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Problem

This is a HW problem from CMU 15-455 (hw10, p1(a)), spring 17 by Ryan O'Donnell.

Assume $L \in {\sf RP \backslash ZPP}$. Define $$ L' = \left\{ (x, y) : \text{either $x \in L$ and $y \notin L$, or vice versa} \right\}, $$ Prove that $L' \notin {\sf RP} \cap {\sf coRP}$.

My Thoughts

The path of this proof is clear: show that $L' \in {\sf RP}$ (or ${\sf coRP}$) implies $L \in {\sf coRP}$, thus $L \in {\sf ZPP}$, contradicts the assumption. Here I consider to prove $L' \in {\sf RP} \implies L \in {\sf coRP}$ first, but the following points give a conclusion that stuck me:

  1. I don't know any property/information about the language $L$ itself, only a checker $A$ is given.
  2. To exploit checker $B$ of $L'$, two different strings are needed. Pairs of identical strings like $(x, x)$ are not in $L'$ thus will always be rejected, gives no information b/c rejection by $B$ can stand for both valid and invalid, although error probability can be arbitrarily small (but never be 0).
  3. If I want to ensure $x \notin L$ with no error, I must choose a string $y \in L$ since only acceptance by $B$ is "absolutely correct". But how can I find such $y$ in such a way that can work for every possible $L$ and for every time it runs?
  4. If I look for a random string $y$ from the street, the probability of success depends on $|L \cap \Sigma^n|$, but it's unknown.

Here the only way I can imagine is to "pre-calculate" such string $y \in L$, then $L' \in {\sf RP} \implies L \in {\sf coRP}$ follows directly. But how can I sure that such $y$ can always be found, within an acceptable time complexity? I'm not sure about this point, so I asked the question How to decide complexity affected by 'magic number'? It seems that this is an acceptable solution, but I'm not sure still. So I ask this question here, and I hope you you can point out some elegant solutions that I haven't found. Thanks!

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1 Answer 1

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To show $L'\notin \mathsf{RP \cup coRP}$ you can essentially follow the proof of $\mathsf{ZPP=RP\cap coRP}$.

Since $L\neq \Sigma^*,\varnothing$ (otherwise it's in ZPP), there exist $y_1\in L, y_2\notin L$. Note that you don't have to find $y_1,y_2$ at runtime, as you can simply hardcode them into your machine for $L'$. If $L'\in RP$, then you can simultaneously run $M_L(x)$ and $M_{L'}(x,y_1)$ where $M_L,M_{L'}$ are the RP machines for $L,L'$ correspondingly. If $L'\in coRP$ then you can simultaneously run $M_L(x), M_{\overline{L'}}\left(x,y_2\right)$.

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  • $\begingroup$ It seems that the part of your answer that goes further than what I described is "hardcode", do I understand it correctly? $\endgroup$
    – Marcythm
    Jul 29 at 7:08
  • $\begingroup$ I don't see a concrete suggestion for a ZPP machine for $L$ in your post, but if the only thing that was bothering you was how to find a fixed string outside/inside the language, then yes - this is addressed by hardcoding them into the machine. $\endgroup$
    – Ariel
    Jul 29 at 7:19
  • $\begingroup$ got it, thanks. (I didn't describe how to construct ZPP from RP + coRP b/c I thought this is obvious, but also thanks for pointing this out.) $\endgroup$
    – Marcythm
    Jul 29 at 9:11

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