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# Chomsky Normal Form in Theory of Computation

#### Theory of Computation It is much easier to work with context-free grammars if the given CFG is its normal form. In this article we discuss the Chomsky Normal Form.

1. Introduction.
2. Chomsky Normal Form.
3. Summary.
4. References.

## Introduction.

Normally it is much easier to work with a context-free grammar if the given context-free grammar is in a normal form.

Normal forms include:

• Chomsky Normal Form, whereby productions are in the form of A → BC or A → a, where A, B and C are variables and a is a terminal symbol.
• Greibach Normal Form, where productions are in the form of A → aα, where α ∈ V* and A ∈ V.

## Chomsky Normal Form.

Rules in a context free grammar G = (V,Σ, R, S) are of the form:

A → w where A is a variable and w is a string over the alphabet V ∪ Σ.
We will show that every context-free grammar G can be converted into a context-free grammar G′ such that L(G) =L(G′) and rules of G′ are of restricted form as defined below.

Definition 1: A context-free grammar G = (V,Σ, R, S) is said to be in chomsky normal form if every rule in R has one of the forms:

1. A → BC where A, B and C are elements of V, B ≠ S, and C ≠ S.
2. A → a where A is an element of V and a is also an element of Σ.
3. S → ϵ where S is the start variable.

Now, we should convince ourselves that for such a grammar G, R contains the rule number 3 iff ϵ ∈ L(G)

Theorem 1: Let Σ be an alphabet and L ⊆ Σ* be a context-free language. Then there exists a context-free grammar in chomsky normal for whose language is L.

Proof: L is a context-free language and there exists a grammar G = (V,Σ, R, S), such that L(G) = L. Now we transform G to a grammar in chomsky normal form in five steps:

1. We eliminate the start variable from the right side of the rules by defining G1= (V1,Σ, R1, S1), where S1 is the start variable, V1 = V ∪ {S1} and R1 = R ∪ {S1 → S}.
This grammar will have the following property:
• The start variable S1 doesn't occur on the right side of any rule in R1
• L(G1) =L(G)
1. An ϵ-rule is a rule in the form of A → ϵ where A is a variable that is not equal to the start variable.
In this step we eliminate all ϵ-rules from G1
Now we consider rules one by one. Let A → ϵ be an example of a rule where A ∈ V1 and A ≠ S1. We change G1 as follows:
• First we remove the rule A → ≠ from the current set R1.
• Then, for each rule in the current set R1 in the form of:
(a). B → A, add rule B → ≠, unless the rule has been deleted from R1. In this way we replace the two-step derivation B ⇒ A ⇒ ≠ by a one-step derivation B ⇒ ≠
(b). B → uAv, where u and v are strings, both of which are not empty. We add the rule B →uv to R1 now observe that we have replaced a two-step derivation B ⇒ uAv ⇒ uv with a single-step derivation B ⇒ uv.
(c). B → uAvAw, where u, v and w are all strings. We add the rules B → uvw, B → uAvw, and B → uvAw to R1. If u = v = w = ϵ and the rule B → ϵ has been deleted from R1, we don't add the rule B → ϵ
(d). We shall treat all rules where A occurs multiple times on the right-hand side in a similar way.

This process is repeated until all ϵ-rules are eliminated. Let R2 be the set of rules after all ϵ-rules are eliminated. We define G2 = (V2,Σ, R2, S2) where V2 = V1 and S2 = S1.

The properties of this grammar are:

• It's start variable S2 doesn't occur on the right-hand side of any rule in R2.
• R2 doesn't contain any ϵ-rule but may contain S2 → ϵ rule.
• L(G2) = L(G1) = L(G)
1. A unit-rule is in the form A → B where A and B are variables. In this step we eliminate all unit rules from G2. To do this we consider all rules .
Let A → B be an example of a rule where A and B are elements of V2. Knowing that B ≠ S2 we change G2 as follows:
• First we remove rule A → B from the current R2 set.
• For each rule in R2 set in the form of B → u where u ∈(V2 ∪ Σ)*, we add the rule A → u to the current set R2 unless it is a unit-rule that has been eliminated. This way, we replace a two-step derivation A ⇒ B ⇒ u with a one-step derivation A ⇒ u.
The process is repeated until all unit-rules are eliminated.
Let R3 be thes set of rules after all the rules have been eliminated. We therefore define G3 = (V3,Σ, R3, S3), where V3 = V2 and S3 = S2.
The above grammar has the following properties:
• The start variable S3 doesn't occur on the right side of any rule in R3.
• R3 doesn't contain any ϵ-rule but may contain S3 → ϵ rule.
• R3 doesn't contain any unit-rule
• L(G3) = L(G2) =L(G1) = L(G)
1. This step involves eliminating all rules with more than two symbols on the right side. For each rule in the set R3 in the form of A → u1u2, ..., uk,, where k ≥ 3 and each ${\mathrm{u}}_{\mathrm{i}}$ is an element of V3 ∪ Σ we change G3 as follows:
• First we remove the rule A → u1 u2, ..., ${\mathrm{u}}_{\mathrm{i}}$ from the current set R3.
• Next we add the following rules to set R3.

