Re: join post
I guess I haven't read those papers, so sorry if I was leading you up the garden path re GTMs. It sounds interesting that the universal prior could work for generalisation of the Turing machine, although I'm not sure what the implications would be. Anyway, it sounds like you've got a research programme :) Cheers On Thu, Nov 27, 2008 at 06:12:49PM -0500, Abram Demski wrote: Russel, The paper does indeed showcase one example of a universal prior that includes non-computable universes... Theorem 4.1. So it's *possible*. Of course it then proceeds to dash hopes for a universal prior over a broader domain, defined by GTMs. So, it would be interesting to know more about the conditions that make universality possible. --Abram -- A/Prof Russell Standish Phone 0425 253119 (mobile) Mathematics UNSW SYDNEY 2052 [EMAIL PROTECTED] Australiahttp://www.hpcoders.com.au --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: join post
On Wed, Nov 26, 2008 at 02:55:08PM -0500, Abram Demski wrote: Russel, I do not see why some appropriately modified version of that theorem couldn't be proven for other settings. As a concrete example let's just use Schmidhuber's GTMs. There would be universal GTMs and a constant cost for conversion and everything else needed for a version of the theorem, wouldn't there be? (I am assuming things, I will look up some details this afternoon... I have the book you refer to, I'll look at the theorem... but I suppose I should also re-read the paper about GTMs before making bold claims...) --Abram IIRC, Schmidhuber's machines were non-prefix Turing machines. As such there may or may not be a probability distribution in the first place. Solomonoff's original proposal using universal Turing machine didn't work because the probability distribution could not be defined. If, however, a probility distribution could be defined, then it would probably end up being equivalent to the S-L universal prior. Cheers -- A/Prof Russell Standish Phone 0425 253119 (mobile) Mathematics UNSW SYDNEY 2052 [EMAIL PROTECTED] Australiahttp://www.hpcoders.com.au --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: join post
Russel, Hmm, can't we simply turn any coding into a prefix-free-coding by prefacing each code for a GTM with a number of 1s indicating the length of the following description, and then a 0 signaling the beginning of the actual description? I am not especially familiar with the prefix issue, so please forgive me if I am wrong... Also, I do not understand why there would be reason to suspect that the probability distribution, once properly defined, would turn out to be equivalent to the S-L prior. GTMs can formally represent more things than TMs, so why would those things not end up in the probability distribution? --Abram Demski On Thu, Nov 27, 2008 at 5:18 AM, Russell Standish [EMAIL PROTECTED] wrote: On Wed, Nov 26, 2008 at 02:55:08PM -0500, Abram Demski wrote: Russel, I do not see why some appropriately modified version of that theorem couldn't be proven for other settings. As a concrete example let's just use Schmidhuber's GTMs. There would be universal GTMs and a constant cost for conversion and everything else needed for a version of the theorem, wouldn't there be? (I am assuming things, I will look up some details this afternoon... I have the book you refer to, I'll look at the theorem... but I suppose I should also re-read the paper about GTMs before making bold claims...) --Abram IIRC, Schmidhuber's machines were non-prefix Turing machines. As such there may or may not be a probability distribution in the first place. Solomonoff's original proposal using universal Turing machine didn't work because the probability distribution could not be defined. If, however, a probility distribution could be defined, then it would probably end up being equivalent to the S-L universal prior. Cheers -- A/Prof Russell Standish Phone 0425 253119 (mobile) Mathematics UNSW SYDNEY 2052 [EMAIL PROTECTED] Australiahttp://www.hpcoders.com.au --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: join post
