Re: [agi] Defining understanding (was Re: Newcomb's Paradox)
- Original Message From: Matt Mahoney [EMAIL PROTECTED] I don't claim that compression is simple. It is not. Text compression is AI-complete. The general problem is not even computable. ...I claim that compression can be used to measure intelligence. I explain in more detail at http://cs.fit.edu/~mmahoney/compression/rationale.html -- Matt Mahoney, [EMAIL PROTECTED] --- It will take me a while to read your paper. However, I want to say that I am skeptical that you would be able to use compression to even measure intelligence. I do think it might be worthwhile to come up with basic elements of intelligence, and these could include correlations of productive output from different algorithms or something like that. But, from there you have to continue to build the system. It would be necessary to show how those elements can be combined to produce higher (or better) intelligence, and the Shannon/Hutter enthusiasts (along with everyone else) simply have not done this. (I think the contemporary advancements in AI are probably due to faster memory access and parallelism as much as any achievement in AI software.) But this means that you are advancing a purely speculative theory without any evidence to support it. Right now I am working on my own religious journey (but mine is seriously religious interestingly enough) writing a polynomial time SAT program. Now let's say that this SAT theory actually worked and was followed by a theory that showed that it could be used both to advance AI and to compress data. You might have a -I told you so- moment. But I might then have a -so what- moment. (I say that in a competitive but cordial way.) Of course intelligence will involve some kind of compression method! But so what? It will also involve some kind of speculative method. Does that mean that we can use speculation to 'measure' intelligence? Well, sure. Someone might be able to devise a psychometric measure of speculative potential or something like that. But this does not translate into an objective measure of intelligence until it is compared with thousands of subjects and integrated into a system that indicates that this particular measure of speculative potential can be correlated with other measures of intelligence and achievement. Sometimes a compression algorithm is just a compression algorithm. Jim Bromer --- agi Archives: http://www.listbox.com/member/archive/303/=now RSS Feed: http://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: http://www.listbox.com/member/?member_id=8660244id_secret=103754539-40ed26 Powered by Listbox: http://www.listbox.com
Re: [agi] Defining understanding (was Re: Newcomb's Paradox)
--- Jim Bromer [EMAIL PROTECTED] wrote: - Original Message From: Matt Mahoney [EMAIL PROTECTED] I don't claim that compression is simple. It is not. Text compression is AI-complete. The general problem is not even computable. ...I claim that compression can be used to measure intelligence. I explain in more detail at http://cs.fit.edu/~mmahoney/compression/rationale.html -- Matt Mahoney, [EMAIL PROTECTED] --- It will take me a while to read your paper. However, I want to say that I am skeptical that you would be able to use compression to even measure intelligence. I do think it might be worthwhile to come up with basic elements of intelligence, and these could include correlations of productive output from different algorithms or something like that. But, from there you have to continue to build the system. It would be necessary to show how those elements can be combined to produce higher (or better) intelligence, and the Shannon/Hutter enthusiasts (along with everyone else) simply have not done this. (I think the contemporary advancements in AI are probably due to faster memory access and parallelism as much as any achievement in AI software.) But this means that you are advancing a purely speculative theory without any evidence to support it. The evidence is described in my paper which you haven't read yet. For building AGI, my proposal is http://www.mattmahoney.net/agi.html Unfortunately, I estimate the cost to be US $1 quadrillion over the next 30 years. But I believe it is coming, because AGI is worth that much. If I use compression anywhere, it will be to evaluate candidate language models for peers in a market that right now does not yet exist. Right now I am working on my own religious journey (but mine is seriously religious interestingly enough) writing a polynomial time SAT program. It is worth $1 million if you succeed, but I wouldn't waste my time on it. http://www.claymath.org/millennium/P_vs_NP/ -- Matt Mahoney, [EMAIL PROTECTED] --- agi Archives: http://www.listbox.com/member/archive/303/=now RSS Feed: http://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: http://www.listbox.com/member/?member_id=8660244id_secret=103754539-40ed26 Powered by Listbox: http://www.listbox.com
Re: [agi] Defining understanding (was Re: Newcomb's Paradox)
