On 10/21/2011 10:10 AM, Bruno Marchal wrote:

On 21 Oct 2011, at 15:08, Stephen P. King wrote:On 10/21/2011 8:14 AM, Bruno Marchal wrote:On 19 Oct 2011, at 05:30, Russell Standish wrote:On Mon, Oct 17, 2011 at 07:03:38PM +0200, Bruno Marchal wrote:This, ISTM, is a completely different, and more wonderful beast,thanthe UD described in your Brussells thesis, or Schmidhuber's '97 paper. This latter beast must truly give rise to a continuum of histories, due to the random oracles you were talking about.All UDs do that. It is always the same beast.On reflection, yes you're correct. The new algorithm you proposed is more efficient than the previous one described in your thesis, as machines are only executed once for each prefix, rather over and over again for each input having the same prefix. But in an environment ofunbounded resources, such as we're considering here, that has noimport.Note that my programs are not prefixed. They are all generated andexecuted. To prefix them is usefulm when they are generated by arandom coin, which I do not need to do.So the histories, we're agreed, are uncountable in number, but OMs (bundles of histories compatible with the "here and now") are surely still countable.This is not obvious for me. For any to computational states whichare in a sequel when emulated by some universal UM,there areinfinitely many UMs, including one dovetailing on the reals, leadingto intermediate states. So I think that the "computationalneighborhoods" are a priori uncoutable. That fits with thetopological semantics of the first person logics (S4Grz, S4Grz1, X,X*, X1, X1*). But many math problems are unsolved there.Hi Bruno and Russel,I would like to better understand what "topological semantics"means. Are you considering relations defined only in set theoreticalsense, ala the closed or open or clopen nature of the sets relativeto each other?I guess you know the topological semantics of intuitionist logic.Instead of interpreting the propositions by sets in a boolean algebra,you interpret it by open set in a topological space. You havesoundness and completeness theorem for that. I think this came fromsemantics for the semantics for the modal logic S4, which mirrors wellintuitionist logic.What about the form of the axiom of choice for the set theory??

`There are multiple versions of set theory depending on the`

`selection of its form of axiom of choice.`

How do you induce compactness?Why do you want the space being compact. The point of the topologicalsemantics is that it works for all topological spaces, like anyboolean algebra can be used for classical logic.

`A space needs to be compact for many reasons including, but not`

`exclusive to, the ability to have a physics in them. Spaces that are not`

`compact will not have, for instance, fixed points that allow for notions`

`such as centers of mass, etc. There are also analycity reasons, and more.`

How is a "space" defined in strictly arithmetic terms?Why do you want to define it in arithmetic. With comp, arithmetic canbe used for the ontology, but the internal epistemology needs muchmore. Remember that the tiny effective sigma_1 arithmetic alreadyemulate all LĂ¶bian machines like PA, ZF, etc. Numbers can use sets tounderstand themselves.

`Yes, that numbers are what the Stone duality based idea assumes,`

`but numbers alone do not induce the "understanding" unless and until`

`sets are defined in distinction to them. This is why it cannot be`

`monistic above the "nothing" level. Tarsky's theorem prevents`

`understanding in number monist theories.`

If we take the no information ensemble,You might recall what you mean by this exactly.and transform it by applying a universal turing machine and collect just the countable output string where the machine halts, then apply another observer function that also happens to be a UTM, the final result will still be a Solomonoff-Levin distribution over the OMs.This is a bit unclear to me. Solomonof-Levin distribution are verynice, they are machine/theory independent, and that is quite in thespirit of comp, but it seems to be usable only in ASSA typeapproach. I do not exclude this can help for providing a role tolittle program, but I don't see at all how it could help for thecomputation of the first person indeterminacy, aka the derivation ofphysics from computer science needed when we assume comp incognitive science. In the work using Solomonof-Levin, the mind-bodyproblem is still under the rug. They don't seem aware of thefirst/third person description.S-L seems to assume 1p = 3p or no 1p at all!I think they just ignore 1p.This result follows from the compiler theorem - composition of a UTM with another one is still a UTM. So even if there is a rich structure to the OMs caused by them being generated in a UD, that structure will be lost in the process of observation. The net effect is that UD* is just as much a "veil" on the ultimate ontology as is the no information ensemble.UD*, or sigma_1 arithmetic, can be seen as an effective(mechanically defined) definition of a zero information. It is theeverything for the computational approach, but it is tiny comparedto the first person view of it by internal observers accounted inthe limit by the UD.How do we define this notion of size?There are many ways. Tiny in the sense of the ordering of sets ofprovable proposition of arithmetic, for example. Robinson theory istiny compared to PA which is tiny compared to ZF, in that sense.Tiny as opposed to ???To big! Or strong. ZF proves much more arithmetical propositions than PA.

Oh, the number of independent but mutually necessary axioms?

Unless I'm missing something here.Lets leave the discussion of the universal prior to another post.In anutshell, though, no matter what prior distribution you put onthe "noinformation" ensemble, an observer of that ensemble will always see the Solomonoff-Levin distribution, or universal prior.I don't think it makes sense to use a universal prior. That would make sense if we suppose there are computable universes, and if we try to measure the probability we are in such structure. This is typical of Schmidhuber's approach, which is still quite similar to physicalism, where we conceive observers as belonging to computable universes. Put in another way, this is typical of using some sort of identity thesis between a mind and a program.I understand your point, but the concept of universal prior is of far more general applicability than Schmidhuber's model. There need not be any identity thesis invoked, as for example in applications such as observers of Rorshach diagrams. And as for identity thesis, you do have a type of identity thesis in the statement that "brains make interaction with other observers relatively more likely" (or something like that).yes, by the duplication (multiplication) of populations ofobservers, like in comp, but also like in Everett.But we also need something that acts to code the "no preferredreference frame" of GR. Everett does not solve this problem, it onlycompounds it. :-(I am not sure I follow you. Everett makes clear that the choice ofbase is a false problem.

`How so? Please point to a discussion of this! Everett is explicitly`

`non-relativistic....`

There has to be some form of identity thesis between brain and mind that prevents the Occam catastrophe, and also prevent the full retreat into solipsism. I think it very much an open problem what that is.This will depend on the degree of similarity between between quantummechanics and the comp physics, which is given entirely by the(quantified) material hypostases (mainly the Z1* and X1* logics).Open but well mathematically circumscribed problem.From what I have studied so far, the (static) relationship betweenQM and COMP physics is the relationship between Boolean algebras andOrthocomplete lattices.Let us hope. Even just that has not been proved. Open problem.I think that a partial solution might require a weakening of thedefinition of a countable model of arithmetic and application of theTennenbaum theorem (that would allow for a variational principle,something like q-deformed theories where q is a measure of therelative recursiveness of the model of the theory).?

`I am still studying the math on this, the problem that I have is`

`that there do not seem to be words for the idea that I see. The closest`

`concept is how a fixed point is defined on a manifold... Crudely, think`

`of a manifold (not a space per se) where every point is a model of`

`arithmetic, the fixed point is the one that is countable and recursive.`

The idea is that every 1p would observe itself, in the Lob sense,to be recursive.?

How does a Lobian machine recognize its properties?

The proof would require showing that a Lobian machine on anon-standard model of arithmetic would *not* be able to "see" itsnon-standardness and thus it would bet that only it is recursive,thus it's Bp&p would be 1p and not 3p truth.? (Bp&p) is the definition of 1-p in, well not really in arithmetic,but in terms of arithmetic. We cannot define "p" (p is true) in anyeffective theory.

This is part of the incompleteness of your result :-( Onward! Stephen

Bruno http://iridia.ulb.ac.be/~marchal/

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