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They paid more than $500 just to acquire revenue free Deep Mind, so it's safe to say that yes Google is spending more than a billion on AI/ML.

Really things might get scary when the DOD decides to direct some of its monstrous military budget at AI (which I'm sure its already doing).

I really think an AI race between states will be inevitable

DARPA provided much of the early funding for autonomous vehicle research.

The "Grand Challenge"s and all that.

Just left a DARPA grant for an AI project. They've been one of the biggest AI funders for some time.

In addition to DARPA there's IARPA, which has the MICrONS project: http://www.iarpa.gov/index.php/research-programs/microns

Why would you think that they are not already doing so?

They've been doing that off and on for half a century. You may be interested in the history in https://en.wikipedia.org/wiki/AI_winter

Maybe they're going to have lots of AI schools and competitions like they do with Go schools.