Disjoint sets $A$ and $B$ are computably inseparable, 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.

Question. 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?

Disjoint sets $A$ and $B$ are computably inseparable, 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.

Question. 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 $P=NP$.

Edit. Mark Sapir points out that if we assume $\text{P}\neq\text{NP}\cap\text{Co-NP}$, then there is are some easy examples, namely, any set $L\in\text{NP}\cap\text{Co-NP}\setminus P$ together with its complement. In light of this (since this particular example won't help me with my intended purpose), let me modify the particular question to:

Question. Under some standard complexity theory hypothesis, such as $\text{P}\neq \text{NP}\cap\text{Co-NP}$, are there disjoint sets in NP with no separation in $\text{NP}\cap\text{Co-NP}$?

And are there other analogues of the phenomenon with other complexity classes?