Comments on: Representations of uncomputable and uncountable sets http://math.andrej.com/2008/02/06/representations-of-uncomputable-and-uncountable-sets/ Mathematics for computers Fri, 04 Jul 2008 17:27:03 +0000 http://wordpress.org/?v=2.3.3 By: Andrej Bauer http://math.andrej.com/2008/02/06/representations-of-uncomputable-and-uncountable-sets/#comment-6380 Andrej Bauer Sun, 02 Mar 2008 00:46:33 +0000 http://math.andrej.com/2008/02/06/representations-of-uncomputable-and-uncountable-sets/#comment-6380 Kaveh: it is meaningless to communicate data if the receiving party does not analyze it in any way. To <emph>know</emph> what given data communicates/represents/names is to be able to do something reasonable with the data, i.e., you want to perform some operations on the data. So this is quite pragmatic, really. To give you an example (that is essentially the same as what I wrote above): suppose we are two banks that communicate credit card numbers to each other. We would like to agree on a protocol which allows us to "feed the input" and "read the output" but we impose no requirement on what we should be able to do with the credit card numbers. Then the following (bizarre) protocol is ok: whenever you want to communicate a credit-card number to me, just send me the string "hi". You have fed the input and I have read the output. As soon as we want to do something with the credit card numbers, this protocol will suck. And that is the whole point of my post: first figure out what it is that you want to do (such as "I want to be able to tell the digits of the credit-card number"), <em>then</em> you choose a representation. You also ask about sufficient conditions that make a representation acceptable. This is a question that was studied by various people. For example, in Type Two Computation, "good representations" are characterized topologically and go under the name <em>admissible</em>. Another possibility is to express the abstract mathematical properties that characterize your data and then compute the representation from that using the realizability interpretation of logic and type theory. That is what <a href="/rz/" rel="nofollow">RZ</a> does. Kaveh: it is meaningless to communicate data if the receiving party does not analyze it in any way. To know what given data communicates/represents/names is to be able to do something reasonable with the data, i.e., you want to perform some operations on the data. So this is quite pragmatic, really.

To give you an example (that is essentially the same as what I wrote above): suppose we are two banks that communicate credit card numbers to each other. We would like to agree on a protocol which allows us to “feed the input” and “read the output” but we impose no requirement on what we should be able to do with the credit card numbers. Then the following (bizarre) protocol is ok: whenever you want to communicate a credit-card number to me, just send me the string “hi”. You have fed the input and I have read the output.

As soon as we want to do something with the credit card numbers, this protocol will suck. And that is the whole point of my post: first figure out what it is that you want to do (such as “I want to be able to tell the digits of the credit-card number”), then you choose a representation.

You also ask about sufficient conditions that make a representation acceptable. This is a question that was studied by various people. For example, in Type Two Computation, “good representations” are characterized topologically and go under the name admissible. Another possibility is to express the abstract mathematical properties that characterize your data and then compute the representation from that using the realizability interpretation of logic and type theory. That is what RZ does.

]]> By: Kaveh http://math.andrej.com/2008/02/06/representations-of-uncomputable-and-uncountable-sets/#comment-6316 Kaveh Wed, 27 Feb 2008 21:30:53 +0000 http://math.andrej.com/2008/02/06/representations-of-uncomputable-and-uncountable-sets/#comment-6316 > but what operations on the set do you want to compute with? I have two questions regarding this condition: 1. Why is it important from pragmatic point of view? For a computational task all we need is to be able to feed the input and read the output (maybe in a uniform way), we don't need to be able to compute any function on representation. 2. Is this a sufficient condition for accepting a representation(/coding/numbering/naming)? As I wrote in 1, what is important is that we should be able to communicate with the computing machine. Does the operations grantee this? > but what operations on the set do you want to compute with?

I have two questions regarding this condition:

1. Why is it important from pragmatic point of view? For a computational task all we need is to be able to feed the input and read the output (maybe in a uniform way), we don’t need to be able to compute any function on representation.

2. Is this a sufficient condition for accepting a representation(/coding/numbering/naming)? As I wrote in 1, what is important is that we should be able to communicate with the computing machine. Does the operations grantee this?

