA|aB,
A→ bAA|aS|a,
B→ aBB|bS|b
where S is the start symbol.
Convert this CFG to CNF.

Consider the production,
S→ bA|aB
This is not in CNF. So replace ’b’ by non-terminal, P and ’a’ by non-terminal, Q. Thsi is by adding the productions,
P→ b, Q→ a.
,Then we get,
S→ P A|QB
P→ b
Q→ a

Now the CFG becomes,
S→ P A|QB
P→ b
Q→ a
A→ P AA|QS|a,
B→ QBB|P S|b

In the above the productions,
A→ P AA,
B→ QBB
are not in CNF. So replace AA by R and BB by T. this is done by adding the productions, R→ AA, T→ BB.
Then we get,
A→ P R,
B→ QT
R→ AA,
T→ BB

Now the CFG becomes,
S→ P A|QB
P→ b
Q→ a
A→ P R|QS|a,
B→ QT|P S|b
R→ AA,
T→ BB
where S is the start symbol.
Note that above CFG is in CNF.

Example 2:
Consider the following CFG,
S→ 1A|0B,
A→ 1AA|0S|0,
B→ 0BB|1
where S is the start symbol.
Convert this CFG to CNF.

Consider the production, S→ 1A|0B,
This is not in CNF. So replace 1 by P and 0 by Q, by adding the productions, P→ 1, Q→ 0,
we get,
S→ P A|QB
P→ 1
Q→ 0

Now the CFG is,
S→ P A|QB
P→ 1
Q→ 0
A→ P AA|QS|0,
B→ QBB|1

In the above CFG, the productions,
A→ P AA
B→ QBB
are not in CNF.
So replace AA by R and BB by T. Then we get,
A→ P R
B→ QT
R→ AA
T→ BB

Now the CFG is,
S→ P A|QB
P→ 1
Q→ 0
A→ P R|QS|0,
B→ QT|1
R→ AA
T→ BB
where S is the start symbol.
Above CFG is in CNF.

Example 3:
Consider the following CFG,
S→ a|b|CSS,
where S is the start symbol.
Convert this CFG to CNF.

In the above, the production, S→ CSS,
is not in CNF. So replace SS by P, we get,
S→ CP
P→ SS

Now the CFG becomes,
S→ a|b|CP
P→ SS
where S is the start symbol.
Above CFG is in CNF.

Example 4:
Consider the following CFG,
S→ abSb|a|aAb,
A→ bS|aAAb,
where S is the start symbol.
Convert this CFG to CNF.

Consider the production, S→ abSb|aAb is not in CNF.
So replace Ab by P , we get,
S→ abSb|aP
P→ Ab

Now the grammar becomes,
S→ abSb|a|aP
P→ Ab
A→ bS|aAP

Now replace Sb by R, we get,
S→ abR|a|aP
P→ Ab
A→ bS|aAP
R→ Sb

Now replace bR with T, we get
S→ aT|a|aP
P→ Ab
A→ bS|aAP
R→ Sb
T→ bR

Now replace aA with U, we get,
S→ aT|a|aP
P→ Ab
A→ bS|UP
R→ Sb
T→ bR
U→ aA

Now replace a with V and b with W, we get,
S→ V T|a|V P
P→ AW

 A→ W S|UP
R→ SW
T→ W R
U→ V A
V→ a
W→ b
where S is the start symbol.
The above CFG is in CNF.

Exercises:
Convert the following CFGs to CNF.
1.
S→ bA|aB
A→ bAA|aS|a
B→ aBB|bS|b
where S is the start symbol.

2.
S→ aAD
A→ aB|bAB
B→ b
D→ d
where S is the start symbol.

3.
S→ aAbB
A→ aA|a
B→ bB|b
where S is the start symbol.

4.
S→ aAbB
A→ Ab|b
B→ Ba|a
where S is the start symbol.

5.
S→ aA|bB
A→ bAA|a
B→ BBa|b
where S is the start symbol.

5.
S→ abSb|ab
where S is the start symbol.

5 Greibach Normal Form (GNF)
A context free grammar, G is in Greibach nromal form, if every production is of the form,
A→ aX
S→ ε, for this production, S should be start symbol.
where a is a single terminal,
X is a set of 0 or more non-terminals.

Example 1:
The following CFG is in Greibach normal form (GNF):
S→ aAB|ε
A→ bC
B→ b
C→ c
where S is the start symbol.

Example 2:
The following CFG is not in Greibach normal form (GNF):
S→ aAB
A→ BbC
B→ b|ε
C→ c
where S is the start symbol.
The above is not in GNF becuause the productions, A→ BbC, B→ ε do not satisfy the conditions of GNF.

