12 ms·

Absolutely. Consider any number which can be expressed as a patterned continued fraction, like pi or e. pi's digits are such that it is very hard to compress, b

Absolutely. Consider any number which can be expressed as a patterned continued fraction, like pi or e. pi's digits are such that it is very hard to compress, but it is absolutely trivial to generate. (http://en.wikipedia.org/wiki/Continued_fraction http://en.wikipedia.org/wiki/Continued_fraction)

Fargren 14y ago
How is the algorithm that generates the digits of pi a compressed form of pi? It gives me all the information about the number in finite amount of time. In randomness theory, the definition of the most compressed form of a sequence S is, given a computer C, the shortest program with output S in C. So the program that generates the output of the simulation of the pendulums is a compression of it's output. Since it's and infinite sequence that can be compressed into a finite program, it is not random from the perspective of information theory or randomness theory.
AnthonyMouse 14y ago
>Consider any number which can be expressed as a patterned continued fraction, like pi or e. pi's digits are such that it is very hard to compress, but it is absolutely trivial to generate. You're confusing theoretical compressibility with the actual compression ratios you get when you put a string through typical compression functions. Here's a different example. Suppose you have ASCII text encrypted with AES with the AES decryption key appended to it. Run that through gzip and you'll probably get output which is larger than the input. But here's a different compression algorithm that will work much better: Decrypt the encrypted data with the key and then compress the plaintext with gzip. It isn't that the original string is of the sort that can't be compressed, it's that you need a compression function which is suited to the input. Pi is like that. Compare this to other strings, like the output of an environmental hardware random number generator, which will almost always be totally incompressible whatsoever.

Fargren 14y ago

How is the algorithm that generates the digits of pi a compressed form of pi? It gives me all the information about the number in finite amount of time. In randomness theory, the definition of the most compressed form of a sequence S is, given a computer C, the shortest program with output S in C. So the program that generates the output of the simulation of the pendulums is a compression of it's output. Since it's and infinite sequence that can be compressed into a finite program, it is not random from the perspective of information theory or randomness theory.

AnthonyMouse 14y ago

>Consider any number which can be expressed as a patterned continued fraction, like pi or e. pi's digits are such that it is very hard to compress, but it is absolutely trivial to generate. You're confusing theoretical compressibility with the actual compression ratios you get when you put a string through typical compression functions. Here's a different example. Suppose you have ASCII text encrypted with AES with the AES decryption key appended to it. Run that through gzip and you'll probably get output which is larger than the input. But here's a different compression algorithm that will work much better: Decrypt the encrypted data with the key and then compress the plaintext with gzip. It isn't that the original string is of the sort that can't be compressed, it's that you need a compression function which is suited to the input. Pi is like that. Compare this to other strings, like the output of an environmental hardware random number generator, which will almost always be totally incompressible whatsoever.