On Thu, Nov 27, 2008 at 02:40:04PM -0500, Abram Demski wrote: Russel, Hmm, can't we simply turn any coding into a prefix-free-coding by prefacing each code for a GTM with a number of 1s indicating the length of the following description, and then a 0 signaling the beginning of the actual description? I am not especially familiar with the prefix issue, so please forgive me if I am wrong... Sure - but you also need to change the machine to recognise your code. Also, I do not understand why there would be reason to suspect that the probability distribution, once properly defined, would turn out to be equivalent to the S-L prior. GTMs can formally represent more things than TMs, so why would those things not end up in the probability distribution? --Abram Demski Its been a while since I read Schmidhuber's papers, but I thought he was talking about machines that continuosly updated their output, but would eventually converge (ie for each bit i of the output, there was a time t_i after which that bit would not change). This seems to be a restriction on the notion of Turing machine to me, but not as restrictive as a prefix machine. -- A/Prof Russell Standish Phone 0425 253119 (mobile) Mathematics UNSW SYDNEY 2052 [EMAIL PROTECTED] Australiahttp://www.hpcoders.com.au --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
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Russel, I just went to look at the paper Hierarchies of generalized Kolmogorov complexities and nonenumerable universal measures computable in the limit-- to find a quote showing that GTMs were a generalization of Turing Machines rather then a restriction. I found such a quote as soon as page 2: This constructive notion of describability is less restrictive then the traditional notion of computability, mainly because we do not insist on the existence of a halting program that computes an upper bound of the convergence time [...] But, as soon as the abstract, I found: Among other things we show [...] that there is no universal approximable distribution [...] So scratch that example! I now remember reading that when I first encountered the paper, but obviously I did not pay much attention or I would have recalled it sooner... I'll look over the proof again and try to figure out whether it applies to even more general models (like priors based on arithmetic describability or analytic describability). --Abram On Thu, Nov 27, 2008 at 5:22 PM, Russell Standish [EMAIL PROTECTED] wrote: On Thu, Nov 27, 2008 at 02:40:04PM -0500, Abram Demski wrote: Russel, Hmm, can't we simply turn any coding into a prefix-free-coding by prefacing each code for a GTM with a number of 1s indicating the length of the following description, and then a 0 signaling the beginning of the actual description? I am not especially familiar with the prefix issue, so please forgive me if I am wrong... Sure - but you also need to change the machine to recognise your code. Also, I do not understand why there would be reason to suspect that the probability distribution, once properly defined, would turn out to be equivalent to the S-L prior. GTMs can formally represent more things than TMs, so why would those things not end up in the probability distribution? --Abram Demski Its been a while since I read Schmidhuber's papers, but I thought he was talking about machines that continuosly updated their output, but would eventually converge (ie for each bit i of the output, there was a time t_i after which that bit would not change). This seems to be a restriction on the notion of Turing machine to me, but not as restrictive as a prefix machine. -- A/Prof Russell Standish Phone 0425 253119 (mobile) Mathematics UNSW SYDNEY 2052 [EMAIL PROTECTED] Australiahttp://www.hpcoders.com.au --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: join post
Russel, The paper does indeed showcase one example of a universal prior that includes non-computable universes... Theorem 4.1. So it's *possible*. Of course it then proceeds to dash hopes for a universal prior over a broader domain, defined by GTMs. So, it would be interesting to know more about the conditions that make universality possible. --Abram --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: join post
Hi Abram, On 26 Nov 2008, at 00:01, Abram Demski wrote: Bruno, Yes, I have encountered the provability logics before, but I am no expert. We will perhaps have opportunity to talk about this. In any given generation, the entity who can represent the truth-predicate of the most other entities will dominate. Why? The notion of the entities adapting their logics in order to better reason about each other is meant to be more of an informal justification than an exact proof, so I'm not worried about stating my assumptions precisely... If I did, I might simply take this to be an assumption rather than a derived fact. But, here is an informal justification. Since the entities start out using first-order logic, it will be useful to solve the halting problem to reach conclusions about what a fellow-creature *won't* ever reach conclusions about. This means a provable predicate will be useful. To support deduction with this predicate, of course, the entities will gain more and more axioms over