Re: [agi] What is the smallest set of operations that can potentially define everything and how do you combine them ?
On Wed, Jul 14, 2010 at 10:35 PM, Michael Swan ms...@voyagergaming.comwrote: I'd argue that mathematical operations are unnecesary, we don't even have integer support inbuilt. I'd disagree. is a mathematical operation, and in combination can become an enormous number of concepts. Sure, I think the brain is more sensibly understood in a programattical sense than mathematical. I say programattical because it probably has 100 billion or so conditional statements, a difficult thing to represent mathematically. Even so, each conditional is going to have maths constructs in it. Sorry, I meant unnecessary to demonstrate that particular point. There's no need to say you have no innate ability to know what 3456/6 is when you are unlikely to have an innate concept of the number 3456 or any other arbitrary number greater than a few hundred to begin with, you can get by with a few lookup tables upon which you get a vague idea what 3456 of something would be, but if I were to show you a sheet of paper with 3000-4000 dots on it, you would be unlikely to be able to tell me whether it was greater or less than 3456. I don't see any way an evaluator of some sort wouldn't be completely necessary for an AGI, sorry for the confusion. Though, you do bring to mind the point that while can be an extremely useful tool for composing other concepts, our internal comparisons do seem to tend more towards the analog than towards the binary, and while you can compose those analog outputs with and - easily enough, you probably want concepts supported as close to natively as is possible. Remember, there are turing-complete one-dimensional systems of cellular automata, but that doesn't make it feasible to port the Linux kernel to them. --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=8660244-6e7fb59c Powered by Listbox: http://www.listbox.com
Re: [agi] What is the smallest set of operations that can potentially define everything and how do you combine them ?
And yet you dream dreams wh. are broad-ranging in subject matter, unlike all programs wh. are extremely narrow-ranging. -- From: Michael Swan ms...@voyagergaming.com Sent: Thursday, July 15, 2010 5:16 AM To: agi agi@v2.listbox.com Subject: Re: [agi] What is the smallest set of operations that can potentially define everything and how do you combine them ? I watched a brain experiment last night that proved that connections between major parts of the brain stop when you are asleep. They put electricity at different brain points, and it went everywhere when the person was a awake, and dissipated when they were asleep. On Thu, 2010-07-15 at 02:13 +0100, Mike Tintner wrote: A demonstration of global connectedness is - associate with anO I get: number, sun, dish, disk, ball, letter, mouth, two fingers, oh, circle, wheel, wire coil, outline, station on metro, hole, Kenneth Noland painting, ring, coin, roundabout connecting among other things - language, numbers, geometry, food, cartoons, paintings, speech, sports, science, technology, art, transport, transportation system, money. Note though the other crucial weakness of the brain wh. impairs global connections - fatigue. To maintain any piece of information in consciousness for long is a strain, (unless it's sexual?). But the above demonstrates IMO why the brain is and has to be an image processor. --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?; Powered by Listbox: http://www.listbox.com --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?; Powered by Listbox: http://www.listbox.com --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=8660244-6e7fb59c Powered by Listbox: http://www.listbox.com
Re: [agi] Comments On My Skepticism of Solomonoff Induction
On Wed, Jul 14, 2010 at 7:46 PM, Abram Demski abramdem...@gmail.com wrote: Jim, There is a simple proof of convergence for the sum involved in defining the probability of a given string in the Solomonoff distribution: At its greatest, a particular string would be output by *all* programs. In this case, its sum would come to 1. This puts an upper bound on the sum. Since there is no subtraction, there is a lower bound at 0 and the sum monotonically increases as we take the limit. Knowing these facts, suppose it *didn't* converge. It must then increase without bound, since it cannot fluctuate back and forth (it can only go up). But this contradicts the upper bound of 1. So, the sum must stop at 1 or below (and in fact we can prove it stops below 1, though we can't say where precisely without the infinite computing power required to compute the limit). --Abram I believe that Solomonoff Induction would be computable given infinite time and infinite resources (the Godel Theorem fits into this category) but some people disagree for reasons I do not understand. If it is not computable then it is not a mathematical theorem and the question of whether the sum of probabilities equals 1 is pure fantasy. If it is computable then the central issue is whether it could (given infinite time and infinite resources) be used to determine the probability of a particular string being produced from all possible programs. The question about the sum of all the probabilities is certainly an interesting question. However, the problem of making sure that the function was actually computable would interfere with this process of determining the probability of each particular string that can be produced. For example, since some strings would be infinite, the computability problem makes it imperative that the infinite strings be partially computed at an iteration (or else the function would be hung up at some particular iteration and the infinite other calculations could not be considered computable). My criticism is that even though I believe the function may be theoretically computable, the fact that each particular probability (of each particular string that is produced) cannot be proven to approach a limit through mathematical analysis, and since the individual probabilities will fluctuate with each new string that is produced, one would have to know how to reorder the production of the probabilities in order to demonstrate that the individual probabilities do approach a limit. If they don't, then the claim that this function could be used to define the probabilities of a particular string from all possible program is unprovable. (Some infinite calculations fluctuate infinitely.) Since you do not have any way to determine how to reorder the infinite probabilities a priori, your algorithm would have to be able to compute all possible reorderings to find the ordering and filtering of the computations that would produce evaluable limits. Since there are trans infinite rearrangements of an infinite list (I am not sure that I am using the term 'trans infinite' properly) this shows that the conclusion that the theorem can be used to derive the desired probabilities is unprovable through a variation of Cantor's Diagonal Argument, and that you can't use Solomonoff Induction the way you have been talking about using it. Since you cannot fully compute every string that may be produced at a certain iteration, you cannot make the claim that you even know the probabilities of any possible string before infinity and therefore your claim that the sum of the probabilities can be computed is not provable. But I could be wrong. Jim Bromer --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=8660244-6e7fb59c Powered by Listbox: http://www.listbox.com
Re: [agi] Comments On My Skepticism of Solomonoff Induction