I had said: But this means that you are advancing a purely speculative theory without any evidence to support it. Matt said: The evidence is described in my paper which you haven't read yet. I did glance at the paper and I don't think I will be able to understand your evidence. Can you give me some clues using plain language. - Matt said: For building AGI, my proposal is http://www.mattmahoney.net/agi.html Unfortunately, I estimate the cost to be US $1 quadrillion over the next 30 years. But I believe it is coming, because AGI is worth that much. If I use compression anywhere, it will be to evaluate candidate language models for peers in a market that right now does not yet exist. - Can you explain what you mean by the statement that you would use compression to evaluate candidate language models? I had said: Right now I am working on my own religious journey (but mine is seriously religious interestingly enough) writing a polynomial time SAT program. Matt said: It is worth $1 million if you succeed, but I wouldn't waste my time on it. http://www.claymath.org/millennium/P_vs_NP/ - I had given up on it as a waste of time, but I decided to look more carefully at it on what I considered the slight possibility that the Lord had actually indicated that I would be able to do it. I have evidence now that I did not have 7 months ago that it may actually work. Jim Bromer --- agi Archives: http://www.listbox.com/member/archive/303/=now RSS Feed: http://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: http://www.listbox.com/member/?member_id=8660244id_secret=103754539-40ed26 Powered by Listbox: http://www.listbox.com
Re: [agi] Defining understanding (was Re: Newcomb's Paradox)
- Original Message From: Matt Mahoney [EMAIL PROTECTED] Your question answering machine is algorithmically complex. A smaller program could describe a procedure for answering the questions, and in that case it could answer questions not in the original set of 1. Here is another example: 3 = 9 7 = 49 8 = 64 12 = 144 2 = 4 6 = ? You could write a program that stores the first 5 training examples in a table, or you could find a smaller program that computes the output as a mathematical function of the input. When you test your programs with 6 = ? which program would give you the answer you expect? Which would you say understands the training set? You can take the position that a machine can never understand anything the way that a human could. I don't care. Call it something else if you want, like AI. -- Matt Mahoney, [EMAIL PROTECTED] --- First of all, just to make sure you understand me, when I said that the 'generator=prediction compression=understanding' theory and the is not worth spending much time on, I did not mean that I thought your ideas were not worth spending time on. I was just criticizing that one idea, not all of your ideas. Secondly, I have already considered the example that you supplied me with. I tried to explain that I have already discussed these ideas in another group. And I have never taken the position that a machine can never understand anything. That is either a straw man argument or it shows that you did not understand what it was that I did say in my last message. Or maybe you did not even bother to read my last message very carefully before you fired off your resply. I don't see any other way of explanation. I do think a truly intelligent computer program would be algorithmically complex. Evidently that would be a difference in our opinions if you have completely accepted the position that you are advocating. I also think that generalization, generalization-like relations or generative procedures are necessary to produce AI. I think that is a similarity in our positions. You can try to find the fundamentals of intelligence, that is of algorithmic intelligence, but that does not mean that you will be able to produce intelligence before you find a theory that is complex enough to explain how artificial intelligence can be produced. That is another weakness of compression=understanding theory. Jim Bromer --- agi Archives: http://www.listbox.com/member/archive/303/=now RSS Feed: http://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: http://www.listbox.com/member/?member_id=8660244id_secret=103754539-40ed26 Powered by Listbox: http://www.listbox.com
Re: [agi] Defining understanding (was Re: Newcomb's Paradox)
--- Jim Bromer [EMAIL PROTECTED] wrote: You can try to find the fundamentals of intelligence, that is of algorithmic intelligence, but that does not mean that you will be able to produce intelligence before you find a theory that is complex enough to explain how artificial intelligence can be produced. That is another weakness of compression=understanding theory. I don't claim that compression is simple. It is not. Text compression is AI-complete. The general problem is not even computable. Perhaps you misunderstand that I think gzip has AI. No, I claim that compression can be used to measure intelligence. I explain in more detail at http://cs.fit.edu/~mmahoney/compression/rationale.html -- Matt Mahoney, [EMAIL PROTECTED] --- agi Archives: http://www.listbox.com/member/archive/303/=now RSS Feed: http://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: http://www.listbox.com/member/?member_id=8660244id_secret=103754539-40ed26 Powered by Listbox: http://www.listbox.com
Re: [agi] Defining understanding (was Re: Newcomb's Paradox)
