From what I remember reading online before biking is allowed on the Appalachian Trail but only in certain places such as where the A.T. coincides for about three miles with the C&O Canal towpath in Maryland, the Virginia Creeper Trail in Virginia.
But some of the Appalachian Trail does not allow bike riding through it I believe for safety reasons but of course you could try biking the Appalachian Trail and the worst that would happen is you maybe get fined or told not do do so and you'd have to push the bike.
However it's very unlikely you'd get caught if you did bike halfway through the Appalachian Trail.