5.1 Conversion to GNF
Every CFG can be converted to Greibach normal form (GNF).
Following are the steps:
Step 1: Convert the given CFG to CNF.
 Rename all non-terminals by A1, A2, ......An, where A1 is the start symbol.

Step 2: Modify the grammar so that every production is of the form,
Ai→ aγ, or
Ai→ Ajγ, where j>i, and
γ is a set of 0 or more non-terminals.
This can be done by performing a number of substitutions.

Now if a production is of the form,
A→ Aγ, it is in left recursive form.
This left recursion can be eliminated by using a new variable, Z.
This is done as follows:
Let there be a production, A→ ADF|Aa|a|b|c
This can be written as,

A→ a|b|c|aZ|bZ|cZ
Z→ DF|a|DFZ|aZ

In a production leftmost symbol of RHS must be a terminal as,
An→ αγ,
otherwise if it is of the form,
An→ Amγ
then replace Am on RHS of production of Am−1 by replacement rule.

Thus in short, to convert a grammar to GNF, first start with grammar in CNF and then apply substitution rule and
eliminate left recursion.

Example 1:
Consider the CFG,
S→ AB|BC
A→ aB|bA|a
B→ bB|cC|b
C→ c
Convert this grammar to Greibach normal form (GNF).

The production, S→ AB|BC is not in GNF.
On applying the substitution rule we get,
S→ aBB|bAB|aB|bBC|cCC|bC

Now the CFG becomes,
S→ aBB|bAB|aB|bBC|cCC|bC
A→ aB|bA|a
B→ bB|cC|b
C→ c

Example 2:
Consider the CFG,
S→ abaSa|aba
Convert this grammar to Greibach normal form (GNF).

Replace a by A using the production, A→ a,
Replace b by B using the production, B→ b,
we get,
S→ aBASA|aBA
A→ a
B→ b
Now the above CFG is in GNF.

Example 3:
Consider the CFG,
S→ AB
A→ aA|bB|b
B→ b
Convert this grammar to Greibach normal form (GNF).

In the above CFG, the production,
S→ AB
is not in GNF.
So replace this by,
S→ aAB|bBB|bB

Now the CFG becomes,
S→ aAB|bBB|bB
A→ aA|bB|b
B→ b
The above CFG is in Greibach normal form.

Example 4:
Consider the CFG,
S→ abSb|aa
Convert this grammar to Greibach normal form (GNF).

The above production is not in GNF.
So introduce two productions,
A→ a
B→ b
Now the CFG becomes,
S→ aBSB|aA
A→ a
B→ b
Above CFG is in GNF.

Example 5:
Consider the CFG,
S→ aSa|bSb|aa|bb
Convert this grammar to Greibach normal form (GNF).

Introduce two productions,
A→ a
B→ b

Then the grammar becomes,
S→ aSP|bSQ|aP|bQ
A→ a
B→ b
Above CFG is in GNF.

Example 6:
Consider the CFG,
S→ AA|a
A→ SS|b
Convert this grammar to Greibach normal form (GNF).

Step 1:
Above CFG is in CNF.
Rename the variables, that is,
Rename S, A by A1, A2.
Then the grammar becomes,
A1→ A2A2|a
A2→ A1A1|b

Step 2: In this, productions must be of the form that
the RHS of a production must start with a terminal followed by nonterminals, or
the RHS of a production starts with a higher numbered variable.
The productions,
A1→ A2A2|a
A2→ b
satisfy this criterion.
Consider the production,
A2→ A1A1
Replace the first A1 on the RHS by its production.
Then,
A2→ A2A2A1|aA1
Now the CFG is,
A1→ A2A2|a
A2→ b
A2→ A2A2A1|aA1
That is,

A1→A2A2|a
A2→ A2A2A1|aA1|b
The production,A2→ A2A2A1|aA1|b contains left recursion.
To eliminate left recursion, this is written as,
A2→ aA1|b|aA1Z|bZ
Z→ A2A1|A2A1Z
Now the CFG is,
A1→ A2A2|a
A2→ aA1|b|aA1Z|bZ
Z→ A2A1|A2A1Z
Consider the production, A1→A2A2
Replace first A2 on the RHS with its production, we get,
A1→ aA1A2|bA2|aA1ZA2|bZA2
Now the CFG is
A1→ aA1A2|bA2|aA1ZA2|bZA2
A2→ aA1|b|aA1Z|bZ
Z→ A2A1|A2A1Z

Consider the production, Z→ A2A1|A2A1Z
Replace first A2 on the RHS with its production,
Z→ aA1A1|aA1A1Z|bA1|bA1Z|aA1ZA1|aA1ZA1Z|bZA1|bZA1Z
Now the CFG is,

A1→ aA1A2|bA2|aA1ZA2|bZA2
A2→ aA1|b|aA1Z|bZ
Z→ aA1A1|aA1A1Z|bA1|bA1Z|aA1ZA1|aA1ZA1Z|bZA1|bZA1Z

Above CFG is in GNF.

