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9,73,241,561,1081,1849,_?_

algebra.com

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Re: 9,73,241,561,1081,1849,_?_

#61

Post by scott_s »

amichail wrote: post #59
scott_s wrote: post #52

Earlier quoted context omitted.

Luckily, mathematicians have already done this for me: http://en.wikipedia.org/wiki/Closed-form_solution Note, that I said "reasonable people." Reasonable people know what is being asked. If, for example, I asked someone how much they weigh, they are not going to respond "I don't know. I weighed myself five minutes ago, but I must have a different weight right now."

Which functions do you allow in closed-form solutions? Do you allow Turing-completeness? If not, why not? What exactly is the measure used to determine the size of a solution?

The explanation was clear what kind of functions are allowed: http://en.wikipedia.org/wiki/Elementary_function_%28differen...

Again, reasonable people. If we had to precisely define every question and interaction, we would never get anything done. Instead we rely on shared assumptions and clarify only when actual confusion occurs. (Yours is not an actual confusion.)

Re: 9,73,241,561,1081,1849,_?_

#62

Post by g__ »

osteele wrote: post #60

The wonderful (satirical, but mathematically sophisticated) book Mathematics Made Difficult introduces difference tables, and uses them to demonstrate how to compute the next number in the sequence [1,2,4,8,16,…] – that number being, of course, 31. This answer is correct in that it is the next item in the lowest-order polynomial that generates the first five terms. This reveals both the strength and weakness of diffe…

This answer is also correct in another way :)

http://threesixty360.wordpress.com/2008/04/30/1-2-4-8-what-c...

Re: 9,73,241,561,1081,1849,_?_

#63

Post by g__ »

bdr wrote: post #22
amichail wrote: post #15

Earlier quoted context omitted.

How will you prove that your answer is correct?

Well, that could be done by brute force. Realistically, though, it's more about whether anyone can beat your answer.

You'd have to determine if a program halts. Unless you restrict to non-Turing-complete language, this is undecidable.

Re: 9,73,241,561,1081,1849,_?_

#64

Post by amichail »

scott_s wrote: post #61
amichail wrote: post #59

Earlier quoted context omitted.

Which functions do you allow in closed-form solutions? Do you allow Turing-completeness? If not, why not? What exactly is the measure used to determine the size of a solution?

The explanation was clear what kind of functions are allowed: http://en.wikipedia.org/wiki/Elementary_function_%28differen... Again, reasonable people. If we had to precisely define every question and interaction, we would never get anything done. Instead we rely on shared assumptions and clarify only when actual confusion occurs. (Yours is not an actual confusion.)

There's no reason to not allow arbitrary programs.

Moreover, even if we agree to use a certain form of closed expression, you will still need to define how we measure the size of such an expression and you will need to find a way to prove that an answer has the smallest possible expression size.

Re: 9,73,241,561,1081,1849,_?_

#65

Post by rsaarelm »

billpg wrote: post #35
bdr wrote: post #14

Earlier quoted context omitted.

Here's one interpretation: What will the simplest program that outputs these numbers output next?

Define "simplest program"?

The shortest program in a low-level language. On the definition of a low-level language, see: http://www.paul-almond.com/WhatIsALowLevelLanguage.htm

Re: 9,73,241,561,1081,1849,_?_

#66

Post by scott_s »

amichail wrote: post #64
scott_s wrote: post #61

Earlier quoted context omitted.

The explanation was clear what kind of functions are allowed: http://en.wikipedia.org/wiki/Elementary_function_%28differen... Again, reasonable people. If we had to precisely define every question and interaction, we would never get anything done. Instead we rely on shared assumptions and clarify only when actual confusion occurs. (Yours is not an actual confusion.)

There's no reason to not allow arbitrary programs. Moreover, even if we agree to use a certain form of closed expression, you will still need to define how we measure the size of such an expression and you will need to find a way to prove that an answer has the smallest possible expression size.

Sure there's a reason: someone said so.

I also said nothing about "smallest possible expression size."

Re: 9,73,241,561,1081,1849,_?_

#67

Post by eru »

jey wrote: post #54
amichail wrote: post #31

Earlier quoted context omitted.

Why would you think that?

It only takes a counter-example (an example of a shorter program) to show that an answer is incorrect, but you could probably pick up a Turing Award or two if you find a way to put a tight bound on the Kolmogorov complexity of a bit string.

It's only difficult in general. Nobody says there aren't any easy instances.

Re: 9,73,241,561,1081,1849,_?_

#68

Post by eru »

g__ wrote: post #63
bdr wrote: post #22

Earlier quoted context omitted.

Well, that could be done by brute force. Realistically, though, it's more about whether anyone can beat your answer.

You'd have to determine if a program halts. Unless you restrict to non-Turing-complete language, this is undecidable.

I can decide if some programs halt. (Just not for all.)

Re: 9,73,241,561,1081,1849,_?_

#69

Post by jey »

eru wrote: post #67
jey wrote: post #54

Earlier quoted context omitted.

It only takes a counter-example (an example of a shorter program) to show that an answer is incorrect, but you could probably pick up a Turing Award or two if you find a way to put a tight bound on the Kolmogorov complexity of a bit string.

It's only difficult in general. Nobody says there aren't any easy instances.

Sure, there's a bunch of trivial/easy ones for programs that do almost nothing, but I really wonder if are there any interesting programs for which tight bounds are known?

Re: 9,73,241,561,1081,1849,_?_

#70

Post by shrughes »

tkahn6 wrote: post #30

Can anyone explain why it takes so long to find the equation differentially? I understand that is subjective, but the 'constants' are only constant from y''' to y''. From y'' to y' the 'constant' is linear (meaning you have to solve for change in y' - y) and the 'constant' is quadratic from y' to y. Basically you can't just integrate each DE and solve for a constant C by finding the value of the lower order DE where…

Huh? What DE?
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