How to convert FA to Right Linear Regular Grammar?

A grammar with at most one variable at the right side of production is called linear grammar.

Example 1

      S→aSb/ε

Example 2

      S→Ab

      A→aAb

      A→ε

Right Linear Grammar

A grammar is right linear grammar where all the non terminals in the right hand sides are at the right end.

For example,

      S->aS/ε

Algorithm for conversion

The algorithm to convert the finite automata (FA) to the right linear grammar is as follows −

Step 1 − Begin the process from the start state.

Step 2 − Repeat the process for each state.

Step 3 − Write the production as the output followed by the state on which the transition is going.

Step 4 − And at last, add € (epsilon) to end the derivation.

Example 1

Let’s consider a Finite automaton (FA) as given below −

Pick the start state A and output is on symbol ‘a’ going to state B

      A→aB

Now we will pick state B and then we will go on each output

      i.e B→aB

      B→bB

      B→ε

Therefore,

Final right linear grammar is as follows −

      A→aB

      B→aB/bB/ε

Example 2

Start with state A which is an initial state the output on symbol ‘a’ goes to A and B and the output on symbol ‘b’ goes to A. Now the production rule of right linear grammar is −

      A→aA/bA/aB

Now pick state B and then go on each output the right linear grammar is −

      B→aB/bB/ε

The final right linear grammar for the given finite automata is −

      A→aA/bA/aB

      B→aB/bB/ε