Example 8:
Consider the CFG,
S→ AB
A→ BS|a
B→ SA|b
Convert this grammar to Greibach normal form (GNF).

Step 1: The given grammar is in Chomsky normal form.
Rename the variables, that is,
Rename S, A, B by A1, A2, A3.
Then the grammar becomes,
A1→ A2A3
A2→ A3A1|a
A3→ A1A2|b

Step 2: In this, productions must be of the form that
the RHS of a production must start with a terminal followed by nonterminals, or
the RHS of a production starts with a higher numbered variable.
The productions,
A1→ A2A3
A2→ A3A1|a
A3→ b
satisfy this criterion.
But the production,
A3→ A1A2 does not satisfy this.
By applying substitution rule, substitute for A1 as
A3→ A2A3A2

substitute for A2 as

 

A3→ A3A1A3A2|aA3A2
Now the CFG is,
A1→ A2,A3
A2→ A3A1|a
A3→ A3A1A3A2|aA3A2|b
Consider the production,
A3→ A3A1A3A2|aA3A2|b
Eliminating left recursion from the above production, we get,
A3→ aA3A2B3|bB3|aA3A2|b
B3→ A1A3A2B3|A1A3A2
Now we have the productions,
A1→ A2A3
A2→ A3A1|a
A3→ aA3A2B3|bB3|aA3A2|b
B3→ A1A3A2B3|A1A3A2
Now all the A3 productions start with terminals.
Using substitution rule, replace A3 in the RHS of the production for A2.
A2→ A3A1|a becomes,
A2→ aA3A2B3A1|bB3A1|aA3A2A1|bA1|a

Now the CFG is,
A1→ A2A3
A2→ aA3A2B3A1|bB3A1|aA3A2A1|bA1|a
A3→ aA3A2B3|bB3|aA3A2|b
B3→ A1A3A2B3|A1A3A2
Using substitution rule, replace A2 in the RHS of the production for A1.
A1→ aA3A2B3A1A3|bB3A1A3|aA3A2A1A3|bA1A3|aA3
Now the CFG is,
A1→ aA3A2B3A1A3|bB3A1A3|aA3A2A1A3|bA1A3|aA3
A2→ aA3A2B3A1|bB3A1|aA3A2A1|bA1|a
A3→ aA3A2B3|bB3|aA3A2|b
B3→ A1A3A2B3|A1A3A2
Using substitution rule, replace A1 in the RHS of the production for B3.
B3→aA3A2B3A1A3A3A2B3|aA3A2B3A1A3A3A2|bB3A1A3A3A2B3|bB3A1A3A3A2|aA3A2A1A3A3A2B3|aA3A2A1A3A3A2|bA1A3A3A2B3|bA1A3A3A2|aA3A3A2B3|aA3A3A2

Now the CFG is,
A1→ aA3A2B3A1A3|bB3A1A3|aA3A2A1A3|bA1A3|aA3
A2→ aA3A2B3A1|bB3A1|aA3A2A1|bA1|a
A3→ aA3A2B3|bB3|aA3A2|b

B3 →aA3A2B3A1A3A3A2B3|aA3A2B3A1A3A3A2|bB3A1A3A3A2B3|bB3A1A3A3A2|aA3A2A1A3A3A2B3|aA3A2A1A3A2A1A3A3A2|bA1A3A3A2B3|bA1A3A3A2|aA3A3A2B3|aA3A3A2
Above CFG is in Greibach normal form.

Exercises:
1.
Consider the CFG,
A→ aBD|bDB|c|AB|AD
Convert this grammar to Greibach normal form (GNF).

2.
Consider the CFG,
S→ AB
A→ BS|b
B→ SA|a
Convert this grammar to Greibach normal form (GNF).

3.
Consider the CFG,
E→ E + T|T
T→ T ∗ F|F
F→ (E)|a
Convert this grammar to Greibach normal form (GNF).

4.
Consider the CFG,
S→ Y Y |0
Y→ SS|1
Convert this grammar to Greibach normal form (GNF).

5.
Consider the CFG,
S→ XY
X→ Y S|1
Y→ SX|0
Convert this grammar to Greibach normal form (GNF).

6.
Consider the CFG,
S→ XY
X→ 0X|1Y |1
Y→ 1
Convert this grammar to Greibach normal form (GNF).

7.
Consider the CFG,
S→ 01S1|11
Convert this grammar to Greibach normal form (GNF).

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FAQs on Normal Forms for CFGs - Theory of Computation - Computer Science Engineering (CSE)