Jim Bromer wrote: Since you cannot fully compute every string that may be produced at a certain iteration, you cannot make the claim that you even know the probabilities of any possible string before infinity and therefore your claim that the sum of the probabilities can be computed is not provable. But I could be wrong. Could be. Theorem 1.7.2 in http://www.vetta.org/documents/disSol.pdf proves that finding just the shortest program that outputs x gives you a probability for x close to the result you would get if you found all of the (infinite number of) programs that output x. Either number could be used for Solomonoff induction because the difference is bounded only by the choice of language. -- Matt Mahoney, matmaho...@yahoo.com From: Jim Bromer jimbro...@gmail.com To: agi agi@v2.listbox.com Sent: Thu, July 15, 2010 8:18:13 AM Subject: Re: [agi] Comments On My Skepticism of Solomonoff Induction On Wed, Jul 14, 2010 at 7:46 PM, Abram Demski abramdem...@gmail.com wrote: Jim, There is a simple proof of convergence for the sum involved in defining the probability of a given string in the Solomonoff distribution: At its greatest, a particular string would be output by *all* programs. In this case, its sum would come to 1. This puts an upper bound on the sum. Since there is no subtraction, there is a lower bound at 0 and the sum monotonically increases as we take the limit. Knowing these facts, suppose it *didn't* converge. It must then increase without bound, since it cannot fluctuate back and forth (it can only go up). But this contradicts the upper bound of 1. So, the sum must stop at 1 or below (and in fact we can prove it stops below 1, though we can't say where precisely without the infinite computing power required to compute the limit). --Abram I believe that Solomonoff Induction would be computable given infinite time and infinite resources (the Godel Theorem fits into this category) but some people disagree for reasons I do not understand. If it is not computable then it is not a mathematical theorem and the question of whether the sum of probabilities equals 1 is pure fantasy. If it is computable then the central issue is whether it could (given infinite time and infinite resources) be used to determine the probability of a particular string being produced from all possible programs. The question about the sum of all the probabilities is certainly an interesting question. However, the problem of making sure that the function was actually computable would interfere with this process of determining the probability of each particular string that can be produced. For example, since some strings would be infinite, the computability problem makes it imperative that the infinite strings be partially computed at an iteration (or else the function would be hung up at some particular iteration and the infinite other calculations could not be considered computable). My criticism is that even though I believe the function may be theoretically computable, the fact that each particular probability (of each particular string that is produced) cannot be proven to approach a limit through mathematical analysis, and since the individual probabilities will fluctuate with each new string that is produced, one would have to know how to reorder the production of the probabilities in order to demonstrate that the individual probabilities do approach a limit. If they don't, then the claim that this function could be used to define the probabilities of a particular string from all possible program is unprovable. (Some infinite calculations fluctuate infinitely.) Since you do not have any way to determine how to reorder the infinite probabilities a priori, your algorithm would have to be able to compute all possible reorderings to find the ordering and filtering of the computations that would produce evaluable limits. Since there are trans infinite rearrangements of an infinite list (I am not sure that I am using the term 'trans infinite' properly) this shows that the conclusion that the theorem can be used to derive the desired probabilities is unprovable through a variation of Cantor's Diagonal Argument, and that you can't use Solomonoff Induction the way you have been talking about using it. Since you cannot fully compute every string that may be produced at a certain iteration, you cannot make the claim that you even know the probabilities of any possible string before infinity and therefore your claim that the sum of the probabilities can be computed is not provable. But I could be wrong. Jim Bromer agi | Archives | Modify Your Subscription --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription:
Re: [agi] How do we Score Hypotheses?
It is no wonder that I'm having a hard time finding documentation on hypothesis scoring. Few can agree on how to do it and there is much debate about it. I noticed though that a big reason for the problems is that explanatory reasoning is being applied to many diverse problems. I think, like I mentioned before, that people should not try to come up with a single universal rule set for applying explanatory reasoning to every possible problem. So, maybe that's where the hold up is. I've been testing my ideas out on complex examples. But now I'm going to go back to simplified model testing (although not as simple as black squares :) ) and work my way up again. Dave On Wed, Jul 14, 2010 at 12:59 PM, David Jones davidher...@gmail.com wrote: Actually, I just realized that there is a way to included inductive knowledge and experience into this algorithm. Inductive knowledge and experience about a specific object or object type can be exploited to know which hypotheses in the past were successful, and therefore which hypothesis is most likely. By choosing the most likely hypothesis first, we skip a lot of messy hypothesis comparison processing and analysis. If we choose the right hypothesis first, all we really have to do is verify that this hypothesis reveals in the data what we expect to be there. If we confirm what we expect, that is reason enough not to look for other hypotheses because the data is explained by what we originally believed to be likely. We only look for additional hypotheses when we find something unexplained. And even then, we don't look at the whole problem. We only look at what we have to to explain the unexplained data. In fact, we could even ignore the unexplained data if we believe, from experience, that it isn't pertinent. I discovered this because I'm analyzing how a series of hypotheses are navigated when analyzing images. It seems to me that it is done very similarly to way we do it. We sort of confirm what we expect and try to explain what we don't expect. We try out hypotheses in a sort of trial and error manor and see how each hypothesis affects what we find in the image. If we confirm things because of the hypothesis, we are likely to keep it. We keep going, navigating the tree of hypotheses, conflicts and unexpected observations until we find a good hypothesis. Something like that. I'm attempting to construct an algorithm for doing this as I analyze specific problems. Dave On Wed, Jul 14, 2010 at 10:22 AM, David Jones davidher...@gmail.comwrote: What do you mean by definitive events? I guess the first problem I see with my approach is that the movement of the window is also a hypothesis. I need to analyze it in more detail and see how the tree of hypotheses affects the hypotheses regarding the es on the windows. What I believe is that these problems can be broken down into types of hypotheses, types of events and types of relationships. then those types can be reasoned about in a general way. If possible, then you have a method for reasoning about any object that is covered by the types of hypotheses, events and relationships that you have defined. How to reason about specific objects should not be preprogrammed. But, I think the solution to this part of AGI is to find general ways to reason about a small set of concepts that can be combined to describe specific objects and situations. There are other parts to AGI that I am not considering yet. I believe the problem has to be broken down into separate pieces and understood before putting it back together into a complete system. I have not covered inductive learning for example, which would be an important part of AGI. I have also not yet incorporated learned experience into the algorithm, which is also important. The general AI problem is way too complicated to consider all at once. I simply can't solve hypothesis generation, comparison and disambiguation while at the same time solving induction and experience-based reasoning. It becomes unwieldly. So, I'm starting where I can and I'll work my way up to the full complexity of the problem. I don't really understand what you mean here: The central unsolved problem, in my view, is: How can hypotheses be conceptually integrated along with the observable definitive events of the problem to form good explanatory connections that can mesh well with other knowledge about the problem that is considered to be reliable. The second problem is finding efficient ways to represent this complexity of knowledge so that the program can utilize it efficiently. You also might want to include concrete problems to analyze for your central problem suggestions. That would help define the problem a bit better for analysis. Dave On Wed, Jul 14, 2010 at 8:30 AM, Jim Bromer jimbro...@gmail.com wrote: On Tue, Jul 13, 2010 at 9:05 PM, Jim Bromer jimbro...@gmail.com wrote: Even if you refined your model until it was just
Re: [agi] How do we Score Hypotheses?
