It will cost a trillion dollars because someone in this thread's already thrown that out as a budgeting figure, and it's a government program, so no matter how much it should cost, a trillion will be the target. But based on studies of government contract projects, we should anticipate a 150% cost overrun, at minimum, so we're up to $1.5 trillion.
It would be cheaper not to have the nuclear powered starship equiped with solid-fueled impulse engines on the saucer section, but the Senator from Utah made sure those were part of the basic design. The onboard AI spends 10% of its CPU time thinking about the stupidity of it.
In any event, the ship just needs a pocket calculator to fly. It's the crew that needs to be AI's. Then they don't need space suits to go out on the hull and replace broken tiles for thousands of years, and don't have to procreate to make sure the ship doesn't run low on "tile-replacers."
But, as was said during Apollo, why use computers when you can build more humans using untrained manual labor?
A trillion dollars was my out-of-the-ass figure and you know that. It was a hypothetical to illustrate that we just don't know what something like that would cost in real dollars because we're nowhere near accomplishing it.
You are right that it only takes a computer of minuscule power so properly guide a ship through space. There's not much involved besides some simple sensors and physics calculations. Just harden the thing against interstellar radiation and make sure it has an adequate energy supply and the damn thing should be able to run forever, or until it crashes into something.