**Context Free Language-**

Before you go through this article, make sure that you have gone through the previous article on **Context Free Language**.

We have discussed-

- Context free language is generated using a context free grammar.
- Each context free language is accepted by a Pushdown automaton.

In this article, we will discuss a decision algorithm of CFL.

**Algorithm To Decide Whether CFL Is Finite Or Not-**

For a given CFG, there exists an algorithm to decide whether its language is finite or not.

**Step-01:**

Reduce the given grammar completely by-

- Eliminating ∈ productions
- Eliminating unit productions
- Eliminating useless productions

**Also Watch-** **How To Reduce Grammar?**

**Step-02:**

- Draw a directed graph whose nodes are variables of the given grammar.
- There exists an edge from node A to node B if there exists a production of the form A → αBβ.

Now, following 2 cases are possible-

**Case-01:**

- Directed graph contains a cycle.
- In this case, language of the given grammar is infinite.

**Case-02:**

- Directed graph does not contain any cycle.
- In this case, language of the given grammar is finite.

**Also Read-** **Algorithm To Decide Whether CFL Is Empty**

**PRACTICE PROBLEMS BASED ON DECIDING WHETHER CFL IS FINITE-**

**Problem-01:**

Check whether language of the following grammar is finite or not-

S → AB / a

A → BC / b

B → CC / c

**Solution-**

**Step-01:**

The given grammar is already completely reduced.

**Step-02:**

We will draw a directed graph whose nodes will be S , A , B , C.

Now,

- Due to the production S → AB, directed graph will have edges S → A and S → B.
- Due to the production A → BC, directed graph will have edges A → B and A → C.
- Due to the production B → CC, directed graph will have edge B → C.

The required directed graph is-

Clearly,

- The directed graph does not contain any cycle.
- Therefore, language of the given grammar is finite.

**Problem-02:**

Check whether language of the following grammar is finite or not-

S → XS / b

X → YZ

Y → ab

Z → XY

**Solution-**

**Step-01:**

The given grammar is already completely reduced.

**Step-02:**

We will draw a directed graph whose nodes will be S , X , Y , Z.

Now,

- Due to the production S → XS / b, directed graph will have edges S → X and S → S.
- Due to the production X → YZ, directed graph will have edges X → Y and X → Z.
- Due to the production Z → XY, directed graph will have edges Z → X and Z → Y.

The required directed graph is-

Clearly,

- The directed graph contain cycles.
- Therefore, language of the given grammar is infinite.

**Next Article-** **CYK Algorithm**

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