Hypotheses are scored using Bayes law. Let D be your observed data and H be your hypothesis. Then p(H|D) = p(D|H)p(H)/p(D). Since p(D) is constant, you can remove it and rank hypotheses by p(D|H)p(H). p(H) can be estimated using the minimum description length principle or Solomonoff induction. Ideally, p(H) = 2^-|H| where |H| is the length (in bits) of the description of the hypothesis. The value is language dependent, so this method is not perfect. -- Matt Mahoney, matmaho...@yahoo.com From: David Jones davidher...@gmail.com To: agi agi@v2.listbox.com Sent: Thu, July 15, 2010 10:22:44 AM Subject: Re: [agi] How do we Score Hypotheses? It is no wonder that I'm having a hard time finding documentation on hypothesis scoring. Few can agree on how to do it and there is much debate about it. I noticed though that a big reason for the problems is that explanatory reasoning is being applied to many diverse problems. I think, like I mentioned before, that people should not try to come up with a single universal rule set for applying explanatory reasoning to every possible problem. So, maybe that's where the hold up is. I've been testing my ideas out on complex examples. But now I'm going to go back to simplified model testing (although not as simple as black squares :) ) and work my way up again. Dave On Wed, Jul 14, 2010 at 12:59 PM, David Jones davidher...@gmail.com wrote: Actually, I just realized that there is a way to included inductive knowledge and experience into this algorithm. Inductive knowledge and experience about a specific object or object type can be exploited to know which hypotheses in the past were successful, and therefore which hypothesis is most likely. By choosing the most likely hypothesis first, we skip a lot of messy hypothesis comparison processing and analysis. If we choose the right hypothesis first, all we really have to do is verify that this hypothesis reveals in the data what we expect to be there. If we confirm what we expect, that is reason enough not to look for other hypotheses because the data is explained by what we originally believed to be likely. We only look for additional hypotheses when we find something unexplained. And even then, we don't look at the whole problem. We only look at what we have to to explain the unexplained data. In fact, we could even ignore the unexplained data if we believe, from experience, that it isn't pertinent. I discovered this because I'm analyzing how a series of hypotheses are navigated when analyzing images. It seems to me that it is done very similarly to way we do it. We sort of confirm what we expect and try to explain what we don't expect. We try out hypotheses in a sort of trial and error manor and see how each hypothesis affects what we find in the image. If we confirm things because of the hypothesis, we are likely to keep it. We keep going, navigating the tree of hypotheses, conflicts and unexpected observations until we find a good hypothesis. Something like that. I'm attempting to construct an algorithm for doing this as I analyze specific problems. Dave On Wed, Jul 14, 2010 at 10:22 AM, David Jones davidher...@gmail.com wrote: What do you mean by definitive events? I guess the first problem I see with my approach is that the movement of the window is also a hypothesis. I need to analyze it in more detail and see how the tree of hypotheses affects the hypotheses regarding the es on the windows. What I believe is that these problems can be broken down into types of hypotheses, types of events and types of relationships. then those types can be reasoned about in a general way. If possible, then you have a method for reasoning about any object that is covered by the types of hypotheses, events and relationships that you have defined. How to reason about specific objects should not be preprogrammed. But, I think the solution to this part of AGI is to find general ways to reason about a small set of concepts that can be combined to describe specific objects and situations. There are other parts to AGI that I am not considering yet. I believe the problem has to be broken down into separate pieces and understood before putting it back together into a complete system. I have not covered inductive learning for example, which would be an important part of AGI. I have also not yet incorporated learned experience into the algorithm, which is also important. The general AI problem is way too complicated to consider all at once. I simply can't solve hypothesis generation, comparison and disambiguation while at the same time solving induction and experience-based reasoning. It becomes unwieldly. So, I'm starting where I can and I'll work my way up to the full complexity of the problem. I don't really understand what you mean here: The central unsolved problem, in my view, is: How can hypotheses be
Re: [agi] How do we Score Hypotheses?
