<p>$\newcommand{\cc}[1]{\mathsf{#1}}$Does a theorem along the following lines hold: If $g(n)$ is a little bigger than $f(n)$, then $\cc{NTIME}(g) \cap \cc{coNTIME}(g) \neq \cc{NTIME}(f) \cap \cc{coNTIME}(f)$?</p>

<p>It's easy to show that $\cc{NP} \cap \cc{coNP} \neq \cc{NEXP} \cap \cc{coNEXP}$, at least. Proof: Assume not. Then $$\cc{NEXP} \cap \cc{coNEXP} \subseteq \cc{NP} \cap \cc{coNP} \subseteq \cc{NP} \cup \cc{coNP} \subseteq \cc{NEXP} \cap \cc{coNEXP},$$ so $\cc{NP} = \cc{coNP}$, and hence (by padding) $\cc{NEXP} = \cc{coNEXP}$. But then our assumption implies that $\cc{NP} = \cc{NEXP}$, contradicting the nondeterministic time hierarchy theorem. QED.</p>

<p>But I don't even see how to separate $\cc{NP} \cap \cc{coNP}$ from $\cc{NSUBEXP} \cap \cc{coNSUBEXP}$, as diagonalization seems tricky in this setting.</p>

CC BY-SA 3.0
2015-07-01T05:43:51.317Z
The intersection is a semantic class, and [we don't know how to prove hierarchy theorems for semantic classes](http://cstheory.stackexchange.com/a/4461/).
Kaveh
<cc.complexity-theory><time-complexity><nondeterminism><time-hierarchy><hierarchy-theorems>
Hierarchy theorem for NTIME intersect coNTIME?
William Hoza
# Hierarchy theorem for NTIME intersect coNTIME?

(Asked by: username_0 on 2015-07-01)

$\newcommand{\cc}[1]{\mathsf{#1}}$Does a theorem along the following lines hold: If $g(n)$ is a little bigger than $f(n)$, then $\cc{NTIME}(g) \cap \cc{coNTIME}(g) \neq \cc{NTIME}(f) \cap \cc{coNTIME}(f)$?

It's easy to show that $\cc{NP} \cap \cc{coNP} \neq \cc{NEXP} \cap \cc{coNEXP}$, at least. Proof: Assume not. Then $$\cc{NEXP} \cap \cc{coNEXP} \subseteq \cc{NP} \cap \cc{coNP} \subseteq \cc{NP} \cup \cc{coNP} \subseteq \cc{NEXP} \cap \cc{coNEXP},$$ so $\cc{NP} = \cc{coNP}$, and hence (by padding) $\cc{NEXP} = \cc{coNEXP}$. But then our assumption implies that $\cc{NP} = \cc{NEXP}$, contradicting the nondeterministic time hierarchy theorem. QED.

But I don't even see how to separate $\cc{NP} \cap \cc{coNP}$ from $\cc{NSUBEXP} \cap \cc{coNSUBEXP}$, as diagonalization seems tricky in this setting.

**Tags:** cc.complexity-theory, time-complexity, nondeterminism, time-hierarchy, hierarchy-theorems

> **Comments:**
> **username_1**: The intersection is a semantic class, and [we don't know how to prove hierarchy theorems for semantic classes](http://cstheory.stackexchange.com/a/4461/).  

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