The uncomfortable question: Domain

[Phryxia]

At first I just want to know whether two silly expressions are same or not. But as I study more and more, it was not silly problem.

In most of the case, implicit domain assumption works. For example when we compare x and (x^0.5)^2, we assume their start domain is same and it's the smallest one(In this example, x >= 0). Since I defined that valid optimization never shrinks the domain, it was enough.

But following questions were arised.

ln(x) * ln(y^2) vs ln(x^2) * ln(y)

If we apply the same approach of previous things, they should be same since our assumption starts from the smallest domain. But how should I do that? Pulling the exponent out shrinks the domain theoretically, therefore it's not valid. But if we know the start domain explicitly (like (x > 0, y > 0), we may pull them out.

Even we ignore the case of different domain, same domain can cause problem.

x = (x^2)^0.5 if x >= 0 otherwise they are different

There is no way to decide whether some transformation shrinks the total domain or not with implicit assumption.

To handle this, we need following things to prepare.

• Represent the set using equality / inequality.
• Decide whether a set is subset of others or not.