:) You say that as if bayesian explanatory reasoning is the only way. There is much debate over bayesian explanatory reasoning and non-bayesian. There are pros and cons to bayesian methods. Likewise, there is the problem with non-bayesian methods because few have figured out how to do it effectively. I'm still going to pursue a non-bayesian approach because I believe there is likely more merit to it and that the short-comings can be overcome. Dave On Thu, Jul 15, 2010 at 10:54 AM, Matt Mahoney matmaho...@yahoo.com wrote: Hypotheses are scored using Bayes law. Let D be your observed data and H be your hypothesis. Then p(H|D) = p(D|H)p(H)/p(D). Since p(D) is constant, you can remove it and rank hypotheses by p(D|H)p(H). p(H) can be estimated using the minimum description length principle or Solomonoff induction. Ideally, p(H) = 2^-|H| where |H| is the length (in bits) of the description of the hypothesis. The value is language dependent, so this method is not perfect. -- Matt Mahoney, matmaho...@yahoo.com -- *From:* David Jones davidher...@gmail.com *To:* agi agi@v2.listbox.com *Sent:* Thu, July 15, 2010 10:22:44 AM *Subject:* Re: [agi] How do we Score Hypotheses? It is no wonder that I'm having a hard time finding documentation on hypothesis scoring. Few can agree on how to do it and there is much debate about it. I noticed though that a big reason for the problems is that explanatory reasoning is being applied to many diverse problems. I think, like I mentioned before, that people should not try to come up with a single universal rule set for applying explanatory reasoning to every possible problem. So, maybe that's where the hold up is. I've been testing my ideas out on complex examples. But now I'm going to go back to simplified model testing (although not as simple as black squares :) ) and work my way up again. Dave On Wed, Jul 14, 2010 at 12:59 PM, David Jones davidher...@gmail.comwrote: Actually, I just realized that there is a way to included inductive knowledge and experience into this algorithm. Inductive knowledge and experience about a specific object or object type can be exploited to know which hypotheses in the past were successful, and therefore which hypothesis is most likely. By choosing the most likely hypothesis first, we skip a lot of messy hypothesis comparison processing and analysis. If we choose the right hypothesis first, all we really have to do is verify that this hypothesis reveals in the data what we expect to be there. If we confirm what we expect, that is reason enough not to look for other hypotheses because the data is explained by what we originally believed to be likely. We only look for additional hypotheses when we find something unexplained. And even then, we don't look at the whole problem. We only look at what we have to to explain the unexplained data. In fact, we could even ignore the unexplained data if we believe, from experience, that it isn't pertinent. I discovered this because I'm analyzing how a series of hypotheses are navigated when analyzing images. It seems to me that it is done very similarly to way we do it. We sort of confirm what we expect and try to explain what we don't expect. We try out hypotheses in a sort of trial and error manor and see how each hypothesis affects what we find in the image. If we confirm things because of the hypothesis, we are likely to keep it. We keep going, navigating the tree of hypotheses, conflicts and unexpected observations until we find a good hypothesis. Something like that. I'm attempting to construct an algorithm for doing this as I analyze specific problems. Dave On Wed, Jul 14, 2010 at 10:22 AM, David Jones davidher...@gmail.comwrote: What do you mean by definitive events? I guess the first problem I see with my approach is that the movement of the window is also a hypothesis. I need to analyze it in more detail and see how the tree of hypotheses affects the hypotheses regarding the es on the windows. What I believe is that these problems can be broken down into types of hypotheses, types of events and types of relationships. then those types can be reasoned about in a general way. If possible, then you have a method for reasoning about any object that is covered by the types of hypotheses, events and relationships that you have defined. How to reason about specific objects should not be preprogrammed. But, I think the solution to this part of AGI is to find general ways to reason about a small set of concepts that can be combined to describe specific objects and situations. There are other parts to AGI that I am not considering yet. I believe the problem has to be broken down into separate pieces and understood before putting it back together into a complete system. I have not covered inductive learning for example, which would be an important part of AGI. I have also not yet incorporated
RE: [agi] OFF-TOPIC: University of Hong Kong Library
Make sure you study that up YKY :) John From: YKY (Yan King Yin, 甄景贤) [mailto:generic.intellige...@gmail.com] Sent: Thursday, July 15, 2010 8:59 AM To: agi Subject: [agi] OFF-TOPIC: University of Hong Kong Library Today, I went to the HKU main library: =) KY agi | https://www.listbox.com/member/archive/303/=now Archives https://www.listbox.com/member/archive/rss/303/ Description: https://www.listbox.com/images/feed-icon-10x10.jpg| https://www.listbox.com/member/?; Modify Your Subscription http://www.listbox.com/ Description: https://www.listbox.com/images/listbox-logo-small.png --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=8660244-6e7fb59c Powered by Listbox: http://www.listbox.com image001.jpgimage002.png
Re: [agi] Comments On My Skepticism of Solomonoff Induction
I think that Solomonoff Induction includes a computational method that produces probabilities of some sort and whenever those probabilities were computed (in a way that would make the function computable) they would sum up to 1. But the issue that I am pointing out is that there is no way that you can determine the margin of error in what is being computed for what has been repeatedly claimed that the function is capable of computing. Since you are not able to rely on something like the theory of limits, you are not able to determine the degree of error in what is being computed. And in fact, there is no way to determine that what the function would compute would be in any way useful for the sort of things that you guys keep talking about. Jim Bromer On Thu, Jul 15, 2010 at 8:18 AM, Jim Bromer jimbro...@gmail.com wrote: On Wed, Jul 14, 2010 at 7:46 PM, Abram Demski abramdem...@gmail.comwrote: Jim, There is a simple proof of convergence for the sum involved in defining the probability of a given string in the Solomonoff distribution: At its greatest, a particular string would be output by *all* programs. In this case, its sum would come to 1. This puts an upper bound on the sum. Since there is no subtraction, there is a lower bound at 0 and the sum monotonically increases as we take the limit. Knowing these facts, suppose it *didn't* converge. It must then increase without bound, since it cannot fluctuate back and forth (it can only go up). But this contradicts the upper bound of 1. So, the sum must stop at 1 or below (and in fact we can prove it stops below 1, though we can't say where precisely without the infinite computing power required to compute the limit). --Abram I believe that Solomonoff Induction would be computable given infinite time and infinite resources (the Godel Theorem fits into this category) but some people disagree for reasons I do not understand. If it is not computable then it is not a mathematical theorem and the question of whether the sum of probabilities equals 1 is pure fantasy. If it is computable then the central issue is whether it could (given infinite time and infinite resources) be used to determine the probability of a particular string being produced from all possible programs. The question about the sum of all the probabilities is certainly an interesting question. However, the problem of making sure that the function was actually computable would interfere with this process of determining the probability of each particular string that can be produced. For example, since some strings would be infinite, the computability problem makes it imperative that the infinite strings be partially computed at an iteration (or else the function would be hung up at some particular iteration and the infinite other calculations could not be considered computable). My criticism is that even though I believe the function may be theoretically computable, the fact that each particular probability (of each particular string that is produced) cannot be proven to approach a limit through mathematical analysis, and since the individual probabilities will fluctuate with each new string that is produced, one would have to know how to reorder the production of the probabilities in order to demonstrate that the individual probabilities do approach a limit. If they don't, then the claim that this function could be used to define the probabilities of a particular string from all possible program is unprovable. (Some infinite calculations fluctuate infinitely.) Since you do not have any way to determine how to reorder the infinite probabilities a priori, your algorithm would have to be able to compute all possible reorderings to find the ordering and filtering of the computations that would produce evaluable limits. Since there are trans infinite rearrangements of an infinite list (I am not sure that I am using the term 'trans infinite' properly) this shows that the conclusion that the theorem can be used to derive the desired probabilities is unprovable through a variation of Cantor's Diagonal Argument, and that you can't use Solomonoff Induction the way you have been talking about using it. Since you cannot fully compute every string that may be produced at a certain iteration, you cannot make the claim that you even know the probabilities of any possible string before infinity and therefore your claim that the sum of the probabilities can be computed is not provable. But I could be wrong. Jim Bromer --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=8660244-6e7fb59c Powered by Listbox: http://www.listbox.com
Re: [agi] How do we Score Hypotheses?
