Re: [agi] Hypercomputation and AGI
On Dec 30, 2008, at 11:45 AM, Steve Richfield wrote: Bingo! You have to tailor the techniques to the problem - more than just solving the equations, but often the representation of quantities needs to be in some sort of multivalued form. What I meant is that if the standard algebraic reduction algorithm is not possible, there are other algorithms you can use to generate a set of equations that can be solved using algebraic reduction. Humans are pretty limited in their ability to manually apply the generate a set of equations that can be solved algorithm(s) because there are too many dimensions, but computers have no problem. I cut my teeth working on these types of solvers (in FORTRAN, yech). I wonder if we aren't really talking about analog computation (i.e. computing with analogues, e.g. molecules) here? Analog computers have been handily out-computing digital computers for a long time. Since digital and analog are the same thing computationally (digital is a subset of analog), and non-digital computers have been generally superior for several decades, this is not relevant. The difference between digital and analog is the signal-to-noise ratio (SNR) that has to be maintained by the computer system. You can simulate with perfect fidelity high SNR computers on low SNR computers (like digital computers) since they are equivalent, trading SNR for frequency. If you apply the formula for converting digital bits to analog SNR (analog SNR = 1.76+6.02*bits), it becomes obvious why things like thermal noise make it impossible to directly implement e.g. a modest 32-bit digital processor as a non-digital equivalent. When most people talk about analog computation, they are really talking about real computers (whether they realize it or not), which are a form of hypercomputer. If it was possible to build such a computer, it would have some strange consequences for physics that are not in evidence. Cheers, J. Andrew Rogers --- 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=123753653-47f84b Powered by Listbox: http://www.listbox.com
Re: [agi] Hypercomputation and AGI
On Jan 1, 2009, at 2:35 PM, J. Andrew Rogers wrote: Since digital and analog are the same thing computationally (digital is a subset of analog), and non-digital computers have been generally superior for several decades, this is not relevant. Gah, that should be *digital* computers have generally been superior for several decades (the last non-digital hold-outs I am aware of were designed in the late 1970s). J. Andrew Rogers --- 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=123753653-47f84b Powered by Listbox: http://www.listbox.com
Re: [agi] Hypercomputation and AGI
Ben, A few points concerning the central argument: --Reading the argument again, I again mistakenly interpreted it the way I had the first time (until I recalled the details of our previous discussion). The presentation of the argument causes me to assume that U is some kind of oracle directly accessible to all the agents in the community, which as we previously discussed causes the argument to fail. I think the argument could be made clearer by emphasizing that this is not supposed to be the case. --I am not clear about your intentions for the YES case. All I can see is you admitting that in the YES case it may be easier for A2 to internally make use of U. And now a more off-the-wall idea. It seems possible to show that beings whose mind ran by Method X (be it finite-state-machine, turing-machine, or hyper-machine) cannot possibly find scientific use for concepts which involve Method X. This is somewhat inexact. One way to formalize it is to take methods to mean different logics (rather than different types of machine), and derive the result as a trivial corollary of tarski's undefinability theorem: entities that use Method X cannot understand it, so of *course* they cannot find a place for it in their science. Perhaps other formalizations of the idea are more interesting. Of course, as AGI people we hope that we *can* understand the mind in some sense. --Abram On Mon, Dec 29, 2008 at 1:45 PM, Ben Goertzel b...@goertzel.org wrote: Hi, I expanded a previous blog entry of mine on hypercomputation and AGI into a conference paper on the topic ... here is a rough draft, on which I'd appreciate commentary from anyone who's knowledgeable on the subject: http://goertzel.org/papers/CognitiveInformaticsHypercomputationPaper.pdf This is a theoretical rather than practical paper, although it does attempt to explore some of the practical implications as well -- e.g., in the hypothesis that intelligence does require hypercomputation, how might one go about creating AGI? I come to a somewhat surprising conclusion, which is that -- even if intelligence fundamentally requires hypercomputation -- it could still be possible to create an AI via making Turing computer programs ... it just wouldn't be possible to do this in a manner guided entirely by science; one would need to use some other sort of guidance too, such as chance, imitation or intuition... -- Ben G -- Ben Goertzel, PhD CEO, Novamente LLC and Biomind LLC Director of Research, SIAI b...@goertzel.org I intend to live forever, or die trying. -- Groucho Marx agi | Archives | Modify Your Subscription -- Abram Demski Public address: abram-dem...@googlegroups.com Public archive: http://groups.google.com/group/abram-demski Private address: abramdem...@gmail.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=123753653-47f84b Powered by Listbox: http://www.listbox.com
Re: [agi] Hypercomputation and AGI
