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@SynergyBuild - Computability refers to how (and whether) a recursive definition can be defined rather than if it is physically possible.
Graham's Number is just a power of three (a very large one, but a power nonetheless). Its recursive definition follows from how Knuth's Up-Arrow notation is recursive exponentiation, which is recursive multiplication, which is recursive addition, which finally is a recursive successor function (the "plus one" function), defined axiomatically to be a primitive recursive function and thus computable.
TREE(3) is the maximum number of entries into a sequence of trees with ever-increasing numbers of nodes (and a bunch of other stuff). We can prove that the sequence does end, but we cannot prove that we can calculate it recursively. We can potentially solve for it by drawing seemingly infinite sequences of trees, but we would have no way of knowing if we found the longest sequence. That is why TREE(3) is not computable.
An example I know better is the busy beaver function. The busy beaver function, abbreviated as BB(n), is the classic example of a noncomputable function. Its solutions (though proved to be defined on all inputs of n) are proven that, for any output of a computable function f evaluated at n, BB(n)>f(n), no computable function can be used to solve (and therefore recursively solve) for any value (technically I skipped a point of the n being above a certain size, but it is not relevant to the point). Hence, only lower bounds can be found for any value of BB(n).
March 14, 2019 4:36 p.m.
@SynergyBuild - While I love me some TREE, it is not computable, and would thus not be valid in a tournament setting. While we know it is finite, we have no way (at the moment) of knowing anything about it. We do not even know its parity. We can compute Graham's Number, as evidenced by the recursive up-arrow notation we use to define it. Thus, it can be computed and even explained, making it allowable in a tournament setting.
March 14, 2019 11:34 a.m.
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