Matt mahoney: I am not sure what you mean by AGI. I consider a measure of intelligence to be the degree to which goals are satisfied in a range of environments. It does not matter what the goals are. They may seem irrational to you. The goal of a smart bomb is to blow itself up at a given target. I would consider bombs that hit their targets more often to be more intelligent. I consider understanding to mean intelligence in this context. You can't say that a robot that does nothing is unintelligent unless you specify its goals. We may consider intelligence as a measure and AGI as a threshold. AGI is not required for understanding. You can measure the degree to which various search engines understand your query, spam filters understand your email, language translators understand your document, vision systems understand images, intrusion detection systems understand network traffic, etc. Each system was designed with a goal and can be evaluated according to how well that goal is met. AIXI allows us to evaluate intelligence independent of goals. An agent understands its input if it can predict it. This can be measured precisely. I think you are thinking of solomonoff induction, AIXI won't answer your questions unless it has the goal of getting reward from you for answering the question. It will do what it predicts will get it reward, not try and output the end of all strings given to it. I propose prediction as a general test of understanding. For example, do you understand the sequence 0101010101010101 ? If I asked you to predict the next bit and you did so correctly, then I would say you understand it. What would happen if I said, I don't have time for silly games, please stop emailing me. Would you consider that I understood it? If I want to test your understanding of X, I can describe X, give you part of the description, and test if you can predict the rest. If I want to test if you understand a picture, I can cover part of it and ask you to predict what might be there. This only works if my goal includes revealing my understanding to you. Will Pearson --- agi Archives: http://www.listbox.com/member/archive/303/=now RSS Feed: http://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: http://www.listbox.com/member/?member_id=8660244id_secret=101455710-f059c4 Powered by Listbox: http://www.listbox.com
Re: [agi] Defining understanding (was Re: Newcomb's Paradox)
--- William Pearson [EMAIL PROTECTED] wrote: Matt mahoney: I propose prediction as a general test of understanding. For example, do you understand the sequence 0101010101010101 ? If I asked you to predict the next bit and you did so correctly, then I would say you understand it. What would happen if I said, I don't have time for silly games, please stop emailing me. Would you consider that I understood it? If it was a Turing test, then probably yes. But a Turing test is not the best way to test for intelligence. Ben Goertzel once said something like pattern recognition + goals = AGI. I am generalizing pattern recognition to prediction and proposing that the two components can be tested separately. For example, a speech recognition system is evaluated by word error rate. But for development it is useful to separate the system into its two main components, an acoustic model and a language model, and test them separately. A language model is just a probability distribution. It does not have a goal. Nevertheless, the model's accuracy can be measured by using it in a data compressor whose goal (implicit in the encoder) is to minimize the size of the output without losing information. The compressed size correlates well with word error rate. Such testing is useful because if the system has a poor word error rate but the language model is good, then the problem can be narrowed down to the acoustic model. Without this test, you wouldn't know. I propose compression as a universal goal for testing the predictor component of AI. More formally, if the system predicts the next symbol with probability p, then that symbol has utility log(p). AIXI provides a formal justification for this approach. In AIXI, an agent and an environment (both Turing machines) exchange symbols interactively. In addition, the environment signals a numeric reward to the agent during each cycle. The goal of the agent is to maximize the accumulated reward. Hutter proved that the optimal (but uncomputable) strategy of the agent is to guess at each step that the environment is modeled by the shortest Turing machine consistent with the interaction so far. Note that this strategy is independent of the goal implied by the reward signal. -- Matt Mahoney, [EMAIL PROTECTED] --- agi Archives: http://www.listbox.com/member/archive/303/=now RSS Feed: http://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: http://www.listbox.com/member/?member_id=8660244id_secret=101455710-f059c4 Powered by Listbox: http://www.listbox.com
Re: [agi] Defining understanding (was Re: Newcomb's Paradox)
- Original Message Matt Mahoney said: Remember that the goal is to test for understanding in intelligent agents that are not necessarily human. What does it mean for a machine to understand something? What does it mean to understand a string of bits? I propose prediction as a general test of understanding. For example, do you understand the sequence 0101010101010101 ? If I asked you to predict the next bit and you did so correctly, then I would say you understand it. If I want to test your understanding of X, I can describe X, give you part of the description, and test if you can predict the rest. If I want to test if you understand a picture, I can cover part of it and ask you to predict what might be there. Understanding = compression. If you can take a string and find a shorter description (a program) that generates