On Wed, Jul 14, 2010 at 10:22 AM, David Jones davidher...@gmail.com wrote: What do you mean by definitive events? I was just trying to find a way to designate obsverations that would be reliably obvious to a computer program. This has something to do with the assumptions that you are using. For example if some object appeared against a stable background and it was a different color than the background, it would be a definitive observation event because your algorithm could detect it with some certainty and use it in the definition of other more complicated events (like occlusion.) Notice that this example would not necessarily be so obvious (a definitive event) using a camera, because there are a number of ways that an illusion (of some kind) could end up as a data event. I will try to reply to the rest of your message sometime later. Jim Bromer --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=8660244-6e7fb59c Powered by Listbox: http://www.listbox.com
Re: [agi] How do we Score Hypotheses?
Jim, even that isn't an obvious event. You don't know what is background and what is not. You don't even know if there is an object or not. You don't know if anything moved or not. You can make some observations using predefined methods and then see if you find matches... then hypothesize about the matches... It all has to be learned and figured out through reasoning. That's why I asked what you meant by definitive events. Nothing is really definitive. It is all hypothesized in a non-monotonic manner. Dave On Thu, Jul 15, 2010 at 12:01 PM, Jim Bromer jimbro...@gmail.com wrote: On Wed, Jul 14, 2010 at 10:22 AM, David Jones davidher...@gmail.comwrote: What do you mean by definitive events? I was just trying to find a way to designate obsverations that would be reliably obvious to a computer program. This has something to do with the assumptions that you are using. For example if some object appeared against a stable background and it was a different color than the background, it would be a definitive observation event because your algorithm could detect it with some certainty and use it in the definition of other more complicated events (like occlusion.) Notice that this example would not necessarily be so obvious (a definitive event) using a camera, because there are a number of ways that an illusion (of some kind) could end up as a data event. I will try to reply to the rest of your message sometime later. Jim Bromer *agi* | Archives https://www.listbox.com/member/archive/303/=now https://www.listbox.com/member/archive/rss/303/ | Modifyhttps://www.listbox.com/member/?;Your Subscription http://www.listbox.com --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=8660244-6e7fb59c Powered by Listbox: http://www.listbox.com
Re: [agi] How do we Score Hypotheses?
Sounds like a good explanation of why a body is essential for vision - not just for POV and orientation [up/left/right/down/ towards/ away] but for comparison and yardstick - you do know when your body or parts thereof are moving -and it's not merely touch but the comparison of other objects still and moving with your own moving hands and body that is important. The more you go into it, the crazier the prospect of vision without eyes in a body becomes. From: David Jones Sent: Thursday, July 15, 2010 5:54 PM To: agi Subject: Re: [agi] How do we Score Hypotheses? Jim, even that isn't an obvious event. You don't know what is background and what is not. You don't even know if there is an object or not. You don't know if anything moved or not. You can make some observations using predefined methods and then see if you find matches... then hypothesize about the matches... It all has to be learned and figured out through reasoning. That's why I asked what you meant by definitive events. Nothing is really definitive. It is all hypothesized in a non-monotonic manner. Dave On Thu, Jul 15, 2010 at 12:01 PM, Jim Bromer jimbro...@gmail.com wrote: On Wed, Jul 14, 2010 at 10:22 AM, David Jones davidher...@gmail.com wrote: What do you mean by definitive events? I was just trying to find a way to designate obsverations that would be reliably obvious to a computer program. This has something to do with the assumptions that you are using. For example if some object appeared against a stable background and it was a different color than the background, it would be a definitive observation event because your algorithm could detect it with some certainty and use it in the definition of other more complicated events (like occlusion.) Notice that this example would not necessarily be so obvious (a definitive event) using a camera, because there are a number of ways that an illusion (of some kind) could end up as a data event. I will try to reply to the rest of your message sometime later. Jim Bromer agi | Archives | Modify Your Subscription agi | Archives | Modify Your Subscription --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=8660244-6e7fb59c Powered by Listbox: http://www.listbox.com
Re: [agi] How do we Score Hypotheses?
