Chomsky Normal Form in 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.

Table of contents.

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)

2. 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)

3. 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)

4. 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 ${\mathrm{A}}_1$, ${\mathrm{A}}_2$, ..., ${\mathrm{A}}_{\mathrm{k-2}}$ 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.
$A\Rightarrow u_1{A}_1\Rightarrow u_1{u}_2{A}_2\Rightarrow ...\Rightarrow u_1{u}_2...u_{k-2}{A}_{k-2}\Rightarrow u_1{u}_2...u_k$

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)

5. 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.