J. Andrew, On 1/1/09, J. Andrew Rogers and...@ceruleansystems.com wrote: On Jan 1, 2009, at 2:35 PM, J. Andrew Rogers wrote: Since digital and analog are the same thing computationally (digital is a subset of analog), and non-digital computers have been generally superior for several decades, this is not relevant. Gah, that should be *digital* computers have generally been superior for several decades (the last non-digital hold-outs I am aware of were designed in the late 1970s). Ignoring the issues or representation and display, I agree. However, consider three interesting cases... 1. I only survived my college differential equations course with the help of a (now antique) EAI analog computer. Therein, I could simply wire it up as the equation stated, with about as many wires as symbols in the equations, without (much) concern for the internal workings of either the computer or the equation, and get out a parametric plot any way I wanted. However, with a digital computer, maybe there is suitable software by now, but I would have to worry about how the computer did things, e.g. how fine the time slices are, etc. Further, I couldn't just throw the equation at the machine with a digital computer much as I could do with the analog computer, though again, maybe software has caught up by now. 2. Related to above and mentioned earlier, electrolytic fish-tank analogs have long been used to characterize electric and magnetic fields. While these may not be as accurate as digital simulation, they are in TRUE walk-around 3-D representation, and changes can be made in seconds with no need to verify that the change indeed reflects the intended change. This is another example where, at the loss of a few down in the noise digits, you can be SURE that the model indeed simulates reality. The same was long true of wind tunnels, until things got SO valuable (and competitive) that it was worth the millions of dollars to go after those last few digits. 3. Conditioning high-speed phenomena. Transistors are now SO fast and have SO much gain that they have become nearly perfect mathematical components. Most people don't think of their TV tuners as being analog computers, but... Steve Richfield --- 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=123753653-47f84b Powered by Listbox: http://www.listbox.com
Re: [agi] Hypercomputation and AGI
On Dec 30, 2008, at 12:51 AM, Steve Richfield wrote: On a side note, there is the clean math that people learn on their way to a math PhD, and then there is the dirty math that governs physical systems. Dirty math is fraught with all sorts of multi- valued functions, fundamental uncertainties, etc. To work in the world of dirty math, you must escape the notation and figure out what the equation is all about, and find some way of representing THAT, which may well not involve simple numbers on the real-number line, or even on the complex number plane. What does dirty math really mean? There are engineering disciplines essentially *built* on solving equations with gross internal inconsistencies and unsolvable systems of differential equations. The modern world gets along pretty admirably suffering the very profitable and ubiquitous consequences of their quasi-solutions to those problems. But it is still a lot of hairy notational math and equations, just applied in a different context that has function uncertainty as an assumption. The unsolvability does not lead them to pull numbers out of a hat, they have sound methods for brute-forcing fine approximations across a surprisingly wide range of situations. When the clean mathematical methods do not apply, there are other different (not dirty) mathematical methods that you can use. Indeed, I have sometimes said the only real education I ever got in AI was spending years studying an engineering discipline that is nothing but reducing very complex systems of pervasively polluted data and nonsense equations to precise predictive models where squeezing out an extra 1% accuracy meant huge profit. None of it is directly applicable, the value was internalizing that kind of systems perspective and thinking about every complex systems problem in those terms, with a lot of experience algorithmically producing predictive models from them. It was different but it was still ordinary math, just math appropriate for the particular problem. The only thing you could really say about it was that it produced a lot of great computer scientists and no mathematicians to speak of (an odd bias, that). With this as background, as I see it, hypercomputation is just another attempt to evade dealing with some hard mathematical problems. The definition of hypercomputation captures some very specific mathematical concepts that are not captured in other conceptual terms. I do not see what is being evaded, since it is more like pointing out the obvious with respect to certain limits implied by the conventional Turing model. Cheers, J. Andrew Rogers --- 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=123753653-47f84b Powered by Listbox: http://www.listbox.com
Re: [agi] Hypercomputation and AGI
2008/12/29 Ben Goertzel b...@goertzel.org: Hi, I expanded a previous blog entry of mine on hypercomputation and AGI into a conference paper on the topic ... here is a rough draft, on which I'd appreciate commentary from anyone who's knowledgeable on the subject: http://goertzel.org/papers/CognitiveInformaticsHypercomputationPaper.pdf I'm still a bit fuzzy about your argument. So I am going to ask a question to hopefully clarify things somewhat. Couldn't you use similar arguments to say that we can't use science to distinguish between finite state machines and Turing machines? And thus question the usefulness of Turing Machines for science? As if you are talking about a finite data sets these can always be represented by a compressed giant look up table. Will --- 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=123753653-47f84b Powered by Listbox: http://www.listbox.com