time; axioms that help solve instances of the halting problem will survive, while axioms that provide incorrect information will not. This means that the provable predicate has a moving target: more and more is provable over time. All right. Eventually it will become useful to abstract away from the details with a true predicate. Here, assuming the mechanist hypothesis (or some weakening), the way the truth predicate is introduced is really what will decide if the soul of the machine will fall in Hell, or get enlightened and go to Heaven. The all encompassing truth is not even nameable or describable by the machines. The true predicate essentially says provable by some sufficiently evolved system. This allows an entity to ignore the details of the entity it is currently reasoning about. If PA (Peano Arithmetic) deduces I can prove that I am consistent from I can prove that ZF (Zermelo Fraenkel Set Theory) proves that I am consistent, then PA goes to hell! If an entity refers to a more powerful entity, even if we trust that more powerful entity, it just an invalid argument of authority. Of course if PA begins to *believe* in the axioms of ZF, then PA becomes ZF, and can assert the consistency of PA without problem. But then, we are no more talking *about* PA, but about ZF. This won't always work-- sometimes it will need to resort to reasoning about provability again. But, it should be a useful concept (after all, we find it to be so). Sure. But truth is really an interrogation mark. We can only search it. Of course, this gives rise to an outlandish number of truth-values (one for each ordinal number), when normally any more than 2 is considered questionable. Not really, because those truth value are, imo, not really truth value, but they quantify a ladder toward infinite credibility, assurance or something. Perhaps security. I agree that the explosion of truth-values is acceptable because they are not really truth-values... but they do not go further and further into absolute confidence, but rather further and further into meaninglessness. Obviously my previous explanation was not adequate. First we have true and false. Dealing with these in an unrestricted manner, we can construct sentences such as this sentence is false. I don't think we can really do that. We cannot, I think. (And I can prove this making the assumption that we are ideally sound universal machines). We need to label these somehow as meaningless or pathological. I think either a fixed-point construction or the revision theory are OK options for doing this; In my opinion, revision theories are useful when a machine begins to bet on an universal environment independent of herself. Above her Godel-Lob-Solovay correct self-reference logic, she will have to develop a non monotonic surface to be able to handle its errors, dreams, etc. It is a bit more close to practical artifiicial intelligence engineering than machine theology, but I am ok with that. perhaps one is better than the other, perhaps they are ultimately equivalent where it matters, I don't know. Anyway, now we are stuck with a new predicate: meaningless. Using this in an unrestricted manner, I can say this sentence is either meaningless or false. I need to rule this out, but I can't label it meaningless, or I will then conclude it is true (assuming something like classical logic). So I need to invent a new predicate, 2-meaningless. Using this in an unrestricted manner again would lead to trouble, so I'll need 3-meaningless and 4-meaningless and finitely-meaningless and countably-meaningless and so on. Indeed. It seems you make the point. Best, Bruno Marchal http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To
Re: join post
Russel,
I do not see why some appropriately modified version of that theorem
couldn't be proven for other settings. As a concrete example let's
just use Schmidhuber's GTMs. There would be universal GTMs and a
constant cost for conversion and everything else needed for a version
of the theorem, wouldn't there be? (I am assuming things, I will look
up some details this afternoon... I have the book you refer to, I'll
look at the theorem... but I suppose I should also re-read the paper
about GTMs before making bold claims...)
--Abram
On Tue, Nov 25, 2008 at 5:41 PM, Russell Standish [EMAIL PROTECTED] wrote:
On Tue, Nov 25, 2008 at 04:58:41PM -0500, Abram Demski wrote:
Russel,
Can you point me to any references? I am curious to hear why the
universality goes away, and what crucially depends means, et cetera.
-Abram Demski
This is sort of discussed in my book Theory of Nothing, but not in
technical detail. Excuse the LaTeX notation below.
