Dishwasher automatically turning on batch feed disposer?
Some dishwashers have an internal food grinder, so there is no need to use an external one. It was common practice to connect the discharge of the dishwasher to the disposer, but not likely that the disposer would be activated by the dishwasher.  In fact, it could be quite dangerous if the disposer was to activate automatically.

It's not impossible for the dishwasher to activate the disposer, but it's not likely (or common, or possibly even safe).  You could verify whether or not the disposer is activated quite easily, simply by running the dishwasher and monitoring the disposer to see if it does indeed come on.  There would also have to be some wiring running between the dishwasher and the disposer, so you could look to see if there is.