Re: [agi] Hypercomputation and AGI
It seems to come down to the simplicity measure... if you can have simplicity(Turing program P that generates lookup table T) simplicity(compressed lookup table T) then the Turing program P can be considered part of a scientific explanation... On Tue, Dec 30, 2008 at 10:02 AM, William Pearson wil.pear...@gmail.comwrote: 2008/12/29 Ben Goertzel b...@goertzel.org: Hi, I expanded a previous blog entry of mine on hypercomputation and AGI into a conference paper on the topic ... here is a rough draft, on which I'd appreciate commentary from anyone who's knowledgeable on the subject: http://goertzel.org/papers/CognitiveInformaticsHypercomputationPaper.pdf I'm still a bit fuzzy about your argument. So I am going to ask a question to hopefully clarify things somewhat. Couldn't you use similar arguments to say that we can't use science to distinguish between finite state machines and Turing machines? And thus question the usefulness of Turing Machines for science? As if you are talking about a finite data sets these can always be represented by a compressed giant look up table. Will --- 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 -- Ben Goertzel, PhD CEO, Novamente LLC and Biomind LLC Director of Research, SIAI b...@goertzel.org I intend to live forever, or die trying. -- Groucho Marx --- 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=123753653-47f84b Powered by Listbox: http://www.listbox.com
Re: [agi] Hypercomputation and AGI
2008/12/30 Ben Goertzel b...@goertzel.org: It seems to come down to the simplicity measure... if you can have simplicity(Turing program P that generates lookup table T) simplicity(compressed lookup table T) then the Turing program P can be considered part of a scientific explanation... Can you clarify what type of language this is in? You mention L-expressions however that is not very clear what that means. lambda expressions I'm guessing. If you start with a language that has infinity built in to its fabric, TMs will be simple, however if you started with a language that only allowed FSM to be specified e.g. regular expressions, you wouldn't be able to simply specify TMs, as you need to represent an infinitely long tape in order to define a TM. Is this analogous to the argument at the end of section 3? It is that bit that is the least clear as far as I am concerned. Will --- 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=123753653-47f84b Powered by Listbox: http://www.listbox.com
Re: [agi] Hypercomputation and AGI
I'm heading off on a vacation for 4-5 days [with occasional email access] and will probably respond to this when i get back ... just wanted to let you know I'm not ignoring the question ;-) ben On Tue, Dec 30, 2008 at 1:26 PM, William Pearson wil.pear...@gmail.comwrote: 2008/12/30 Ben Goertzel b...@goertzel.org: It seems to come down to the simplicity measure... if you can have simplicity(Turing program P that generates lookup table T) simplicity(compressed lookup table T) then the Turing program P can be considered part of a scientific explanation... Can you clarify what type of language this is in? You mention L-expressions however that is not very clear what that means. lambda expressions I'm guessing. If you start with a language that has infinity built in to its fabric, TMs will be simple, however if you started with a language that only allowed FSM to be specified e.g. regular expressions, you wouldn't be able to simply specify TMs, as you need to represent an infinitely long tape in order to define a TM. Is this analogous to the argument at the end of section 3? It is that bit that is the least clear as far as I am concerned. Will --- 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 -- Ben Goertzel, PhD CEO, Novamente LLC and Biomind LLC Director of Research, SIAI b...@goertzel.org I intend to live forever, or die trying. -- Groucho Marx --- 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=123753653-47f84b Powered by Listbox: http://www.listbox.com
Re: [agi] Hypercomputation and AGI
J. Andrew, On 12/30/08, J. Andrew Rogers and...@ceruleansystems.com wrote: On Dec 30, 2008, at 12:51 AM, Steve Richfield wrote: On a side note, there is the clean math that people learn on their way to a math PhD, and then there is the dirty math that governs physical systems. Dirty math is fraught with all sorts of multi-valued functions, fundamental uncertainties, etc. To work in the world of dirty math, you must escape the notation and figure out what the equation is all about, and find some way of representing THAT, which may well not involve simple numbers on the real-number line, or even on the complex number plane. What does dirty math really mean? There are engineering disciplines essentially *built* on solving equations with gross internal inconsistencies and unsolvable systems of differential equations. The modern world gets along pretty admirably suffering the very profitable and ubiquitous consequences of their quasi-solutions to those problems. But it is still a lot of hairy notational math and equations, just applied in a different context that has function uncertainty as an assumption. The unsolvability does not lead them to pull numbers out of a hat, they have sound methods for brute-forcing fine approximations across a surprisingly wide range of situations. When the clean mathematical methods do not apply, there are other different (not dirty) mathematical methods that you can use. The