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, asmachines are only executed once for each prefix, rather over andoveragain for each input having the same prefix. But in an environmentofunbounded 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,leading to 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 have`

`soundness and completeness theorem for that. I think this came from`

`semantics for the semantics for the modal logic S4, which mirrors well`

`intuitionist logic.`

What about the form of the axiom of choice for the set theory?

?

How do you induce compactness?

`Why do you want the space being compact. The point of the topological`

`semantics is that it works for all topological spaces, like any`

`boolean algebra can be used for classical logic.`

How is a "space" defined in strictly arithmetic terms?

`Why do you want to define it in arithmetic. With comp, arithmetic can`

`be used for the ontology, but the internal epistemology needs much`

`more. Remember that the tiny effective sigma_1 arithmetic already`

`emulate all LĂ¶bian machines like PA, ZF, etc. Numbers can use sets to`

`understand themselves.`

If we take the no information ensemble,You might recall what you mean by this exactly.and transform it by applying auniversal turing machine and collect just the countable outputstringwhere 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 derivationof physics 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 the first/third person description.S-L seems to assume 1p = 3p or no 1p at all!

I think they just ignore 1p.

This result follows fromthe compiler theorem - composition of a UTM with another one isstilla 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 of`

`provable proposition of arithmetic, for example. Robinson theory is`

`tiny 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.`

Unless I'm missing something here.Lets leave the discussion of the universal prior to anotherpost. In anutshell, though, no matter what prior distribution you put onthe "noinformation" ensemble, an observer of that ensemble will alwaysseethe 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 computableuniverses. Put in another way, this is typical of using some sortofidentity thesis between a mind and a program.I understand your point, but the concept of universal prior is offarmore general applicability than Schmidhuber's model. There neednot beany 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 of`

`base is a false problem.`

There has to be some form of identity thesis between brain and mindthat prevents the Occam catastrophe, and also prevent the fullretreatinto solipsism. I think it very much an open problem what that is.This will depend on the degree of similarity between betweenquantum mechanics and the comp physics, which is given entirely bythe (quantified) material hypostases (mainly the Z1* and X1*logics). Open but well mathematically circumscribed problem.From what I have studied so far, the (static) relationshipbetween QM and COMP physics is the relationship between Booleanalgebras and Orthocomplete 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).

?

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

?

The proof would require showing that a Lobian machine on a non-standard model of arithmetic would *not* be able to "see" its non-standardness and thus it would bet that only it is recursive, thusit'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 any`

`effective theory.`

Bruno http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.