the string, then use that program to predict subsequent symbols correctly, then I would say you understand the string (or its origin). This is what Hutter's universal intelligent agent does. The significance of AIXI is not a solution to AI (AIXI is not computable), but that it defines a mathematical framework for intelligence. -- Matt Mahoney, [EMAIL PROTECTED] --- I have heard this argument in another discussion group and it is too weak to spend much time on. I am not debating that prediction is an aspect of intelligence. And I agree that we need better ways to gauge 'understanding' for computer programs. But, Understanding=compression. That is really pretty far out there. This conclusion is based on an argument like: One would be able to predict everything if he was able to understand everything (or at least everything predictable). This argument, however, is clearly a fantasy. So we come up with a weaker version. If someone was able to predict a number of events accurately this would be a sign that he must understand something about those events. This argument might work when talking about people, but it does not quite work the way you seem to want it to. You cannot just paste a term of intelligence like 'prediction' onto a mechanical process and reason that the mechanical process may then be seen as equivalent to a mental process. Suppose someone wrote a computer program with 1 questions and the program was able to 'predict' the correct answer for every single one of those questions. Does the machine understand the questions? Of course not. The person who wrote the program understands the subject of those questions, but you cannot conclude that the program understood the subject matter. Ok, so at this point the argument might be salvaged by saying that if the program understood the answers to questions that it had never seen before that this would have to be considered understanding. But that is not good enough either because we know that even if a program is not tested for every single mathematical calculation it could possibly make, there is no evidence that it actually understands anything about its calculator functions in any useful way other than producing a result. While I am willing to agree that computation may be one measure of understanding in persons, calculators are usually very limited, and therefore there must be a great deal more to understanding than producing the correct result based on a generalized procedure. When an advanced intelligent program learns something new it would be able to apply that new knowledge in ways that were not produced stereotypically through the generalizations or generative functions that it was programmed with. This last step is difficult to express perfectly, but if the reader doesn't appreciate the significance of this paragraph so far, it won't matter anyway. Technically, a program that was able to learn in the manner that I am thinking of would have to use some programmed generalizations (or generalization-like relations) in that learning. So my first sentence in this paragraph was actually an imperfect simplification. But, the point is, that an advanced program, like the kind that I am thinking of, will also be able to form its own generalizations (and generalization-like relations) about learned information that is not fully predictable at the time the program was launched. (I am not using 'predictable' here in a contrary argument, I am using it to show that a new concept could be learned in a non-stereotyped way relative to some conceptual context.) This means that understanding, even an understanding of a process that is strictly formalized, may have effects on the understanding of other subject matter, and the understanding of other subject matter may have other effects on the new insight. Insight may be applied to a range of instances. However, the measure of insightfulness must also be related to some more sophisticated method of analysis; one that is effectively based on relative conceptual boundaries. So if you want
Re: [agi] Defining understanding (was Re: Newcomb's Paradox)
--- Jim Bromer [EMAIL PROTECTED] wrote: But, Understanding=compression. That is really pretty far out there. This conclusion is based on an argument like: One would be able to predict everything if he was able to understand everything (or at least everything predictable). This argument, however, is clearly a fantasy. So we come up with a weaker version. If someone was able to predict a number of events accurately this would be a sign that he must understand something about those events. This argument might work when talking about people, but it does not quite work the way you seem to want it to. You cannot just paste a term of intelligence like 'prediction' onto a mechanical process and reason that the mechanical process may then be seen as equivalent to a mental process. Suppose someone wrote a computer program with 1 questions and the program was able to 'predict' the correct answer for every single one of those questions. Does the machine understand the questions? Of course not. The person who wrote the program understands the subject of those questions, but you cannot conclude that the program understood the subject matter. Your question answering machine is algorithmically complex. A smaller program could describe a procedure for answering the questions, and in that case it could answer