On screenshots, the point of view is equivalent to the absolute positions and their relative positions using absolute(screen x and y) measurements. You don't need a robot to learn about how AGI works and figure out how to solve some problems. It would be a terrible mistake to spend years, or even weeks for that matter, on robotics before getting started. Dave On Thu, Jul 15, 2010 at 1:09 PM, Mike Tintner tint...@blueyonder.co.ukwrote: Sounds like a good explanation of why a body is essential for vision - not just for POV and orientation [up/left/right/down/ towards/ away] but for comparison and yardstick - you do know when your body or parts thereof are moving -and it's not merely touch but the comparison of other objects still and moving with your own moving hands and body that is important. The more you go into it, the crazier the prospect of vision without eyes in a body becomes. *From:* David Jones davidher...@gmail.com *Sent:* Thursday, July 15, 2010 5:54 PM *To:* agi agi@v2.listbox.com *Subject:* Re: [agi] How do we Score Hypotheses? Jim, even that isn't an obvious event. You don't know what is background and what is not. You don't even know if there is an object or not. You don't know if anything moved or not. You can make some observations using predefined methods and then see if you find matches... then hypothesize about the matches... It all has to be learned and figured out through reasoning. That's why I asked what you meant by definitive events. Nothing is really definitive. It is all hypothesized in a non-monotonic manner. Dave On Thu, Jul 15, 2010 at 12:01 PM, Jim Bromer jimbro...@gmail.com wrote: On Wed, Jul 14, 2010 at 10:22 AM, David Jones davidher...@gmail.comwrote: What do you mean by definitive events? I was just trying to find a way to designate obsverations that would be reliably obvious to a computer program. This has something to do with the assumptions that you are using. For example if some object appeared against a stable background and it was a different color than the background, it would be a definitive observation event because your algorithm could detect it with some certainty and use it in the definition of other more complicated events (like occlusion.) Notice that this example would not necessarily be so obvious (a definitive event) using a camera, because there are a number of ways that an illusion (of some kind) could end up as a data event. I will try to reply to the rest of your message sometime later. Jim Bromer *agi* | Archives https://www.listbox.com/member/archive/303/=now https://www.listbox.com/member/archive/rss/303/ | Modifyhttps://www.listbox.com/member/?;Your Subscription http://www.listbox.com *agi* | Archives https://www.listbox.com/member/archive/303/=now https://www.listbox.com/member/archive/rss/303/ | Modifyhttps://www.listbox.com/member/?;Your Subscription http://www.listbox.com *agi* | Archives https://www.listbox.com/member/archive/303/=now https://www.listbox.com/member/archive/rss/303/ | Modifyhttps://www.listbox.com/member/?;Your Subscription http://www.listbox.com --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=8660244-6e7fb59c Powered by Listbox: http://www.listbox.com
Re: [agi] OFF-TOPIC: University of Hong Kong Library
Ok Off topic, but not as far as you might think. YKY has posted in Creating Artificial Intelligence on a collaborative project. It is quite important to know *exactly* where he is. You see Taiwan uses the classical character set, The People's Republic uses a simplified character set. Hong Kong was handed back to China in I think 1997. It is still outside the Great Firewall and (I presume) uses classical characters, although I don't really know. If we are to discuss transliteration schemes, translation and writing Chinese (PRC or Taiwan) on Western keyboards, it is important for us to know. I have just bashed up a Java program to write Arabic. You input Roman Buckwalter and it has an internal conversion table. The same thing could in principle be done for a load of character sets. In Chinese you would have to input two Western keys simultaneously. That can be done. I know HK is outside the Firewall because that is where Google has its proxy server. Is YKY there, do you know? - Ian Parker 2010/7/15 John G. Rose johnr...@polyplexic.com Make sure you study that up YKY :) John *From:* YKY (Yan King Yin, 甄景贤) [mailto:generic.intellige...@gmail.com] *Sent:* Thursday, July 15, 2010 8:59 AM *To:* agi *Subject:* [agi] OFF-TOPIC: University of Hong Kong Library Today, I went to the HKU main library: =) KY *agi* | Archives https://www.listbox.com/member/archive/303/=now [image: Description: https://www.listbox.com/images/feed-icon-10x10.jpg]https://www.listbox.com/member/archive/rss/303/| Modify https://www.listbox.com/member/?; Your Subscription [image: Description: https://www.listbox.com/images/listbox-logo-small.png]http://www.listbox.com/ *agi* | Archives https://www.listbox.com/member/archive/303/=now https://www.listbox.com/member/archive/rss/303/ | Modifyhttps://www.listbox.com/member/?;Your Subscription http://www.listbox.com --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=8660244-6e7fb59c Powered by Listbox: http://www.listbox.com image002.pngimage001.jpg
Re: [agi] Comments On My Skepticism of Solomonoff Induction
Jim, Yes this is true provable: there is no way to compute a correct error bound such that it converges to 0 as the computation of algorithmic probability converges to the correct number. More specifically--- we can approximate the algorithmic probability from below, computing better lower bounds which converge to the correct number, but we cannot approximate it from above, as there is no procedure (and can never be any procedure) which creates closer and closer upper bounds converging to the correct number. (We can produce upper bounds that get closer and closer w/o getting arbitrarily near, and we can produce numbers which do approach arbitrarily near to the correct number in the limit but sometimes dip below for a time; but we can't have both features.) The question of whether the function would be useful for the sorts of things we keep talking about ... well, I think the best argument that I can give is that MDL is strongly supported by both theory and practice for many *subsets* of the full program space. The concern might be that, so far, it is only supported by *theory* for the full program space-- and since approximations have very bad error-bound properties, it may never be supported in practice. My reply to this would be that it still appears useful to approximate Solomonoff induction, since most successful predictors can be viewed as approximations to Solomonoff induction. It approximates solomonoff induction appears to be a good _explanation_ for the success of many systems. What sort of alternatives do you have in mind, by the way? --Abram On Thu, Jul 15, 2010 at 11:50 AM, Jim Bromer jimbro...@gmail.com wrote: I think that Solomonoff Induction includes a computational method that produces probabilities of some sort and whenever those probabilities were computed (in a way that would make the function computable) they would sum up to 1. But the issue that I am pointing out is that there is no way that you can determine the margin of error in what is being computed for what has been repeatedly claimed that the function is capable of computing. Since you are not able to rely on something like the theory of limits, you are not able to determine the degree of error in what is being computed. And in fact, there is no way to determine that what the function would compute would be in any way useful for the sort of things that you guys keep talking about. Jim Bromer On Thu, Jul 15, 2010 at 8:18 AM, Jim Bromer jimbro...@gmail.com wrote: On Wed, Jul 14, 2010 at 7:46 PM, Abram Demski abramdem...