]]> By: Andrej Bauer http://math.andrej.com/2008/02/06/representations-of-uncomputable-and-uncountable-sets/#comment-6119 Andrej Bauer Mon, 11 Feb 2008 09:42:32 +0000 http://math.andrej.com/2008/02/06/representations-of-uncomputable-and-uncountable-sets/#comment-6119 You are quite correct that I was a bit hasty in dragging in the Verification principle. What actually matters is whether the laws of physics predict existence or non-existence of computable streams, not what we can verify in finite time. Also note that there might exist funky experiments that test for existence of non-computable streams in a probabilistic way without actually exhibiting a non-computable stream. However, there does seem to be a difference between "There exist binary streams with only finitely many 0s" and "There exist uncomputable binary streams", namely that I can produce a mechanism which outputs an infinite stream od 1s and <em>prove</em> from currently accepted laws of physics that it does not contain any 0s (modulo idealizations such as "the mechanism will not break or run out of energy and time"). Is it possible to build a mechanism which outputs a binary stream and prove, using currently accepted laws of physics, that it outputs a non-computable stream? Is it possible to prove from currently accepted laws of physics that no such mechanism exists? I suspect the current laws of physics do not say much about either option. You are quite correct that I was a bit hasty in dragging in the Verification principle. What actually matters is whether the laws of physics predict existence or non-existence of computable streams, not what we can verify in finite time. Also note that there might exist funky experiments that test for existence of non-computable streams in a probabilistic way without actually exhibiting a non-computable stream.

However, there does seem to be a difference between “There exist binary streams with only finitely many 0s” and “There exist uncomputable binary streams”, namely that I can produce a mechanism which outputs an infinite stream od 1s and prove from currently accepted laws of physics that it does not contain any 0s (modulo idealizations such as “the mechanism will not break or run out of energy and time”). Is it possible to build a mechanism which outputs a binary stream and prove, using currently accepted laws of physics, that it outputs a non-computable stream? Is it possible to prove from currently accepted laws of physics that no such mechanism exists? I suspect the current laws of physics do not say much about either option.

]]> By: Sridhar http://math.andrej.com/2008/02/06/representations-of-uncomputable-and-uncountable-sets/#comment-6118 Sridhar Mon, 11 Feb 2008 04:11:29 +0000 http://math.andrej.com/2008/02/06/representations-of-uncomputable-and-uncountable-sets/#comment-6118 (Where I wrote above "nth program on itself", I meant of course "nth program on input n") (Where I wrote above “nth program on itself”, I meant of course “nth program on input n”)

]]> By: Sridhar http://math.andrej.com/2008/02/06/representations-of-uncomputable-and-uncountable-sets/#comment-6117 Sridhar Mon, 11 Feb 2008 02:40:27 +0000 http://math.andrej.com/2008/02/06/representations-of-uncomputable-and-uncountable-sets/#comment-6117 I'm not sure it's fair to say that physicists actually would, on the basis of something like the verification principle, take the question of the possibility of physical generation of non-computable streams to be meaningless. The fact that we can only observe finite prefixes of a binary stream leaves us in a similar position regarding "There exist binary streams with only finitely many 0s" as with "There exist uncomputable binary streams", does it not? And yet, every physicist would say the former question is settled (by the inductive grounds that lead us to believe certain streams are all 1s). Similarly, we could at least potentially imagine concluding, on inductive grounds, that some physical process actually yields, say, a binary stream whose nth value differs from that of the nth program on itself whenever such computation halts. And thus be led to conclude the physical existence of uncomputable binary streams. I don't think such a situation is terribly likely, but only because I, for reasons that are probably more superstition than logic, doubt the world works that way, and thus don't think the requisite sort of empirical evidence will ever be found. However, it seems clear that the methods of scientific practice themselves present no necessary barrier to drawing such a conclusion. I’m not sure it’s fair to say that physicists actually would, on the basis of something like the verification principle, take the question of the possibility of physical generation of non-computable streams to be meaningless. The fact that we can only observe finite prefixes of a binary stream leaves us in a similar position regarding “There exist binary streams with only finitely many 0s” as with “There exist uncomputable binary streams”, does it not? And yet, every physicist would say the former question is settled (by the inductive grounds that lead us to believe certain streams are all 1s).

Similarly, we could at least potentially imagine concluding, on inductive grounds, that some physical process actually yields, say, a binary stream whose nth value differs from that of the nth program on itself whenever such computation halts. And thus be led to conclude the physical existence of uncomputable binary streams. I don’t think such a situation is terribly likely, but only because I, for reasons that are probably more superstition than logic, doubt the world works that way, and thus don’t think the requisite sort of empirical evidence will ever be found. However, it seems clear that the methods of scientific practice themselves present no necessary barrier to drawing such a conclusion.

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