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This is also required to get the answer exactly right. It's just that the problem domains they chose -- symbolic integration and differential equations -- can

This is also required to get the answer exactly right. It's just that the problem domains they chose -- symbolic integration and differential equations -- can be checked easily. That is, if I have a guess F for the integral of f, I just need to symbolically differentiate F and see whether I get f.* So the paper does "guess and check," and only returns answers that pass the check, i.e. are exactly right. There are some problems where all the checks fail, i.e. the paper can't generate an answer. And there are also questions where Matlab / Maple can't find an answer either. * It's a little more complicated; e.g. if f = cos^2, and the derivative of F is 1 - sin^2, I need to recognize they're the same. But it's safe, in the sense that if I can't determine whether they're the same, I can reject the solution and move on to the next one.

whatshisface 7y ago
Wait a minute, doesn't that mean your checker is doing symbolic algebra?
- jfkebwjsbx 7y ago
  Yes, what is the issue?
- thomasahle 7y ago
  Yes, but the algebra required to check the equations is much less than what's required to solve them.
- martincmartin 7y ago
  Yes, but symbolic differentiation is straight forward. There are simple rules to apply to differentiate any expression: the chain rule, the product rule, the quotient rule, derivatives of known single argument functions, etc. Integration is a completely different ball game. Given an arbitrary expression, most likely it's not symbolically integrable, i.e. it's integral can't be expressed as a combination of known functions. That's why there are lots of functions defined as "the integral of blah." Even given an expression that's known to be symbolically integrable, it's quite the puzzle to figure out what the integral is. There's no straight forward algorithm, as with differentiation. It boils down to "try integration by parts; try this; try that; maybe do some clever algebraic manipulation to get it into a form where you can try one of those things; look it up in a book of integrals to see if it's there, or something similar is there; if all else fails, guess and check."

whatshisface 7y ago

Wait a minute, doesn't that mean your checker is doing symbolic algebra?

jfkebwjsbx 7y ago
Yes, what is the issue?
thomasahle 7y ago
Yes, but the algebra required to check the equations is much less than what's required to solve them.
martincmartin 7y ago
Yes, but symbolic differentiation is straight forward. There are simple rules to apply to differentiate any expression: the chain rule, the product rule, the quotient rule, derivatives of known single argument functions, etc. Integration is a completely different ball game. Given an arbitrary expression, most likely it's not symbolically integrable, i.e. it's integral can't be expressed as a combination of known functions. That's why there are lots of functions defined as "the integral of blah." Even given an expression that's known to be symbolically integrable, it's quite the puzzle to figure out what the integral is. There's no straight forward algorithm, as with differentiation. It boils down to "try integration by parts; try this; try that; maybe do some clever algebraic manipulation to get it into a form where you can try one of those things; look it up in a book of integrals to see if it's there, or something similar is there; if all else fails, guess and check."