100 Prisoners problem, Hash, SRH

https://en.wikipedia.org/wiki/100_prisoners_problem

So the best known strategy is to start with your number, and then go to the number in the box. This will always lead into a loop because eventually you will open a box with your number and that will send you back to the start.

So this strategy in this context will always end up with self reference. And self -reference means a loop.

So that is isomorphic with halting problem (for a finite program to never stop it must end up executing the same commands again and again), the liar paradox where the state keeps alternating forever, and all paradoxes because they end up applying to themselves recursively.

Now is the case of this puzzle the loops have a very interesting property. 30% of loops are less than 51 steps in length.

Now SRH is normally noted as limiting the power of systems where we are looking for the greatest expressive power. But in this case Self-Reference limits the system in a favourable way (for the prisoners) by a huge amount improving their changes of escape from 1/10^30 to 1/3. This is a demonstration of the extraordinary power of Self-Reference to constrain systems.

I guess the challenge now is to find a way in Self-Reference works to increase the expression of a system. Or prove that were it do so would end up in contradiction.

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Crudely/Informally suppose a system S were able to determine expressive power with a function E(x). That means something like the size of the domain. Godel kind of says this when he says his theorems are true for any system sufficiently expressive to "allow" for self reference, that is the domain includes itself (maybe not directly but via isomorphism and mapping of one domain to another - so for example any ordered countable domain becomes a domain for anything that operates on Natural Numbers).

So E belongs to S. E(S) must be greater or equal to E(E). That is whatever expressive power E gives itself cannot be less than the whole system for which E belongs. S must be able to express E for example. Scrap this.

Take two systems S0 and S1 where S1 can express itself and S0 cannot then E(S0) < E(S1). SRH actually says that self-reference leads to reduced expressivity. That seems to be absurd. Surely being able to reference oneself is greater expressivity. But you don't have Godel limitations without self-reference. So very crudely are we saying that including self-reference is 1 step forward and 2 steps back?

Firstly you need a mega system U which is capable of ex[pressing S0 and S1 so you can have a function E that can apply to them both. The very existence of E(S0) and E(S1) statements implies a mega system U that can express both S0, S1 and E. So really we are just talking about subsets of U. S0 is a subset that excludes self-reference and S1 enough to reference itself.

So E(U) > E(S1) > E(S0) is the common sense. and SRH says that E(U) > E(S0) > E(S1) because self-reference actually acts to limit the system.

Now it might seems that U can be S1. You take any self-referential system as S1 and take a subset that excludes self-reference as S0. But SRH says that E(S1) < E(S0) so if SRH is true then S1 is not sufficient to express an E which can determine the expressive power of itself and a subset.

Does this make any sense at all????

S1 is sufficient to reference itself so why can't it determine its own expressive power? SRH would say it can't!

Is there a contradiction in determining your own expressive power? This is going very Godel and provability. Can't you create a "I am false" type statement which says it can't express itself. Tarski also that True/False cannot be expressed within a system without contradiction. Once you give a self-referential system the ability to determine expressability then you open up the possibility of it saying legitimate things that are impossible.

A subset of E would be L(x).. and been here before. L(x) looks at statement x and determines whether it can refer to itself. L(L) is a tautology then as to give any result is to apply self-reference as so much be true.

... run out of time ...