Basically any mapping O(x) from the set of infinite binary strings
{0,1}\infty (equivalently the set of reals [0,1) ) to the integers
induces a probability distribution relative to the uniform measure dx
over {0,1}\infty
P(x) = \int_{y\in O^{-1}(x)} dx
In the case where O(x) is a universal prefix machine, P(x) is just the
usual Solomonoff-Levin universal prior, as discussed in chapter 3 of
Li and Vitanyi. In the case where O(x) is not universal, or perhaps
even not a machine at all, the important Coding theorem (Thm 4.3.3 in
Li and Vitanyi) no longer holds, so the distribution is no longer
universal, however it is still a probability distribution (provided
O(x) is defined for all x in {0,1}\infty) that depends on the choice
of observer map O(x).
Hope this is clear.
--
A/Prof Russell Standish Phone 0425 253119 (mobile)
Mathematics
UNSW SYDNEY 2052 [EMAIL PROTECTED]
Australiahttp://www.hpcoders.com.au
--~--~-~--~~~---~--~~
You received this message because you are subscribed to the Google Groups
Everything List group.
To post to this group, send email to [EMAIL PROTECTED]
To unsubscribe from this group, send email to [EMAIL PROTECTED]
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Re: join post
Bruno, I am glad for the opportunity to discuss these things with someone who knows something about these issues. In my opinion, revision theories are useful when a machine begins to bet on an universal environment independent of herself. Above her Godel-Lob-Solovay correct self-reference logic, she will have to develop a non monotonic surface to be able to handle its errors, dreams, etc. It is a bit more close to practical artifiicial intelligence engineering than machine theology, but I am ok with that. I am interested in nonmonotonic logics as an explanation of how we can have concepts that don't just reduce to first-order theories-- specifically, concepts such as number that fall prey to Godelian incompleteness. In other words, I think that we use nonmonotonic logic is at least a partial answer to what I called the little puzzle. First we have true and false. Dealing with these in an unrestricted manner, we can construct sentences such as this sentence is false. I don't think we can really do that. We cannot, I think. (And I can prove this making the assumption that we are ideally sound universal machines). I'm not claiming that we can *consistently* construct such sentences, just that we can try to construct them, and then run into problems when we try to reason about them. Luckily we have what you called a nonmonotonic surface so we draw back and either give up or try from different angles. --Abram On Wed, Nov 26, 2008 at 10:54 AM, Bruno Marchal [EMAIL PROTECTED] wrote: Hi Abram, On 26 Nov 2008, at 00:01, Abram Demski wrote: Bruno, Yes, I have encountered the provability logics before, but I am no expert. We will perhaps have opportunity to talk about this. In any given generation, the entity who can represent the truth-predicate of the most other entities will dominate. Why? The notion of the entities adapting their logics in order to better reason about each other is meant to be more of an informal justification than an exact proof, so I'm not worried about stating my assumptions precisely... If I did, I might simply take this to be an assumption rather than a derived fact. But, here is an informal justification. Since the entities start out using first-order logic, it will be useful to solve the halting problem to reach conclusions about what a fellow-creature *won't* ever reach conclusions about. This means a provable predicate will be useful. To support deduction with this predicate, of course, the entities will gain more and more axioms over time; axioms that help solve instances of the halting problem will survive, while axioms that provide incorrect information will not. This means that the provable predicate has a moving target: more and more is provable over time. All right. Eventually it will become useful to abstract away from the details with a true predicate. Here, assuming the mechanist hypothesis (or some weakening), the way the truth predicate is introduced is really what will decide if the soul of the machine will fall in Hell, or get enlightened and go to Heaven. The all encompassing truth is not even nameable or describable by the machines. The true predicate