Proving the Riemann Hypothesis without disclosing anything apart from you confirmed it

Consider the adhering to assertion from Scott Aaronson's blog :

Supposing you do confirm the Riemann. Theory, it's feasible to encourage. a person of that, without. disclosing anything apart from the reality. that you confirmed it. It's additionally feasible. to write the evidence down in such a means. that somebody else can validate it,. with really high self-confidence, having just. seen 10 or 20 littles the evidence.

Can any person clarify where this outcome originates from?

2019-05-07 13:40:24
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Answers: 3

The principle behind it is Zero knowledge proof. Wikipedia has an excellent write-up concerning it.

The comparable inquiry was asked in paper "How to Prove a Theorem So No One Else Can Claim It" by M. Blum. Additionally this write-up reviews the inquiry.

2019-05-09 09:04:57

The proper solution has actually currently been offered by Akhil Mathew in the remarks over.

The subject comes from the area of intricacy concept in computer science. In intricacy concept, there exists a fascinating principle for the trouble of determining whether an offered word comes from an offered official language or otherwise : interactive proof systems. These systems design the communication in between source - minimal verifier (claim, you or me) and also an almighty prover (claim, your much, much smarter older sis). The objective of the communication is that the prover encourages the verifier from the reality that the offered word is or is not a component of the language such that:

  • virtually undoubtedly, the verfier can just be encouraged from truth solution and also
  • the verifier can just be misleaded to think the contrary within a really tiny margin of mistake.

There are a lot of academic outcomes relative to these interactive systems. These resuts include the follwoing 2 declarations (offered informally):

  • every little thing that can be confirmed by such an interactive protocoll can additionally be confirmed making use of an interactive protocoll that encourages the verifier (within tiny margin of mistake) yet it does not disclose any kind of details concerning the evidence itself. (Zero-Knowledge-Proof, "Everything conclusive is conclusive in absolutely no - expertise" by Ben - Or et alia, 1988)
  • every evidence can be revised such that examination of simply a constant variety of little bits from this evidence encourages the verifier within just a tiny margin of mistake (PCP-Theorem, "Proof confirmation and also the solidity of approximation troubles" by Arora et alia, 1992, and also a variety of various other documents)

Both of these principles and also outcomes are very non - unimportant and also past the extent of this discussion forum.

Certainly, in the quote over Scott Aaronson is simply making use of some everyday's trouble to highlight these principles. To make use of these outcomes officially, would certainly need to transform the job of "confirming the Riemann Hypothesis" to a choice trouble of official languages, as is typical in intricacy concept.

EDIT : There remains in reality a tiny alteration in the version in between both outcomes mentioned over which I left out. First, the communication in between one prover and also the verifier can be generalised to numerous provers and also a verifier. Next, there is an outcome that locates that the instance of numerous provers can be equivalently reformulated as adheres to : The provers are gotten rid of from the method, instead there is a solitary (perhaps long) string which works as a created evidence for words trouble. Communication is currently considering a randomly picked little this evidence, and also the verifier might pick the area of these little bits arbitrarily. This is called a Probabilistic Checkable Proof (therefore, PCP). So, in this circumstance, the "written evidence" of the Riemann Hypothesis would certainly be taken the evidence string.

2019-05-09 09:04:23

The factor is that SHORTPROOF is an NP - full trouble : offered a sentence of size $n$ in the language of some official evidence system (ZFC, Peano math, etc), does it have an evidence of size at the majority of some dealt with polynomial in $n$, such as $(2n)^{100}$? It's in NP due to the fact that for a practical official system you can examine an offered evidence rather quickly. This trouble was taken into consideration in Goedel's letter to von Neumann that unconditionally mentioned what we currently call the $P \neq NP$ inquiry (the heart of the trouble, the universality of concrete NP - full troubles, had not been recognized till much later).

Any kind of NP - full trouble has an absolutely no - expertise evidence method for showing remedies of circumstances, as an example that we have a SHORTPROOF of the Riemann hypothesis. These are "evidence that disclose absolutely nothing apart from their very own legitimacy".

The duty of the PCP theory is to show that the evidence methods (interactive challenge/response games) can be really reliable for any kind of specified degree of self-confidence. The chance that the prover actually does have a SHORTPROOF of Riemann, considered that we adhere to the method and also the prover wins, goes to the very least 99 percent, or whatever defined level of assurance.

2019-05-08 09:14:52