questions not in the original set of 1. Here is another example: 3 = 9 7 = 49 8 = 64 12 = 144 2 = 4 6 = ? You could write a program that stores the first 5 training examples in a table, or you could find a smaller program that computes the output as a mathematical function of the input. When you test your programs with 6 = ? which program would give you the answer you expect? Which would you say understands the training set? You can take the position that a machine can never understand anything the way that a human could. I don't care. Call it something else if you want, like AI. -- Matt Mahoney, [EMAIL PROTECTED] --- agi Archives: http://www.listbox.com/member/archive/303/=now RSS Feed: http://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: http://www.listbox.com/member/?member_id=8660244id_secret=103754539-40ed26 Powered by Listbox: http://www.listbox.com
Re: [agi] Defining understanding (was Re: Newcomb's Paradox)
Matt Mahoney wrote: Remember that the goal is to test for understanding in intelligent agents that are not necessarily human. What does it mean for a machine to understand something? What does it mean to understand a string of bits? Have you considered testing intelligent agents by simply observing what they do when left alone? If it has understanding, wouldn't it do something? And wouldn't it's choice be revealing? Just a thought. --- agi Archives: http://www.listbox.com/member/archive/303/=now RSS Feed: http://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: http://www.listbox.com/member/?member_id=8660244id_secret=101455710-f059c4 Powered by Listbox: http://www.listbox.com
Re: [agi] Defining understanding (was Re: Newcomb's Paradox)
--- Stan Nilsen [EMAIL PROTECTED] wrote: Matt Mahoney wrote: Remember that the goal is to test for understanding in intelligent agents that are not necessarily human. What does it mean for a machine to understand something? What does it mean to understand a string of bits? Have you considered testing intelligent agents by simply observing what they do when left alone? If it has understanding, wouldn't it do something? And wouldn't it's choice be revealing? Just a thought. What it does depends on its goals, in addition to understanding. Suppose a robot just sits there, doing nothing. Maybe it understands its environment but doesn't need to do anything because its batteries are charged. -- Matt Mahoney, [EMAIL PROTECTED] --- agi Archives: http://www.listbox.com/member/archive/303/=now RSS Feed: http://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: http://www.listbox.com/member/?member_id=8660244id_secret=101455710-f059c4 Powered by Listbox: http://www.listbox.com
Re: [agi] Defining understanding (was Re: Newcomb's Paradox)
Matt Mahoney wrote: --- Stan Nilsen [EMAIL PROTECTED] wrote: Matt Mahoney wrote: Remember that the goal is to test for understanding in intelligent agents that are not necessarily human. What does it mean for a machine to understand something? What does it mean to understand a string of bits? Have you considered testing intelligent agents by simply observing what they do when left alone? If it has understanding, wouldn't it do something? And wouldn't it's choice be revealing? Just a thought. What it does depends on its goals, in addition to understanding. Suppose a robot just sits there, doing nothing. Maybe it understands its environment but doesn't need to do anything because its batteries are charged. -- Matt Mahoney, [EMAIL PROTECTED] If the batteries are charged and it waits around for an order from it's master, then it will always be a robot and not an AGI. If it understands it's environment, it is not an AGI - there are too many mysteries in the big environment to understand it. If nothing else, it ought to be looking for a way to engage itself for someone or somethings benefit - else it probably doesn't understand existence. --- agi Archives: http://www.listbox.com/member/archive/303/=now RSS Feed: http://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: http://www.listbox.com/member/?member_id=8660244id_secret=101455710-f059c4 Powered by Listbox: http://www.listbox.com
Re: [agi] Defining understanding (was Re: Newcomb's Paradox)
--- Stan Nilsen [EMAIL PROTECTED] wrote: Matt Mahoney wrote: --- Stan Nilsen [EMAIL PROTECTED] wrote: Matt Mahoney wrote: Remember that the goal is to test for understanding in intelligent agents that are not necessarily human. What does it mean for a machine to understand something? What does it mean to understand a string of bits? Have you considered testing intelligent agents by simply observing what they do when left alone? If it has understanding, wouldn't it do something? And wouldn't it's choice be revealing? Just a thought. What it does depends on its goals, in addition to understanding. Suppose a robot just sits there, doing nothing. Maybe it understands its environment but doesn't need to do anything because its batteries are charged. -- Matt Mahoney, [EMAIL PROTECTED] If the batteries are charged and it waits around for an order from it's master, then it will always be a robot and not an AGI. If it understands it's environment, it is not an AGI - there are too many mysteries in the big environment to understand it. If nothing else, it ought to be looking for a way to engage itself for someone or somethings benefit - else it probably doesn't understand existence. I am not sure what you mean