@gmail.comwrote: Jim, There is a simple proof of convergence for the sum involved in defining the probability of a given string in the Solomonoff distribution: At its greatest, a particular string would be output by *all* programs. In this case, its sum would come to 1. This puts an upper bound on the sum. Since there is no subtraction, there is a lower bound at 0 and the sum monotonically increases as we take the limit. Knowing these facts, suppose it *didn't* converge. It must then increase without bound, since it cannot fluctuate back and forth (it can only go up). But this contradicts the upper bound of 1. So, the sum must stop at 1 or below (and in fact we can prove it stops below 1, though we can't say where precisely without the infinite computing power required to compute the limit). --Abram I believe that Solomonoff Induction would be computable given infinite time and infinite resources (the Godel Theorem fits into this category) but some people disagree for reasons I do not understand. If it is not computable then it is not a mathematical theorem and the question of whether the sum of probabilities equals 1 is pure fantasy. If it is computable then the central issue is whether it could (given infinite time and infinite resources) be used to determine the probability of a particular string being produced from all possible programs. The question about the sum of all the probabilities is certainly an interesting question. However, the problem of making sure that the function was actually computable would interfere with this process of determining the probability of each particular string that can be produced. For example, since some strings would be infinite, the computability problem makes it imperative that the infinite strings be partially computed at an iteration (or else the function would be hung up at some particular iteration and the infinite other calculations could not be considered computable). My criticism is that even though I believe the function may be theoretically computable, the fact that each particular probability (of each particular string that is produced) cannot be proven to approach a limit through mathematical analysis, and since the individual probabilities will fluctuate with each new string that is produced, one would have to know how to reorder the production of the probabilities in order to demonstrate
Re: [agi] Comments On My Skepticism of Solomonoff Induction
We all make conjectures all of the time, but we don't often don't have anyway to establish credibility for the claims that are made. So I wanted to examine one part of this field, and the idea that seemed most natural for me was Solomonoff Induction. I have reached a conclusion about the subject and that conclusion is that all of the claims that I have seen made about Solomonoff Induction are rationally unfounded including the one that you just made when you said: We can produce upper bounds that get closer and closer w/o getting arbitrarily near, and we can produce numbers which do approach arbitrarily near to the correct number in the limit but sometimes dip below for a time; but we can't have both features. Your inability to fully recognize or perhaps acknowledge that you cannot use Solomonoff Induction to make this claim is difficult for me to comprehend. While the fields of compression and probability have an impressive body of evidence supporting them, I simply have no reason to think the kind of claims that have been made about Solomonoff Induction have any merit. By natural induction I feel comfortable drawing the conclusion that this whole area related to algorithmic information theory is based on shallow methods of reasoning. It can be useful, as it was for me, just as means of exploring ideas that I would not have otherwise explored. But its usefulness comes in learning how to determine its lack of merit. I will write one more thing about my feelings about computability, but I will start a new thread and just mention the relation to this thread. Jim Bromer On Thu, Jul 15, 2010 at 2:45 PM, Abram Demski abramdem...@gmail.com wrote: Jim, Yes this is true provable: there is no way to compute a correct error bound such that it converges to 0 as the computation of algorithmic probability converges to the correct number. More specifically--- we can approximate the algorithmic probability from below, computing better lower bounds which converge to the correct number, but we cannot approximate it from above, as there is no procedure (and can never be any procedure) which creates closer and closer upper bounds converging to the correct number. (We can produce upper bounds that get closer and closer w/o getting arbitrarily near, and we can produce numbers which do approach arbitrarily near to the correct number in the limit but sometimes dip below for a time; but we can't have both features.) The question of whether the function would be useful for the sorts of things we keep talking about ... well, I think the best argument that I can give is that MDL is strongly supported by both theory and practice for many *subsets* of the full program space. The concern might be that, so far, it is only supported by *theory* for the full program space-- and since approximations have very bad error-bound properties, it may never be supported in practice. My reply to this would be that it still appears useful to approximate Solomonoff induction, since most successful predictors can be viewed as approximations to Solomonoff induction. It approximates solomonoff induction appears to be a good _explanation_ for the success of many systems. What sort of alternatives do you have in mind, by the way? --Abram On Thu, Jul 15, 2010 at 11:50 AM, Jim Bromer jimbro...@gmail.comwrote: I think that Solomonoff Induction includes a computational method that produces probabilities of some sort and whenever those probabilities were computed (in a way that would make the function computable) they would sum up to 1. But the issue that I am pointing out is that there is no way that you can determine the margin of error in what is being computed for what has been repeatedly claimed that the function is capable of computing. Since you are not able to rely on something like the theory of limits, you are not able to determine the degree of error in what is being computed. And in fact, there is no way to determine that what the function would compute would be in any way useful for the sort of things that you guys keep talking about. Jim Bromer On Thu, Jul 15, 2010 at 8:18 AM, Jim Bromer jimbro...@gmail.com wrote: On Wed, Jul 14, 2010 at 7:46 PM, Abram Demski abramdem...@gmail.comwrote: Jim, There is a simple proof of convergence for the sum involved in defining the probability of a given string in the Solomonoff distribution: At its greatest, a particular string would be output by *all* programs. In this case, its sum would come to 1. This puts an upper bound on the sum. Since there is no subtraction, there is a lower bound at 0 and the sum monotonically increases as we take the limit. Knowing these facts, suppose it *didn't* converge. It must then increase without bound, since it cannot fluctuate back and forth (it can only go up). But this contradicts the upper bound of 1. So, the sum must stop at 1 or below (and in fact we can prove it stops below 1,
RE: [agi] OFF-TOPIC: University of Hong Kong Library
-Original Message- From: Ian Parker [mailto:ianpark...@gmail.com] Ok Off topic, but not as far as you might think. YKY has posted in Creating Artificial Intelligence on a collaborative project. It is quite important to know exactly where he is. You see Taiwan uses the classical character set, The People's Republic uses a simplified character set. The classical character set is much more artistic but more difficult to learn thus the simplified is becoming popular. Like a social tendency of K-complexity minimalistic language langour. Less energy expended since less bits required for the symbols. Hong Kong was handed back to China in I think 1997. It is still outside the Great Firewall and (I presume) uses classical characters, although I don't really know. If we are to discuss transliteration schemes, translation and writing Chinese (PRC or Taiwan) on Western keyboards, it is important for us to know. I have just bashed up a Java program to write Arabic. You input Roman Buckwalter and it has an internal conversion table. The same thing could in principle be done for a load of character sets. In Chinese you would have to input two Western keys simultaneously. That can be done. I always wondered - do language translators map from one language to another or do they map to a universal language first. And if there is a universal language what is it or.. what are they? I know HK is outside the Firewall because that is where Google has its proxy server. Is YKY there, do you know? Uhm yes. He's been followed by the government censors into the HK library. They're thinking about sending him to re-education camp for being caught red-handed reading AI4U. John --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=8660244-6e7fb59c Powered by Listbox: http://www.listbox.com
Re: [agi] How do we Score Hypotheses?