essentially says provable by some sufficiently evolved system. This allows an entity to ignore the details of the entity it is currently reasoning about. If PA (Peano Arithmetic) deduces I can prove that I am consistent from I can prove that ZF (Zermelo Fraenkel Set Theory) proves that I am consistent, then PA goes to hell! If an entity refers to a more powerful entity, even if we trust that more powerful entity, it just an invalid argument of authority. Of course if PA begins to *believe* in the axioms of ZF, then PA becomes ZF, and can assert the consistency of PA without problem. But then, we are no more talking *about* PA, but about ZF. This won't always work-- sometimes it will need to resort to reasoning about provability again. But, it should be a useful concept (after all, we find it to be so). Sure. But truth is really an interrogation mark. We can only search it. Of course, this gives rise to an outlandish number of truth-values (one for each ordinal number), when normally any more than 2 is considered questionable. Not really, because those truth value are, imo, not really truth value, but they quantify a ladder toward infinite credibility, assurance or something. Perhaps security. I agree that the explosion of truth-values is acceptable because they are not really truth-values... but they do not go further and further into absolute confidence, but rather further and further into meaninglessness. Obviously my previous explanation was not adequate. First we have true and false. Dealing with these in an unrestricted manner, we can construct sentences such as this sentence is false. I don't think we can really do that. We cannot, I think. (And I can prove this making the assumption
Re: join post
Russel, Can you point me to any references? I am curious to hear why the universality goes away, and what crucially depends means, et cetera. -Abram Demski On Tue, Nov 25, 2008 at 5:44 AM, Russell Standish [EMAIL PROTECTED] wrote: On Mon, Nov 24, 2008 at 11:52:55AM -0500, Abram Demski wrote: As I said, I'm also interested in the notion of probability. I disagree with Solomonoff's universal distribution (http://en.wikipedia.org/wiki/Ray_Solomonoff), because it assumes that the universe is computable. I cannot say whether the universe we actually live in is computable or not; however, I argue that, regardless, an uncomputable universe is at least conceivable, even if it has a low credibility. So, a universal probability distribution should include that possibility. Solomonoff-Levin distributions can be found for non computable universes (ie for non Turing complete observers). However, the notion of universality disappears, and the actual probability distribution obtained critically depends on the actual interpretation of data chosen. -- A/Prof Russell Standish Phone 0425 253119 (mobile) Mathematics UNSW SYDNEY 2052 [EMAIL PROTECTED] Australiahttp://www.hpcoders.com.au --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: join post
On Tue, Nov 25, 2008 at 04:58:41PM -0500, Abram Demski wrote:
Russel,
Can you point me to any references? I am curious to hear why the
universality goes away, and what crucially depends means, et cetera.
-Abram Demski
This is sort of discussed in my book Theory of Nothing, but not in
technical detail. Excuse the LaTeX notation below.
Basically any mapping O(x) from the set of infinite binary strings
{0,1}\infty (equivalently the set of reals [0,1) ) to the integers
induces a probability distribution relative to the uniform measure dx
over {0,1}\infty
P(x) = \int_{y\in O^{-1}(x)} dx
In the case where O(x) is a universal prefix machine, P(x) is just the
usual Solomonoff-Levin universal prior, as discussed in chapter 3 of
Li and Vitanyi. In the case where O(x) is not universal, or perhaps
even not a machine at all, the important Coding theorem (Thm 4.3.3 in
Li and Vitanyi) no longer holds, so the distribution is no longer
universal, however it is still a probability distribution (provided
O(x) is defined for all x in {0,1}\infty) that depends on the choice
of observer map O(x).
Hope this is clear.
--
A/Prof Russell Standish Phone 0425 253119 (mobile)
Mathematics
UNSW SYDNEY 2052 [EMAIL PROTECTED]
Australiahttp://www.hpcoders.com.au
--~--~-~--~~~---~--~~
You received this message because you are subscribed to the Google Groups
Everything List group.
