Translation:My mother is a waitress at a Chinese restaurant.
It's almost definitely not programing. Programming for natural languages is extremely difficult and even Google Translate still can't get it right.
Instead, they have a database in which non-programmers simply enter acceptable answers one by one. They may try to think of all they can, then they rely on us contributing the rest by using the "suggest" button.
(There may be a tiny bit of programming since I've seen it misapply English contractions several times. Proving that it is indeed hard to do right.)
They are different, but the way in which they differ depends on context in both speech and writing. In this sentence "in A Chinese restaurant" means there is no attempt to describe which restaurant, whereas "in THE Chinese restaurant" would only be said where you are referring to a particular restaurant that is already known or specified by the context e.g. in a previous sentence or in the town or place that you were discussing. Without any context you would normally say "in A Chinese restaurant", so I suppose that is why Duo prefers that.