If you're going to go for a ternary system, wouldn't full cart (1), empty cart (0), no cart (-1) make sense?
The problem I see with any "full cart/empty cart/no cart" system would be that it would necessitate a continuous flow of carts to prevent a constant "0" entry. What happens if a cart is late? The entire calculation goes haywire and you get a garbled mess as your result. Personally, I think filled="0" part-filled="1" system would work best, with empty carts returned to the "top" of the computer to be re-used.
That said, I do think we should at least attempt a ternary logic or higher computer. It would elegantly reflect dwarven logic in the sense that a statment could be True, False, or Hell-if-I-Know.
There's already been some documentation for three-state logic problems, but I'll boil it down.
In Boolean (and binary) logic, we assume that a given statement is either True or False. In
Łukasiewicz logic, we accept that there are things that cannot be proven true or false and represent it as another value of validity in the statement. So True, False, and Unknown. (Depending on the question asked, Unknown can also mean irrelavent, both, or maybe)
So if we say that "Urist is a Dwarf", that would True. "Elves are cool" would be False. But something like "There are Goblins outside" is a definite Maybe. We simply don't know one way or the other, and can't determine the answer without additional information.
If this was implemented with carts, a Filled/part-filled/empty system would work best, to prevent the continuous null entries that using "no-cart" as a valid entry.