To find the regular expression (R.E.) for a finite automaton (FA) given its transition equations, we can follow these steps:
Step 1: Assign a variable to each state of the FA. In this case, we have four states: q1, q2, q3, and q4.
Step 2: Write down the equations for each state transition.
- q4 = q3 1
- q3 = q2 0
- q2 = q1 1 + q2 1 + q4 1
- q1 = ε + q1 0 + q3 0 + q4 0
Step 3: Eliminate ε-transitions (ε is the empty string) by substitution.
- q1 = 0 + q3 0 + q4 0
- q2 = q1 1 + q2 1 + q4 1
- q3 = q2 0
- q4 = q3 1
Step 4: Substitute the variables with regular expressions to eliminate the equations.
- q1 = 0 + (q2 0) 0 + (q3 1) 0
- q2 = (q1 1 + q2 1 + q4 1)
- q3 = (q2 0)
- q4 = (q3 1)
Step 5: Solve the resulting system of equations using the method of regular expression algebra.
To simplify the expressions, let's assign a new variable for each subexpression:
- A = (q2 0)
- B = (q3 1)
- C = (q1 1 + q2 1 + q4 1)
Now we can substitute these variables in the equations:
- q1 = 0 + A0 + B0
- q2 = C
- q3 = A
- q4 = B
Solving for q1:
q1 = 0 + A0 + B0
= 0 + A + B
= A + B
Solving for q2:
q2 = C
Solving for q3:
q3 = A
Solving for q4:
q4 = B
Finally, combining the solutions, we can express the regular expression for the FA as:
R.E. = (A + B)(C)*A*B
This is the regular expression that represents the language accepted by the given finite automaton.
