# How to show that if 0 < $\alpha$ < 1, RP$_\alpha$ = RP using reliability amplification [duplicate]

Let $$0 \le \alpha \le 1$$. The complexity classs $$\mathsf{RP}_\alpha$$ consists of all languages $$L$$ for which we can find a probabilistic polynomial time Turing machine which satisfies the following conditions:

• $$w \in L \Rightarrow \Pr[\text{accept}] \geq \alpha$$.
• $$w \notin L \Rightarrow \Pr[\text{accept}] = 0$$.

We define $$\mathsf{RP}$$ to be $$\mathsf{RP}_{1/2}$$.

1. How would I show that if $$0 < \alpha < 1$$ then $$\mathsf{RP}_\alpha = \mathsf{RP}$$ using reliability amplification?
2. How would I show that when $$\alpha = 1$$, $$\mathsf{RP}_\alpha = \mathsf{P}$$?
• This is a very standard exercise. You can see it worked out in lecture notes and textbooks. Apr 29 '21 at 6:43
• I believe you can solve question 2 on your own. Apr 29 '21 at 6:44