On Wed, Jul 14, 2010 at 10:22 AM, David Jones davidher...@gmail.com wrote: I don't really understand what you mean here: The central unsolved problem, in my view, is: How can hypotheses be conceptually integrated along with the observable definitive events of the problem to form good explanatory connections that can mesh well with other knowledge about the problem that is considered to be reliable. The second problem is finding efficient ways to represent this complexity of knowledge so that the program can utilize it efficiently. You also might want to include concrete problems to analyze for your central problem suggestions. That would help define the problem a bit better for analysis. Dave I suppose a hypotheses is a kind of concepts. So there are other kinds of concepts that we need to use with hypotheses. A hypotheses has to be conceptually integrated into other concepts. Conceptual integration is something of greater complexity than shallow deduction or probability chains. While reasoning chains are needed in conceptual integration, conceptual integration is to a chain of reasoning what a multi dimension structure is to a one dimensional chain. I will try to come up with some examples. Jim Bromer --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=8660244-6e7fb59c Powered by Listbox: http://www.listbox.com
Re: [agi] Comments On My Skepticism of Solomonoff Induction
Jim, The statements about bounds are mathematically provable... furthermore, I was just agreeing with what you said, and pointing out that the statement could be proven. So what is your issue? I am confused at your response. Is it because I didn't include the proofs in my email? --Abram On Thu, Jul 15, 2010 at 7:30 PM, Jim Bromer jimbro...@gmail.com wrote: We all make conjectures all of the time, but we don't often don't have anyway to establish credibility for the claims that are made. So I wanted to examine one part of this field, and the idea that seemed most natural for me was Solomonoff Induction. I have reached a conclusion about the subject and that conclusion is that all of the claims that I have seen made about Solomonoff Induction are rationally unfounded including the one that you just made when you said: We can produce upper bounds that get closer and closer w/o getting arbitrarily near, and we can produce numbers which do approach arbitrarily near to the correct number in the limit but sometimes dip below for a time; but we can't have both features. Your inability to fully recognize or perhaps acknowledge that you cannot use Solomonoff Induction to make this claim is difficult for me to comprehend. While the fields of compression and probability have an impressive body of evidence supporting them, I simply have no reason to think the kind of claims that have been made about Solomonoff Induction have any merit. By natural induction I feel comfortable drawing the conclusion that this whole area related to algorithmic information theory is based on shallow methods of reasoning. It can be useful, as it was for me, just as means of exploring ideas that I would not have otherwise explored. But its usefulness comes in learning how to determine its lack of merit. I will write one more thing about my feelings about computability, but I will start a new thread and just mention the relation to this thread. Jim Bromer On Thu, Jul 15, 2010 at 2:45 PM, Abram Demski abramdem...@gmail.comwrote: Jim, Yes this is true provable: there is no way to compute a correct error bound such that it converges to 0 as the computation of algorithmic probability converges to the correct number. More specifically--- we can approximate the algorithmic probability from below, computing better lower bounds which converge to the correct number, but we cannot approximate it from above, as there is no procedure (and can never be any procedure) which creates closer and closer upper bounds converging to the correct number. (We can produce upper bounds that get closer and closer w/o getting arbitrarily near, and we can produce numbers which do approach arbitrarily near to the correct number in the limit but sometimes dip below for a time; but we can't have both features.) The question of whether the function would be useful for the sorts of things we keep talking about ... well, I think the best argument that I can give is that MDL is strongly supported by both theory and practice for many *subsets* of the full program space. The concern might be that, so far, it is only supported by *theory* for the full program space-- and since approximations have very bad error-bound properties, it may never be supported in practice. My reply to this would be that it still appears useful to approximate Solomonoff induction, since most successful predictors can be viewed as approximations to Solomonoff induction. It approximates solomonoff induction appears to be a good _explanation_ for the success of many systems. What sort of alternatives do you have in mind, by the way? --Abram On Thu, Jul 15, 2010 at 11:50 AM, Jim Bromer jimbro...@gmail.comwrote: I think that Solomonoff Induction includes a computational method that produces probabilities of some sort and whenever those probabilities were computed (in a way that would make the function computable) they would sum up to 1. But the issue that I am pointing out is that there is no way that you can determine the margin of error in what is being computed for what has been repeatedly claimed that the function is capable of computing. Since you are not able to rely on something like the theory of limits, you are not able to determine the degree of error in what is being computed. And in fact, there is no way to determine that what the function would compute would be in any way useful for the sort of things that you guys keep talking about. Jim Bromer On Thu, Jul 15, 2010 at 8:18 AM, Jim Bromer jimbro...@gmail.com wrote: On Wed, Jul 14, 2010 at 7:46 PM, Abram Demski abramdem...@gmail.comwrote: Jim, There is a simple proof of convergence for the sum involved in defining the probability of a given string in the Solomonoff distribution: At its greatest, a particular string would be output by *all* programs. In this case, its sum would come to 1. This puts an upper bound on the sum. Since