dirty line is rather fuzzy, but you know you've crossed it when instead of locations, things have probability spaces, when you are trying to numerically solve systems of simultaneous equations and it always seems that at least one of them produces NANs, etc. Algebra was designed for the real world as we experience it, and works for most engineering problems, but often runs aground in theoretical physics, at least until you abandon the idea of a 1:1 correspondence between states and variables. Indeed, I have sometimes said the only real education I ever got in AI was spending years studying an engineering discipline that is nothing but reducing very complex systems of pervasively polluted data and nonsense equations to precise predictive models where squeezing out an extra 1% accuracy meant huge profit. None of it is directly applicable, the value was internalizing that kind of systems perspective and thinking about every complex systems problem in those terms, with a lot of experience algorithmically producing predictive models from them. It was different but it was still ordinary math, just math appropriate for the particular problem. Bingo! You have to tailor the techniques to the problem - more than just solving the equations, but often the representation of quantities needs to be in some sort of multivalued form. The only thing you could really say about it was that it produced a lot of great computer scientists and no mathematicians to speak of (an odd bias, that). Yea, but I'd bet that you got pretty good at numerical analysis ;-) With this as background, as I see it, hypercomputation is just another attempt to evade dealing with some hard mathematical problems. The definition of hypercomputation captures some very specific mathematical concepts that are not captured in other conceptual terms. I do not see what is being evaded, ... which is where the break probably is. If someone is going to claim that Turing machines are incapable of doing something, then it seems important to state just what that something is. since it is more like pointing out the obvious with respect to certain limits implied by the conventional Turing model. I wonder if we aren't really talking about analog computation (i.e. computing with analogues, e.g. molecules) here? Analog computers have been handily out-computing digital computers for a long time. One analog computer that produced tide tables, now in a glass case at the NOAA headquarters, performed well for ~100 years until it was finally replaced by a large CDC computer - and probably now with a PC. Some magnetic systems engineers still resort to fish tank analogs rather than deal with software. Steve Richfield --- 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=123753653-47f84b Powered by Listbox: http://www.listbox.com
[agi] Hypercomputation and AGI
Hi, I expanded a previous blog entry of mine on hypercomputation and AGI into a conference paper on the topic ... here is a rough draft, on which I'd appreciate commentary from anyone who's knowledgeable on the subject: http://goertzel.org/papers/CognitiveInformaticsHypercomputationPaper.pdf This is a theoretical rather than practical paper, although it does attempt to explore some of the practical implications as well -- e.g., in the hypothesis that intelligence does require hypercomputation, how might one go about creating AGI? I come to a somewhat surprising conclusion, which is that -- even if intelligence fundamentally requires hypercomputation -- it could still be possible to create an AI via making Turing computer programs ... it just wouldn't be possible to do this in a manner guided entirely by science; one would need to use some other sort of guidance too, such as chance, imitation or intuition... -- Ben G -- Ben Goertzel, PhD CEO, Novamente LLC and Biomind LLC Director of Research, SIAI b...@goertzel.org I intend to live forever, or die trying. -- Groucho Marx --- 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=123753653-47f84b Powered by Listbox: http://www.listbox.com
Re: [agi] Hypercomputation and AGI
On Dec 29, 2008, at 10:45 AM, Ben Goertzel wrote: I expanded a previous blog entry of mine on hypercomputation and AGI into a conference paper on the topic ... here is a rough draft, on which I'd appreciate commentary from anyone who's knowledgeable on the subject: http://goertzel.org/papers/CognitiveInformaticsHypercomputationPaper.pdf This is a theoretical rather than practical paper, although it does attempt to explore some of the practical implications as well -- e.g., in the hypothesis that intelligence does require hypercomputation, how might one go about creating AGI? I come to a somewhat surprising conclusion, which is that -- even if intelligence fundamentally requires hypercomputation -- it could still be possible to create an AI via making Turing computer programs ... it just wouldn't be possible to do this in a manner guided entirely by science; one would need to use some other sort of guidance too, such as chance, imitation or intuition... As more of a meta-comment, the whole notion of hypercomputation seems to be muddled, insofar as super-recursive algorithms may be a limited example of it. I was doing a lot of work with inductive Turing machines several years ago, and most of the differences seemed to be definitional e.g. what constitutes an algorithm or answer. For most practical purposes, the price of implementing them in conventional discrete space is the introduction of some (usually acceptable) error. But if they approximate to the point of functional convergence on a normal Turing machine... As best I have been able to tell, and I have not really been paying attention because the arguments seem to mostly be people talking past each other, is that ITMs raise some interesting philosophical questions regarding hypercomputation. We cannot implement a *strict* hypercomputer, but to what extent does it count if we can asymptotically converge on the functional consequences of a hypercomputer using a normal computer? It suspect it will be hard to evict the belief in Penrosian magic from the error bars in any case. Cheers, J. Andrew Rogers --- 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=123753653-47f84b Powered by Listbox: http://www.listbox.com
