I think you're imagining some setup in which a neural net is directly communicating with an external rule engine that generates a set of legal moves for each piece in each board configuration. That's not how it works. You know how people say that Neural Nets are "black boxes"? That's what they mean. You can't just plug into arbitrary steps of their training with arbitrary external processes. You have to setup everything right at the input and then wait patiently at the output, until what you want comes out. Then you can plug that in to whatever reasoning loop you want to set up.
So like I say, it's not a trivial thing and it doesn't generalise very well to arbitrary problems. You need to match the architecture to the problem. For example, that's why ConvNets have been studied extensively for vision- their architecture is specifically tailored to vision problems.
Hence also the doubts expressed in the article about the claims to general game-playing ability. Yeah, you might imagine that it's possible to create a specialised architecture to play any particular game (or even sets of games with similar setups, like chess and shogi)- but it's not clear how exactly you'd do that automatically. Handcrafting a thousand architectures for a thousand different games is not very "general purpose", is it?