To post to this group, send email to [EMAIL PROTECTED]
To unsubscribe from this group, send email to [EMAIL PROTECTED]
For more options, visit this group at
http://groups.google.com/group/everything-list?hl=en
-~--~~~~--~~--~--~---
Re: join post
Bruno, Yes, I have encountered the provability logics before, but I am no expert. In any given generation, the entity who can represent the truth-predicate of the most other entities will dominate. Why? The notion of the entities adapting their logics in order to better reason about each other is meant to be more of an informal justification than an exact proof, so I'm not worried about stating my assumptions precisely... If I did, I might simply take this to be an assumption rather than a derived fact. But, here is an informal justification. Since the entities start out using first-order logic, it will be useful to solve the halting problem to reach conclusions about what a fellow-creature *won't* ever reach conclusions about. This means a provable predicate will be useful. To support deduction with this predicate, of course, the entities will gain more and more axioms over time; axioms that help solve instances of the halting problem will survive, while axioms that provide incorrect information will not. This means that the provable predicate has a moving target: more and more is provable over time. Eventually it will become useful to abstract away from the details with a true predicate. The true predicate essentially says provable by some sufficiently evolved system. This allows an entity to ignore the details of the entity it is currently reasoning about. This won't always work-- sometimes it will need to resort to reasoning about provability again. But, it should be a useful concept (after all, we find it to be so). Of course, this gives rise to an outlandish number of truth-values (one for each ordinal number), when normally any more than 2 is considered questionable. Not really, because those truth value are, imo, not really truth value, but they quantify a ladder toward infinite credibility, assurance or something. Perhaps security. I agree that the explosion of truth-values is acceptable because they are not really truth-values... but they do not go further and further into absolute confidence, but rather further and further into meaninglessness. Obviously my previous explanation was not adequate. First we have true and false. Dealing with these in an unrestricted manner, we can construct sentences such as this sentence is false. We need to label these somehow as meaningless or pathological. I think either a fixed-point construction or the revision theory are OK options for doing this; perhaps one is better than the other, perhaps they are ultimately equivalent where it matters, I don't know. Anyway, now we are stuck with a new predicate: meaningless. Using this in an unrestricted manner, I can say this sentence is either meaningless or false. I need to rule this out, but I can't label it meaningless, or I will then conclude it is true (assuming something like classical logic). So I need to invent a new predicate, 2-meaningless. Using this in an unrestricted manner again would lead to trouble, so I'll need 3-meaningless and 4-meaningless and finitely-meaningless and countably-meaningless and so on. --Abram On Mon, Nov 24, 2008 at 5:03 PM, Bruno Marchal [EMAIL PROTECTED] wrote: On 24 Nov 2008, at 21:52, Abram Demski wrote: Hi Bruno, I am not sure I follow you here. All what Godel's incompleteness proves is that no machine, or no axiomatisable theory can solve all halting problems. The undecidability is always relative to such or such theory or machine prover. For self-modifying theorem prover, the undecidable sentence can evolve. (extensionaly, and yet remain the same intensionally) I agree with everything you say there, so I'm not sure where you aren't following me. I definitely agree with the idea of a self-modifying theorem prover that becomes stronger over time-- I think it is the right model. What I am saying in the paragraph you quoted is that one way out is to claim that when we add axioms to strengthen our system, we can choose either the axiom or the negation arbitrarily, since either is consistent with the system so far. I've argued with people who explicitly claim this. My opinion is that there is only one correct choice for each addition. I agree, for a category of pure machine, not yet confronted to some bigger or older universal machine. In the case of the halting problem, we want to reflect the actual truth about halting; in the (equivalent) domain of undecidable numerical statements, we still want the actual truth. Well some of them, like you, equivalent or bigger machine, will never know, unless you are infinitely patient. Insolubility here is hard, but it makes the mathematical world necessarily ever surprising. You can hope for a theory of everything, which preserves the mystery, which makes it even more mysterious. Also, I should mention that the arithmetical hierarchy shows that some problems are more undecidable than halting: if we had a halting oracle, we would still be unable to decide those problems.