A → u1A1
A1 → u2A2
A2 → u3A3
.
.
.
Ak−3 → uk−2Ak−2
Ak−2 → uk−1uk

Where ${\mathrm{A}}_{1}$, ${\mathrm{A}}_{2}$, ..., ${\mathrm{A}}_{k-2}$ are new variables added to the current set V3.
In this way we replace the one-step derivation A ⇒ u1 u2, ..., ${\mathrm{u}}_{\mathrm{i}}$ by the (k-1)-step derivation.
$A⇒{u}_{1}{A}_{1}⇒{u}_{1}{u}_{2}{A}_{2}⇒...⇒{u}_{1}{u}_{2}...{u}_{k-2}{A}_{k-2}⇒{u}_{1}{u}_{2}...{u}_{k}$

Let R4 and V4 be the set of rules and variables respectively, when we eliminate all rules with more than two symbols on the right. We define G4 = (V4,Σ, R4, S4), where S4 = S3.
The above grammar has the following properties:

• The start variable S4 doesn't occur on the right side of any rule in R4.
• R4 doesn't contain any ϵ-rule but mat contain S3 → ϵ rule.
• R4 doesn't contain any unit-rule.
• R4 doesn't contain any rule with more than two symbols on the right hand side.
• L(G4) = L(G3) = L(G2) = L(G1) = L(G)
1. Now we eliminate all rules with the form A → ${u}_{1}{u}_{2}$ where u1 and u2 are not both variables.
For each rule in the current set R4 in the form of A → ${u}_{1}{u}_{2}$ where u1 and u2 are not both contained in V4 we modify G3 as follows:
• If u1 ∈ Σ and u2 ∈ V4, we replace the rule A → ${u}_{1}{u}_{2}$ in the current R4 set by the two rules A → ${U}_{1}{u}_{2}$ and ${U}_{1}\to {u}_{1}$ where ${U}_{1}$ is a new variable added to the current set V4. In this way we replace the one-step derivation A ⇒ u1u2 with a two-step derivation A ⇒ U1u2 ⇒ u1u2.
• If u1 ∈ V4 and u2 ∈ Σ, we replace the rule A → u1u2 in the current set R4 by the two rules A → ${u}_{1}{U}_{2}$ and ${u}_{2}\to {u}_{2}$ where ${U}_{2}$ is a new variable added to the current V4 set. This way we replace a one-step derivation A ⇒ u1u2 with a two-step derivation A ⇒ u1U2 ⇒ u1u2
• If u1 ∈ Σ, u2 ∈ Σ, and u1 ≠ u2, we replace A → ${u}_{1}{u}_{2}$ in the current R4 set by the three rules, A → ${U}_{1}$${U}_{2}$, ${U}_{1}$${u}_{2}$, and ${U}_{2}$${u}_{2}$ where U1 and U2 are new variables added to the current V4 set. In this way we replace the one-step derivation A ⇒ u1u2 by a three-step derivation A ⇒ U1U2 ⇒ u1U2 ⇒ u1u2.
• If u1 ∈ Σ, u2 ∈ Σ, and u1 = u2, we replace the rule A → u1u2 = u1u1 in the current R4 set by the two rules A → ${U}_{1}$${U}_{1}$ and ${U}_{1}$${u}_{1}$. where ${U}_{1}$ is a new variable added to the current V4 set. In this way we replace a one-step derivation A ⇒ u1u2 = u1u1 with a three-step derivation A ⇒ U1U1 ⇒ u1U1 ⇒ u1u1.

Let R5 be the set of rules and V5 the set of variables after the above step is completed. We define G5 = (V5,Σ, R5, S5) where S5 = S4.
The above grammar has the following properties:

• The start variable S5 doesn't occur on the right side of any rule in R5.
• R5 doesn't have any ϵ-rule but may have S5 → ϵ rule.
• R5 doesn't have any unit-rule.
• R5 doesn't have any rule with more than two symbols on the right side.
• R5 doesn't have any rule in the form of $A\to {u}_{1}{u}_{2}$ where ${u}_{1}$ and ${u}_{2}$ are not both V5 variables.
• L(G5) = L(G4) = L(G3) = L(G2) = L(G1) = L(G).

And since the G5 grammar is in Chomsky normal form we have completed our proof.

## Summary.

It is easier to work with CFGs if a given CFG is in its normal form, following this we have discussed the Chomsky normal form.

Chomsky Normal Form is a form whereby productions are in the form of A → BC or A → a, where A, B and C are variables and a is a terminal symbol.

#### Erick Lumunge

Erick is a passionate programmer with a computer science background who loves to learn about and use code to impact lives positively.

Chomsky Normal Form in Theory of Computation