Given an arbitrary language $L$ with an algorithm $A$ that places $L$ in $co-RP$ what other properties does the Algorithm $A$ need to have to so that $L$ is in $P$?
Considering $L$ is in $co-RP$ we have a guarantee that:
if $x \in L$ then $A$ accepts $x$ guaranteed $(P(A(x)=1) = 1)$
if $x \not\in L$ then $A$ accepts $x$ with a propability $f(n)$ $(P(A(x)=1) = f(n))$ with $n = |x|$ and $f(n) \geq 0$
What property does $f$ need to have so that $L$ is in $P$?
As $P \subseteq ZPP$ and $ZPP = RP \cap co-RP$ there needs to be a property to $f$ that places $L$ also in $RP$. As such there needs to be a way to construct an algorithm $B$ for $L$ that discards $x \not\in L$ guaranteed in polynomial time. Under what conditions for $A$ is this possible? For what $f$ is this the case? Is there any other property $A$ needs to have so that $L$ is in $P$?