Solving Regular Expression Equation R = Q + RP

We are asked to find the unique solution for the regular expression equation $R = Q + RP$, given that P does not contain the empty string ($\epsilon$).

Equation Analysis

The given equation is:

We can use substitution to solve for R. Since P does not contain $\epsilon$, the term $RP^n$ will eventually become empty as n increases.

Step-by-Step Solution

1. Substitute R: Substitute the expression for R from the equation into itself: $R = Q + (Q + RP)P$
2. Expand: Distribute P: $R = Q + QP + RPP$
3. Continue Substitution: Substitute R again: $R = Q + QP + (Q + RP)PP$ $R = Q + QP + QPP + RPPP$
4. Generalize Pattern: Continuing this process, we get a series: $R = Q + QP + QPP + QPPP + \dots + Q P^{n-1} + R P^n$ $R = Q \sum_{i=0}^{n-1} P^i + R P^n$
5. Limit Consideration: Since P does not contain $\epsilon$, the term $R P^n$ approaches the empty set as $n$ approaches infinity. The infinite sum $\sum_{i=0}^{\infty} P^i$ is equivalent to the regular expression $P^*$.
6. Final Solution: Therefore, the equation simplifies to: $R = Q P^*$

This method shows that the unique solution to the equation $R = Q + RP$, given P does not contain $\epsilon$, is $R = QP^*$.