What sort of test can show that a subject demonstrates an understanding of abstract concepts?
So far, from what I've seen, if a test can be written then software can be written to solve the test.
I don't understand why this is an issue though. Testing intelligence was never the hard part of AI. There are so many tasks that computers currently suck at that we would be happy if they were solved, regardless what label you gave the solution. And I don't think many people could see a computer doing tasks like having conversations or solving difficult problems and deny that it is intelligence. Even if there is no formal test to perform that is 100% certain.
How so? To me, it appears completely open-ended. You could sit there forever asking questions and never reach a definitive result.
OTOH, looks like not very much was actually formalized.
Some rough and imperfect, but still useful, definitions of intelligence could be the ability to make good predictions based on past data, the ability to solve optimization problems well, and learning ability.