by AGI. I consider a measure of intelligence to be the degree to which goals are satisfied in a range of environments. It does not matter what the goals are. They may seem irrational to you. The goal of a smart bomb is to blow itself up at a given target. I would consider bombs that hit their targets more often to be more intelligent. I consider understanding to mean intelligence in this context. You can't say that a robot that does nothing is unintelligent unless you specify its goals. We may consider intelligence as a measure and AGI as a threshold. AGI is not required for understanding. You can measure the degree to which various search engines understand your query, spam filters understand your email, language translators understand your document, vision systems understand images, intrusion detection systems understand network traffic, etc. Each system was designed with a goal and can be evaluated according to how well that goal is met. AIXI allows us to evaluate intelligence independent of goals. An agent understands its input if it can predict it. This can be measured precisely. -- Matt Mahoney, [EMAIL PROTECTED] --- agi Archives: http://www.listbox.com/member/archive/303/=now RSS Feed: http://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: http://www.listbox.com/member/?member_id=8660244id_secret=101455710-f059c4 Powered by Listbox: http://www.listbox.com
Re: [agi] Defining understanding (was Re: Newcomb's Paradox)
--- Jim Bromer [EMAIL PROTECTED] wrote: Matt Mahoney said, A formal explanation of a program P would be a equivalent program Q, such that P(x) = Q(x) for all x. Although it is not possible to prove equivalence in general, it is sometimes possible to prove nonequivalence by finding x such that P(x) != Q(x), i.e. Q fails to predict what P will output given x. But I have a few problems with this although his one example was ok. One, there are explanations of ideas that cannot be expressed using the kind of formality he was talking about. Secondly, there are ideas that are inadequate when expressed only using the methods of formality he mentioned, Third, an explanation needs to be used relative to some other purpose. For example, making a prediction of how long something will fall to the ground is a start, but if a person understands Newton's law of gravity, he will be able to utilize it in other gravities as well. And he may be able to relate it to real world situations where precise measurements are not available. And he might apply his knowledge of Newton's laws to see the dimensional similarities (of length, mass, force and so on) between different kinds of physical formulas. Remember that the goal is to test for understanding in intelligent agents that are not necessarily human. What does it mean for a machine to understand something? What does it mean to understand a string of bits? I propose prediction as a general test of understanding. For example, do you understand the sequence 0101010101010101 ? If I asked you to predict the next bit and you did so correctly, then I would say you understand it. If I want to test your understanding of X, I can describe X, give you part of the description, and test if you can predict the rest. If I want to test if you understand a picture, I can cover part of it and ask you to predict what might be there. Understanding = compression. If you can take a string and find a shorter description (a program) that generates the string, then use that program to predict subsequent symbols correctly, then I would say you understand the string (or its origin). This is what Hutter's universal intelligent agent does. The significance of AIXI is not a solution to AI (AIXI is not computable), but that it defines a mathematical framework for intelligence. -- Matt Mahoney, [EMAIL PROTECTED] --- agi Archives: http://www.listbox.com/member/archive/303/=now RSS Feed: http://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: http://www.listbox.com/member/?member_id=8660244id_secret=101455710-f059c4 Powered by Listbox: http://www.listbox.com
[agi] Defining understanding (was Re: Newcomb's Paradox)
--- Stan Nilsen [EMAIL PROTECTED] wrote: I'm not understanding why an *explanation* would be ambiguous? If I have a process / function that consistently transforms x into y, then doesn't the process serve as a non-ambiguous explanation of how y came into being? (presuming this is the thing to be explained.) A formal explanation of a program P would be a equivalent program Q, such that P(x) = Q(x) for all x. Although it is not possible to prove equivalence in general, it is sometimes possible to prove nonequivalence by finding x such that P(x) != Q(x), i.e. Q fails to predict what P will output given x. Prediction can be used as a test of understanding lots of things. For example, if I wanted to test whether you understand Newton's law of gravity, I would ask you to predict how long it will take an object of a certain mass to fall from a certain height. If I wanted to test whether you understand French, I could give you a few lines of text in French and ask you to predict what the next word will be. -- Matt Mahoney, [EMAIL PROTECTED] --- agi Archives: http://www.listbox.com/member/archive/303/=now RSS Feed: http://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: http://www.listbox.com/member/?member_id=8660244id_secret=101455710-f059c4 Powered by Listbox: http://www.listbox.com
