I found a simple proof that NP in ZPP implies PH in ZPP and then an even simpler one.

Assume NP in ZPP. This implies NP in BPP so PH is also in BPP. So we need only show BPP in ZPP.

BPP is in ZPP

^{NP}follows directly by Lautemann's proof that BPP is in Σ

_{2}

^{P}or by the fact that BPP is in MA is in S

_{2}

^{P}is in ZPP

^{NP}. By assumption, BPP in ZPP

^{NP}implies BPP in ZPP

^{ZPP}= ZPP.

And this is even simpler.

ZPP = RP∩co-RP in NP∩co-NP. Σ

_{2}

^{P}= NP

^{NP}in NP

^{ZPP}(by assumption) in NP

^{NP∩co-NP}= NP in ZPP. You can get the higher levels of the hierarchy by an easy induction.

As can be seen from your second argument, you practically use that ZPP is in co-NP and thus NP is in co-NP, which implies (as you have sketched) that NP=PH.

ReplyDeleteYou can also say that NP is in ZPP implies that NP is in BPP (since ZPP is in BPP), so by Karp-Lipton theorem PH=Σ2, and so

ReplyDeletePH=Σ2=NP^NP which, by assumption is in ZPP^ZPP=ZPP.

NP in ZPP implies NP^NP in NP^ZPP but not directly that NP^ZPP in ZPP^ZPP.

DeleteIn proposition 1.3 in https://www.semanticscholar.org/paper/The-Satanic-Notations%3A-Counting-Classes-beyond-%23p-1-Hemaspaandra/c01bebe8da499eb6a585c36cde9a71a406db41e6 it is shown

ReplyDelete#.NP=#.P^NP iff NP=UP^NP iff NP=coNP.

NP in ZPP implies NP=coNP and so ZPP=NP=UP^NP=UP^ZPP holds and so is ZPP low for UP providing ZPP=UP?

Thank you.

In proposition 1.3 in https://dl.acm.org/doi/abs/10.1145/203610.203611 it is stated #.NP=#.P^NP iff NP=UP^NP iff NP=coNP.

ReplyDeleteNP in ZPP implies NP=coNP and so ZPP=NP=UP^NP=UP^ZPP holds and so is ZPP low for UP providing ZPP=UP?

Full question.

ReplyDeleteExplicitly I want to identify relation between UP and ZPP since we do not even know ZPP is in SPP although we believe it to be true.

In proposition 1.3 in https://dl.acm.org/doi/abs/10.1145/203610.203611 it is stated #.NP=#.P^NP iff NP=UP^NP iff NP=coNP.

Theorem 7.2 in On counting and approximation (springer.com) states span-P is approximable in BPP^NP.

NP in ZPP implies span-P is approximable in ZPP.

1. span-P is in #(P^NP) and soI wonder if NP=ZPP would we get span-P in #P and so by theorem 4.9 UP=NP=ZPP?

Ordinarily is ZPP low for NP? So we get:

2. NP in ZPP implies NP=coNP and so ZPP=NP=UP^NP=UP^ZPP holds and so is ZPP low for UP providing NP=ZPP=UP?

So could NP=ZPP have any implications to UP and ZPP.

Assume further UP=coUP and so UPH collapses to UP. We already assumed NP=ZPP holds. So UP is in ZPP and in NP.

3. At least would it say something about the lowness UP making ZPP^UP in UP?

4. coUP=UP in EP in NP=ZPP=coNP holds under assumptions. EP is known to be in SPP. Is EP known to be in UPH under the assumptions?