Re: join post
Hi Abram, welcome. On 24 Nov 2008, at 17:52, Abram Demski wrote (in part): The little puzzle is this: Godel's theorem tells us that any sufficiently strong logic does not have a complete set of deduction rules; the axioms will fail to capture all truths about the logical entities we're trying to talk about. But if these entities cannot be completely axiomized, then in what sense are they well-defined? How is logic logical, if it is doomed to be incompletely specified? One way out here is to say that numbers (which happen to be the logical entities that Godel showed were doomed to incompleteness, though of course the incompleteness theorem has since been generalized to other domains) really are incompletely specified: the axioms are incomplete in that they fail to prove every sentence about numbers either true or false, but they are complete in that the ones they miss are in some sense actually not specified by our notion of number. I don't like this answer, because it is equivalent to saying that the halting problem really has no answer in the cases where it is undecidable. I am not sure I follow you here. All what Godel's incompleteness proves is that no machine, or no axiomatisable theory can solve all halting problems. The undecidability is always relative to such or such theory or machine prover. For self-modifying theorem prover, the undecidable sentence can evolve. (extensionaly, and yet remain the same intensionally) For such machine the self-stopping problem become absolutely-yet- relatively-to-them undecidable. Actually I am very happy with this, because , assuming comp, this could explain why humans fight on this question since so long. And we can bet it is not finished! Tarski 's theorem is even more religious, in the computationalist setting. It means that the concept of truth (about a machine) acts already like a god for that machine. No (sound) machine can givee a name to its own proof predicate. See my paper on the arithmetical interpretation of Plotinus to know more. But the main reason of that paper is the failing of Aristotle materialism to address the mind-body problem. This is what we talk in the MGA thread in case you want catch the train. You can see my url for the papers if interested in the foundation of physics and mind. Best, 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 [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: join post
Hi Bruno, I am not sure I follow you here. All what Godel's incompleteness proves is that no machine, or no axiomatisable theory can solve all halting problems. The undecidability is always relative to such or such theory or machine prover. For self-modifying theorem prover, the undecidable sentence can evolve. (extensionaly, and yet remain the same intensionally) I agree with everything you say there, so I'm not sure where you aren't following me. I definitely agree with the idea of a self-modifying theorem prover that becomes stronger over time-- I think it is the right model. What I am saying in the paragraph you quoted is that one way out is to claim that when we add axioms to strengthen our system, we can choose either the axiom or the negation arbitrarily, since either is consistent with the system so far. I've argued with people who explicitly claim this. My opinion is that there is only one correct choice for each addition. In the case of the halting problem, we want to reflect the actual truth about halting; in the (equivalent) domain of undecidable numerical statements, we still want the actual truth. Also, I should mention that the arithmetical hierarchy shows that some problems are more undecidable than halting: if we had a halting oracle, we would still be unable to decide those problems. Schmidhuber's super-omegas are a perfect example. http://www.idsia.ch/~juergen/kolmogorov.html But you probably knew that already. For such machine the self-stopping problem become absolutely-yet- relatively-to-them undecidable. Actually I am very happy with this, because , assuming comp, this could explain why humans fight on this question since so long. And we can bet it is not finished! This argument is given in longer form elsewhere? Perhaps that paper you mention later on? Tarski 's theorem is even more religious, in the computationalist setting. It means that the concept of truth (about a machine) acts already like a god for that machine. No (sound) machine can givee a name to its own proof predicate. To extend the model of the self-modifying theorem prover... the scenario I use when thinking about truth is a population of entities which, among other things, need to reason properly about each other. (I could perhaps reduce this to one entity that, among other things, needs to reason about itself; but that is needlessly recursive.) The logic that these entities use evolves over time. In any given generation, the entity who can represent the truth-predicate of the most other entities will dominate. The question, then, is: what logic will the population eventually converge to? I think fair arguments could be given for the fixed-point or revision theories in this scenario, but like I said, both create reference gaps... so some creature could dominate over these by inventing a predicate to fill the gap. That creature will then have its own reference gaps, and yet more gap-filling predicates will be created. My current thinking is that each gap-filling predicate will correspond to an ordinal number, so that the maximal logic will be one with a gap-filling predicate for each ordinal number. No gap will be left, because if there was such a gap, it would correspond to an ordinal number larger than all ordinal numbers, which is impossible. Of course, this gives rise to an outlandish number of truth-values (one for each ordinal number), when normally any more than 2 is considered questionable. --Abram Demski --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: join post
