Disjoint sets $A$ and $B$ are [computably inseparable](http://en.wikipedia.org/wiki/Recursively_inseparable_sets), if there
is no computable separating set, a computable set $C$ containing $A$ and disjoint from $B$. The
existence of c.e. computably inseparable sets is a fundamental
phenomenon explaining and unifing many arguments in computability theory.

Perhaps the easiest example of a c.e. computably inseparable pair of sets is the following, 
where $\varphi_e$ is the function computed by program $e$.
$$A=\{e\mid
\varphi_e(0)\downarrow =0\},\qquad B=\{e\mid
\varphi_e(0)\downarrow=1\}.$$
To see this, suppose $C$ is decidable, $A\subset C$ and $B\cap
C=\emptyset$. Since $C$ is computable, we may design via the recursion theorem a program $e$ such that
$\varphi_e(0)\downarrow=1$ just in case $e\in C$, and otherwise
$\varphi_e(0)=0$, and this gives an immediate contradiction. Another computably inseparable pair is the set of theorems versus the set of negations of theorems of PA, or your favorite consistent theory containing arithmetic. 

<b>Question.</b> What is the computational-complexity-theoretic
analogue of computable inseparability?

Specifically, if $P\neq NP$, then are there disjoint NP sets with
no separating set in $P$? 

This plainly fails if $P=NP$. Is there
another analogue of the phenomenon with other complexity classes?