What is understanding of natural language and how can we test or measure it? What is the definition of the understanding of (written) natural language and how can we test or measure this understanding?

Natural language understanding is neglected part of natural language processing in computer science. Exact definitions and tests of understanding are necessary for producing artificial general intelligence (AGI) - e.g. https://link.springer.com/chapter/10.1007/978-3-319-41649-6_13 article argues that artificial general intelligence (e.g. software piece) should learn and self-modify itself but such development is possible only if AGI can estimate the quality/fitness of the new self-modification - whether this self-modification is better than existing version of AGI and whether this self-modification can be applied? Obviously - if we want to apply machine learning and AGI to understanding of the natural language then we should be able to test and measure the understanding.

There is article https://link.springer.com/chapter/10.1007/978-3-319-41649-6_11 about understanding specifically but it is somehow narrow minded to the particular QA task. Maybe cognitive science have better tests?

For the reference - there are some AGI systems of "cognitive architectures" under development (wikipedia has list of them) - like OpenCog, NARS, Soar and some others.
Natural language understanding systems can be based on discourse representation theories, which represent the meaning of English sentences as first-order logical predicates. Attempto Controlled English is one example of a natural language understanding system that relies on discourse representation theories.

Similarly, there are several implementations of semantic parsers that convert natural-language texts into formal representations of their meanings.