# The mathematics of awareness

selfAdjoint
Staff Emeritus
Gold Member
Dearly Missed
Phoenixthoth, the Goertzel link was fascinating, if a little shallow and handwaving. Consider this passage:

Let us go back to the LEGO metaphor. It would be easy to build a computable LEGO universe following Kampis's instructions. For the set of all LEGO structures is countable, and may therefore be mapped into the set of binary sequences, in a one-to-one manner. And each binary sequence may be represented as a Turingmachine program, i.e. as a map from binary sequences to binary sequences. Therefore, using Turing machines, each LEGO structure could be interpreted as a function acting on other LEGO structures. The only problem with this arrangement is that it does not satisfy clause (c) of the definition of component-system. Not every LEGO structure is realizable by our dynamics. Only some computable subset of LEGO structures is realizable.

But now -- and here is where my thinking differs from Kampis's -- suppose one adds a random element to one's Turing machine. Suppose each component of the Turing machine is susceptible to errors! Then, in fact, every possible LEGO structure becomes realizable! Structures may have negligibly small probability, but never zero probability! This is an example of a component-system which is computable by a stochastic Turing machine.

I don't see that he shows that a Turing machine with a stochastic component can generate all possible Lego configurations. Surely it can generate solutions that are not in the computable universe (at least, modulo some theorem that says a stochastic-modified Turing machine can't be modeled on a regular Turing machine)
but he hasn't shown the the universe of Lego constructions is contained in the universe of stochastic-Turing solutions, or even that they intersect. The stochastic solutions might all be malformed in Lego terms.

Kastin's Lego constructions remind me strongly of cellular automata, and I wonder if they might be modeled as such. There is apparently a large literatur on what cellular automata can achieve, and I can't resist referencing a new result: http://www.cscs.umich.edu/~crshalizi/weblog/375.html

Any way, very interesting, thanks for the link. BTW I found your other link on the sorites problem interesting too, although I don't know that the solution of the mathematical vague boundary problem solves the physical one.

I came across "agents" or "introspective agents" in my surfing.

Here is a site that fascinated me and maybe you'll enjoy:
http://cs.wwc.edu/~aabyan/Colloquia/Aware/aware2.html [Broken]

So if a type 2 agent is allowed to be called self-aware by the community, then I'd just have to prove or disprove that a self-referential wff is a type 2 agent. And... I'm a little confused by the beginning of that article. Is <SB, v> the agent? What is the agent? Does that mean it is a model for the set of beliefs?

Ok, so now I wonder what Type an agent would be if the set of beliefs is a single self-referential wff. Seems like that would depend on whether it is a tautology and for the sr wffs I've come across, they are tautologies.

Last edited by a moderator:
selfAdjoint
Staff Emeritus
Gold Member
Dearly Missed
Phoen, I tried to link to it and got this
The XML page cannot be displayed
Cannot view XML input using XSL style sheet. Please correct the error and then click the Refresh button, or try again later.

--------------------------------------------------------------------------------

Parameter entity must be defined before it is used. Error processing resource 'http://www.w3.org/TR/MathML2/dtd/xhtml-math1... [Broken]

%xhtml-prefw-redecl.mod;
-^

Never saw that one before.

Last edited by a moderator:
selfAdjoint said:
Phoen, I tried to link to it and got this

Never saw that one before.
That's too bad.

I just now clicked on the link I gave and bamn! there it was.

Hurkyl
Staff Emeritus
Science Advisor
Gold Member
I just wanted to mention... it almost sounds as if the author thinks that turing machines cannot be self-referential, but the recursion theorem says, in effect, that any program for a turing machine can obtain a string containing itself.

More precisely:

Let T be a turing machine that takes two things as input: a string description a turing machine, and an arbitrary string.

Then, there exists a turing machine M that when given input S behaves exactly as if you ran T on the input <M>, S.

Which we interpret as saying a turing machine can "obtain its own description".