Rob Ives -- another of the masters of paper automata -- has created Logic Goats. The OR goat will nod if you press either the first button or the second. The AND goat will only nod if you press the first and the second buttons.
If we could get a NAND and NOR goats and string them all together in complex chains, we might be able to build a Goat-based computer!
Check out, buy and download the Logic Goat Automata at the Flying Pig web site.