On 24 Nov 2008, at 21:52, Abram Demski wrote: Hi Bruno, I am not sure I follow you here. All what Godel's incompleteness proves is that no machine, or no axiomatisable theory can solve all halting problems. The undecidability is always relative to such or such theory or machine prover. For self-modifying theorem prover, the undecidable sentence can evolve. (extensionaly, and yet remain the same intensionally) I agree with everything you say there, so I'm not sure where you aren't following me. I definitely agree with the idea of a self-modifying theorem prover that becomes stronger over time-- I think it is the right model. What I am saying in the paragraph you quoted is that one way out is to claim that when we add axioms to strengthen our system, we can choose either the axiom or the negation arbitrarily, since either is consistent with the system so far. I've argued with people who explicitly claim this. My opinion is that there is only one correct choice for each addition. I agree, for a category of pure machine, not yet confronted to some bigger or older universal machine. In the case of the halting problem, we want to reflect the actual truth about halting; in the (equivalent) domain of undecidable numerical statements, we still want the actual truth. Well some of them, like you, equivalent or bigger machine, will never know, unless you are infinitely patient. Insolubility here is hard, but it makes the mathematical world necessarily ever surprising. You can hope for a theory of everything, which preserves the mystery, which makes it even more mysterious. Also, I should mention that the arithmetical hierarchy shows that some problems are more undecidable than halting: if we had a halting oracle, we would still be unable to decide those problems. Schmidhuber's super-omegas are a perfect example. That is why the computationalist hypothesis is non trivial. It gives a prominent role to Sigma_1 completeness. The oracle can still play an important role from inside, but even this is not sure. (and then comp gives importance to sigma_1 completeness relatively to an oracle). http://www.idsia.ch/~juergen/kolmogorov.html But you probably knew that already. For such machine the self-stopping problem become absolutely-yet- relatively-to-them undecidable. Actually I am very happy with this, because , assuming comp, this could explain why humans fight on this question since so long. And we can bet it is not finished! This argument is given in longer form elsewhere? Perhaps that paper you mention later on? I have not publish this. I explain it informally in my french thesis long version. But it is obvious I think, especially assuming comp. It is implicit in my Plotinus paper (still on my first page of my url I think). Finite machine are limited. But finite machine which believes in the induction axioms can know that they are limited, and can build theories explaining those limitation. Formally this gives autonomous progression. The correct one have to converge to some non convergence possible in the horizon ... Tarski 's theorem is even more religious, in the computationalist setting. It means that the concept of truth (about a machine) acts already like a god for that machine. No (sound) machine can givee a name to its own proof predicate. To extend the model of the self-modifying theorem prover... the scenario I use when thinking about truth is a population of entities which, among other things, need to reason properly about each other. (I could perhaps reduce this to one entity that, among other things, needs to reason about itself; but that is needlessly recursive.) The logic that these entities use evolves over time. Relatively to some universal machine. I see subtle difficulties I don't want to bore you with. In any given generation, the entity who can represent the truth-predicate of the most other entities will dominate. Why? The question, then, is: what logic will the population eventually converge to? If the entities believes in classical logic, and if they believe in induction, they will converge toward the self-reference logic of Solovay (G and G*, or GL and GLS nowadays). I think fair arguments could be given for the fixed-point or revision theories in this scenario, but like I said, both create reference gaps... so some creature could dominate over these by inventing a predicate to fill the gap. That creature will then have its own reference gaps, and yet more gap-filling predicates will be created. I think so. My current thinking is that each gap-filling predicate will correspond to an ordinal number, so that the maximal logic will be one with a gap-filling predicate for each ordinal number. No gap will be left, because if there was such a gap, it would correspond to an ordinal number larger than all ordinal numbers, which is impossible. Of course, this
