The Axiom computer is the main computer that controls The Axiom. Even though at first it seems that the Axiom computer controls the robots, it seems not because the captain could not issue a voice command to the computer to stop Auto. That may imply that the computer just didn't have the authorization to affect the autopilot but that does not mean that the computer was not responsible to waking up the other robots on the starliner when they were needed.
When the computer's inability affect Auto is noticed it may first seems that the computer was only a reference database and had no any actual control over the starliner, but some details clearly show that that is not the case.
When Auto was turned off it was the computer that made the Axiom to hyper jump and return on Earth. Also when the captain pushed the green it is likely that computer closed the doors of the bridge and presented the message from Shelby Forthright about the Operation Recolonize. It is also clear that it was the computer who opened EVEs cavity after captain has "allowed" that to happen. The computer also seem to have the control over the escape pods and the probe ship.
These points clearly indicate that it was really the computer who ran most of the functions of the Axiom and that Auto was responsible for piloting the starliner, but it seems that he had domination over the computer since he was able to disconnect it from captain's quarter and that's why the captain was unable to order the computer to let him out of the room.
The computer is even more programming based than Auto, but has more humane voice than Auto has (Because the computer had a real voice actor, Sigourney Weaver). The computer only acted in specific beforehand programmed situations.