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We know that at the end computation should be done by physical systems which follow laws of physics. I know there are some researches that study the phase transition phenomenon in physics and try to connect it with some properties in complexity theory (such as P and NP famous problem ). Just a quick review for example the phase transition happens from 2-SAT problem to 3-SAT problem. The first one is in P and the second one is NP-Complete.

My question is that: Is there any study that shows the mapping of Polynomial Hierarchy (PH) and multi-phase systems? Is there any mapping between PH-Complete problems and real physical system states? If so, are all levels of these hierarchy stable?

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    $\begingroup$ If there are PH-complete problems, then the polynomial hierarchy collapses. Proof: Consider a problem $P$ which is complete for PH. Since $P \in $ PH, it appears in some level $\Sigma^p_k$. Since any problem $Q \in $ PH reduces to $P$, it follows that $Q \in \Sigma^p_k$. $\endgroup$
    – Pål GD
    Feb 17, 2013 at 21:24
  • $\begingroup$ I forgot the conclusion. Hence PH $ \subseteq \Sigma^p_k$. $\endgroup$
    – Pål GD
    Feb 17, 2013 at 21:30
  • $\begingroup$ @PålGD: we know that If the polynomial hierarchy has any complete problems, then it has only finitely many distinct levels. considering this fact are there any similar analogies in physical systems? $\endgroup$
    – Reza
    Feb 17, 2013 at 22:18

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