Re: [agi] Hypercomputation and AGI
Well, some of the papers in the references of my paper give formal mathematical definitions of hypercomputation, though my paper is brief and conceptual and not of that nature. So although the generic concept may be muddled, there are certainly some fully precise variants of it. This paper surveys various formally defined varieties of hypercomputing, though I haven't read it closely.. http://www.amirrorclear.net/academic/papers/many-forms.pdf Anyway the argument in my paper is pretty strong and applies to any variant with power beyond that of ordinary Turing machines, it would seem... -- ben g On Mon, Dec 29, 2008 at 4:18 PM, J. Andrew Rogers and...@ceruleansystems.com wrote: On Dec 29, 2008, at 10:45 AM, Ben Goertzel wrote: I expanded a previous blog entry of mine on hypercomputation and AGI into a conference paper on the topic ... here is a rough draft, on which I'd appreciate commentary from anyone who's knowledgeable on the subject: http://goertzel.org/papers/CognitiveInformaticsHypercomputationPaper.pdf This is a theoretical rather than practical paper, although it does attempt to explore some of the practical implications as well -- e.g., in the hypothesis that intelligence does require hypercomputation, how might one go about creating AGI? I come to a somewhat surprising conclusion, which is that -- even if intelligence fundamentally requires hypercomputation -- it could still be possible to create an AI via making Turing computer programs ... it just wouldn't be possible to do this in a manner guided entirely by science; one would need to use some other sort of guidance too, such as chance, imitation or intuition... As more of a meta-comment, the whole notion of hypercomputation seems to be muddled, insofar as super-recursive algorithms may be a limited example of it. I was doing a lot of work with inductive Turing machines several years ago, and most of the differences seemed to be definitional e.g. what constitutes an algorithm or answer. For most practical purposes, the price of implementing them in conventional discrete space is the introduction of some (usually acceptable) error. But if they approximate to the point of functional convergence on a normal Turing machine... As best I have been able to tell, and I have not really been paying attention because the arguments seem to mostly be people talking past each other, is that ITMs raise some interesting philosophical questions regarding hypercomputation. We cannot implement a *strict* hypercomputer, but to what extent does it count if we can asymptotically converge on the functional consequences of a hypercomputer using a normal computer? It suspect it will be hard to evict the belief in Penrosian magic from the error bars in any case. Cheers, J. Andrew Rogers --- 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 -- Ben Goertzel, PhD CEO, Novamente LLC and Biomind LLC Director of Research, SIAI b...@goertzel.org I intend to live forever, or die trying. -- Groucho Marx --- 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=123753653-47f84b Powered by Listbox: http://www.listbox.com
Re: [agi] Hypercomputation and AGI
On Dec 29, 2008, at 1:22 PM, Ben Goertzel wrote: Well, some of the papers in the references of my paper give formal mathematical definitions of hypercomputation, though my paper is brief and conceptual and not of that nature. So although the generic concept may be muddled, there are certainly some fully precise variants of it. My comment was not really against the argument you make in the paper, nor do I disagree with your definition of hypercomputation. (BTW, run spellcheck.) I was referring to the somewhat anomalous difficulty of deciding whether or not some computational models truly meet that definition as a practical matter. Anyway the argument in my paper is pretty strong and applies to any variant with power beyond that of ordinary Turing machines, it would seem... No disagreement with that, which is why I called it a meta- comment. :-) Super-recursive algorithms, inductive Turing machines, and related computational models can be made to sit in a somewhat fuzzy place with respect to whether or not they are hypercomputers or normal Turing machines. A Turing machine that asymptotically converges on producing the same result as a hypercomputer is an interesting case insofar as the results they produce may be close enough that you can consider the difference to be below the noise floor, and if they are functionally equivalent using that somewhat unusual definition then you effectively have equivalence to a hypercomputer without the hypercomputer. Not strictly by definition, but within some strictly implied error bound for the purposes of comparing output (which is all we usually care about). The concept of non-isotropic distributions of random numbers has always interested me for much the same reason, since there seems to be a similar concept at work there. Cheers, J. Andrew Rogers --- 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=123753653-47f84b Powered by Listbox: http://www.listbox.com