1. What are the different normal forms for Context-Free Grammars (CFGs)?
Ans. The different normal forms for CFGs are: - Chomsky Normal Form (CNF): In this normal form, all production rules have the form A -> BC or A -> a, where A, B, and C are non-terminal symbols, and a is a terminal symbol. The only exception is the start symbol, which can also have a production rule of the form S -> ε, where ε represents the empty string. - Greibach Normal Form (GNF): In this normal form, all production rules have the form A -> aγ, where A is a non-terminal symbol, a is a terminal symbol, and γ is a string of non-terminal symbols. The only exception is the start symbol, which can have a production rule of the form S -> ε. - Backus-Naur Form (BNF): BNF is a widely used notation for CFGs, where production rules are represented using angle brackets (<>), and non-terminal symbols are enclosed in angle brackets. For example, <expression> ::= <term> + <expression>. - Extended Backus-Naur Form (EBNF): EBNF extends BNF by allowing the use of additional symbols and constructs, such as square brackets ([...]) for optional elements, curly braces ({...}) for repetition, and parentheses for grouping. EBNF is commonly used in defining the syntax of programming languages. - Augmented Backus-Naur Form (ABNF): ABNF is an extension of BNF that includes additional features, such as defining the length and range of repetitions, specifying case insensitivity, and allowing for comments.
2. How does Chomsky Normal Form (CNF) simplify Context-Free Grammars?
Ans. Chomsky Normal Form (CNF) simplifies Context-Free Grammars by imposing two restrictions on the production rules: 1. All production rules must have exactly two non-terminal symbols or one terminal symbol on the right-hand side. This allows for a clear separation between non-terminal symbols and terminal symbols. 2. The start symbol should not appear on the right-hand side of any production rule, except for the start symbol itself in the case of an empty string production rule. By enforcing these restrictions, CNF eliminates ambiguity in CFGs and makes them more amenable to parsing algorithms. It simplifies the analysis and manipulation of CFGs, allowing for efficient parsing and grammar transformations.
3. How is Greibach Normal Form (GNF) different from Chomsky Normal Form (CNF)?
Ans. Greibach Normal Form (GNF) differs from Chomsky Normal Form (CNF) in the following ways: 1. In GNF, the right-hand side of the production rules can have a terminal symbol as the first character, followed by a string of non-terminal symbols. In CNF, the right-hand side can have either two non-terminal symbols or one terminal symbol. 2. GNF allows for the production of empty strings (ε) by the start symbol itself. In CNF, the production of an empty string is only allowed by the start symbol. 3. GNF is more permissive than CNF in terms of the structure of production rules, allowing for a wider range of grammatical structures. CNF imposes stricter rules to simplify the analysis and manipulation of CFGs. 4. CNF is more widely used in formal language theory and parsing algorithms, while GNF is less common but can be useful in certain applications or transformations of grammars.
4. How is Backus-Naur Form (BNF) used to represent Context-Free Grammars?
Ans. Backus-Naur Form (BNF) is a notation used to represent Context-Free Grammars (CFGs) in a concise and readable manner. In BNF, production rules are represented using angle brackets (<>), and non-terminal symbols are enclosed in angle brackets. For example, consider a CFG for arithmetic expressions: <expression> ::= <term> + <expression> | <term> - <expression> | <term> <term> ::= <factor> * <term> | <factor> / <term> | <factor> <factor> ::= ( <expression> ) | <number> In this BNF representation, each production rule is separated by a vertical bar (|), indicating alternative choices for deriving a non-terminal symbol. The symbols on the right-hand side can be either non-terminal symbols enclosed in angle brackets or terminal symbols (e.g., +, -, *, /, (, ), <number>). BNF provides a clear and concise way to specify the grammar rules for a language, making it easier to understand and analyze the syntax of a programming language or other formal languages.
5. What are the additional features provided by Extended Backus-Naur Form (EBNF)?
Ans. Extended Backus-Naur Form (EBNF) extends the notation of Backus-Naur Form (BNF) by introducing additional symbols and constructs, including: 1. Square brackets ([...]): Square brackets are used to denote optional elements in a production rule. For example, [optional] represents an optional element that can be present or absent in a derivation. 2. Curly braces ({...}): Curly braces are used to denote repetition of elements in a production rule. For example, {expression} represents zero or more repetitions of the non-terminal symbol "expression". 3. Parentheses: Parentheses are used for grouping elements in a production rule. They help establish the precedence and associativity of operators or expressions. EBNF allows for more expressive and compact grammar specifications, making it easier to define complex grammars with repetition, optional elements, and nested structures. It is commonly used in defining the syntax of programming languages and other formal languages.
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