> which is believed to be different from NP and hence a strict superset of P
“a strict superset of P” doesn’t really follow from that. P is also believed to be different from NP, and P is certainly not a strict superset of P.
Of course, I assume you just misspoke slightly, and that whatever it is that you actually meant to say, is correct.
E.g. maybe you meant to say “coNP is believed to both be strict superset of P (as is also believed about NP), and distinct from NP.”
I think it is believed that the intersection of NP and coNP is also a strict superset of P, but that this is believed with less confidence than that NP and coNP are distinct?
I imagine I’m not telling you anything you don’t already know, but for some reason I wrote the following parenthetical, and I’d rather leave it in this comment than delete it.
(If P=NP, then, as P=coP, then NP=P=coP=coNP , but this is considered unlikely.
It is also possible (in the sense of “no-one has yet found a way to prove otherwise” that coNP = NP without them being equal to P.
As a separate alternative, it is also possible (in the same sense) that they are distinct, but that their intersection is equal to P.
So, the possibilities:
P=NP=coNP,
P≠cocapNP=NP=coNP,
P=cocapNP≠NP,coNP,
All 4 are different,
(cocapNP is the intersection of NP and coNP)
Where the last of these is I think considered most likely?
)
“a strict superset of P” doesn’t really follow from that. P is also believed to be different from NP, and P is certainly not a strict superset of P.
Of course, I assume you just misspoke slightly, and that whatever it is that you actually meant to say, is correct.
E.g. maybe you meant to say “coNP is believed to both be strict superset of P (as is also believed about NP), and distinct from NP.”
I think it is believed that the intersection of NP and coNP is also a strict superset of P, but that this is believed with less confidence than that NP and coNP are distinct?
I imagine I’m not telling you anything you don’t already know, but for some reason I wrote the following parenthetical, and I’d rather leave it in this comment than delete it.
(If P=NP, then, as P=coP, then NP=P=coP=coNP , but this is considered unlikely.
It is also possible (in the sense of “no-one has yet found a way to prove otherwise” that coNP = NP without them being equal to P.
As a separate alternative, it is also possible (in the same sense) that they are distinct, but that their intersection is equal to P.
So, the possibilities: P=NP=coNP, P≠cocapNP=NP=coNP, P=cocapNP≠NP,coNP, All 4 are different, (cocapNP is the intersection of NP and coNP)
Where the last of these is I think considered most likely? )