Zero Knowledge in the Cave

[Jim Choate]

Big problem, you've got to -prove- that there is only one of the
explorers (ie he has no twin) -and- that -only you and he are there-.

Your assumption that he is not lying is a major failing of the scientific
process - See the Magnificent Randy for more details and examples.

Responces to your specific questions below.

On Thu, 15 May 2003, Nomen Nescio wrote:

> Zero Knowledge in the Cave
>
> There is a cave with a large entry room.  From this room lead two
> passageways, 1 and 2.  Each of 1 and 2 branches into a myriad of smaller
> passages, twisting and turning through the massive rock formation.
> The passageways go on for miles and have never been fully explored.
>
> One of the big questions has been whether passageways 1 and 2 ever
> connect up.  Is there a way of getting from 1 to 2?  Many have
> searched, but none have ever succeeded.  Most people believe that
> no connection will ever be found.
>
> At least, no one used to.  Now an explorer comes to you and claims to
> have found a passage from 1 to 2, not a very long one, either.  He will
> prove it to you, but to you alone.  Being a secretive type, he wants no
> one else to know.  If you accompany him to the cave, he will prove the
> existence of the passageway to you.
>
> But there's a problem.  You carry a video camera and record everything
> that you see.  If he shows you the existence of the passage, you will
> be able to show the video tape to others, and they will learn of its
> existence as well.
>
> Not to worry, he says.  Come with me.  So you enter the large entry room
> of the cave together.
>
> Now the simplest thing to do in order to demonstrate the existence of the
> connection would be for him to leave through passage 1 and return through
> passage 2.  He could easily do this.  However, your film record of the
> event would prove to anyone else who saw it that there was a connection.
> Another way must be found.  The explorer tells you what to do.
>
> Following his instructions, you leave the entry room for a few minutes,
> while the explorer enters one of the passageways.  You then re-enter the
> room, and loudly call out one of the passageway numbers, either 1 or 2.
> In a few minutes, the explorer comes out of the requested passageway.
> You then leave the cave and repeat the process many times.  Each time, the
> reporter enters one of the passageways unknown to you; when you return and
> name one of them, he is able without fail to exit from the named passage.
>
> You reason that if there were no connection between the passageways,
> the only way the explorer could come out the passage that you named
> would be if he had gone in that same one.  He would have to guess
> which one you were going to choose, and if he were right, he could
> come out that one.  But you have repeated the test dozens of times.
> The chances that someone could guess right so often is infinitisimal.
> The only logical explanation is that the passageway does exist.
>
> Excited, you return to the tavern where you met the explorer and show
> the other patrons your tape.  But to your surprise, they just laugh.
> They don't deny that the tape is real, that the explorer did come out
> of the passageway you named.  But they don't believe in the connection.
>
> Instead, they claim you are in league with the explorer in an attempt to
> perpetrate a fraud.  You have simply predetermined together the sequence
> of numbers you would call out.  Each of you has memorized the sequence,
> and so each time the explorer is able to anticipate the number you will
> call next.  He enters that passage and is able, after a suitable pause,
> to exit from that same one when you call its number.
>
> You leave the bar, frustrated.  You are convinced that the connection
> exists, but even though the tape shows all of the evidence that was so
> convincing to you, no one else finds it persuasive.  The explorer has
> achieved his goal of proving the existence of the connection to you and
> you alone.
>
> Questions for the student:
>
> 1. How could you have done things differently, to produce a tape that
> would be convincing to others?

You can't, video is too easily manipulated. It is not strong enough for
any sort of real-world 'proof' without extensive corroborating evidence.

> 2. What counter-measures and conditions could the explorer have put in
> place to prevent you from getting a convincing tape in this manner?

This is a moot point since the possibilities are endless.

In short, your entire scenario is swiss cheese.


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