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$RTIME(T(n))$ is defined as:

Every language $L$ for which there is a probabilistic $TM$ $M$ running in $T(n)$ time such that $x \in L ⇒ Pr[M(x) = 1] ≥ 2/3 $

$x \notin L ⇒ Pr[(x) = 1] = 0$

Where, $ZTIME(T(n))$ defined as:

All the languages $L$ s.t. there is an expected-time $O(T (n))$ $TM$ $M$ such that for every input $x$, whenever $M$ halts on $x$, the output $M(x)$ it produces is exactly $f_{L}(x)$.

Why does $ZTIME(T(n))$ defined over expected-time and $RTIME(T(n))$ defined over the running time?

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closed as off-topic by Emil Jeřábek, Jan Johannsen, Mohammad Al-Turkistany, Jeffε, András Salamon Feb 8 '17 at 12:00

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Because if you define Zero-error time as running in strictly $O(T(n))$, then you would just get deterministic $O(T(n))$-time. (Why? Because then every choice of random bits suffices, so just pick the string $0^{T(n)}$ and simulate the machine to get the right answer). In particular, if you define $ZPP$ as zero-error with running time strictly bounded by a polynomial you just get your good old class $ZPP=P$. Simply put, if you have randomness but zero error, then you have to allow for some leeway in computation time, or you won't get an interesting computational class.

By contrast, you are free to define $RP$ as operating in expected polynomial time instead of strictly bounded time, but it won't change anything; you just get the normal class $RP$ (Why? Excercise!)

Hence $ZPP$ is to $RP$ a bit like $P$ is to $NP$ because if you take $P$ and $NP$ and say

  • The running time is expected polynomial time instead of strictly polynomially bounded, and
  • The error is one-sided bounded by $\frac{1}{3}$ instead of one-sided unbounded

then you obtain the classes $ZPP$ and $RP$. According to Arora and Barak, that makes the theorem $ZPP=RP\cap coRP$ surprising, because the analogous statement, $P=NP\cap coNP$, is not known (or expected).

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