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April 15th, 2014, 07:21 AM   #1
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generating expressions instead of deriving solutions

generating expressions instead of deriving solutions

All possible expressions could be generated and each one tried out as a possible solution of a given problem. Once in a while we would find it to be an actual solution. This process of generating expressions as mere combinations of symbols and then verifying each one against problems, is one way of directly finding solutions instead of deriving them logically from the statement of the problem.

For instance instead of a series or algorithmic solution to find the square root of a number, we could simply try different numbers one by one, square each one, and we would find out the number that is the correct square root. It requires very little knowledge to do so, and computer programs could go on doing that for many if not all the world's mathematical problems, in principle, at least the problems that can be stated as a search for an expression.

It is a "brute force" method but is it correct or is there a flaw in this reasoning.
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April 15th, 2014, 07:39 AM   #2
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Well, it would be a slow process, and some things are not amenable to it. If the question, for instance, is something as simple as "what is the nth triangular number?", how would you know if you happened upon the right answer except to calculate the nth triangular number anyway? Perhaps you could give an example or two of when it would be useful.

I suspect the example you gave, ie finding a root, is the sort of thing where something like what you are saying would be the least useless. (Pardon my bluntness. Read on.) Squaring is easy, but finding square roots is relatively a lot more difficult. But even here, you won't want to do things fully randomly. Starting with a reasonable guesstimate and then adjusting your next guess based on how close you got would be vastly more sensible.

Even trial factoring should be done with some strategy. Take some huge number. It will have anywhere from 1 to a whole bunch of factors, which can be of any size relative to to number being factored. So there is a lot more "play" and you could in principle start anywhere and, if that number fails, jump to anywhere else w/o necessarily losing any useful info at all. But existing factoring programs do make use of mathematics to streamline the search.
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April 15th, 2014, 08:02 AM   #3
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I think I've read about a similar idea attributed to ancient Greeks, but I could not find a confirmation. The suggestion was to make a number of scribed write all possible predictions for the future, and a group of wise men to select meaningful statements.

In modern mathematics, this was studied in computational complexity theory. If, for example, the size of the solution is limited by the size of the problem, then there are $2^n$ potential solutions where $n$ is that size (if we assume solutions are written in binary code). Now, exponential function grows extremely fast, and from some point, $2^n$ becomes prohibitively large even if measured, say, in microseconds. It is easy to find a realistic problem for which $2^n$ exceeds the number of elementary particles in the universe and the number of microseconds the universe existed. So, even though guessing and checking a solution is theoretically possible for some problems, it is not feasible.

Another difficulty is that a problem can be only semi-decidable. That is, given a problem instance, if there exist a solution, then you will eventually find it through a systematic search, but if no solution exists, then there is no way to ascertain this fact. Some problems have been proved to be of this sort. For example, given an arbitrary potential theorem statement, even if it is only about arithmetic, you could search for all possible proofs, and if there is one, then you will eventually find it. But there is no way algorithm to decide if a given statement has a proof or not. Thus, in the worst case you will look for a non-existing proof forever.
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