A pressure plate on a slab will not open a trapdoor, but will close an open one. Doors work as expected, pressure plate on a slab will open them. wood and stone pressure plates act the same, and all types of slabs the same.
What I expected to happen was...:
I thought that a pressure plate on a slab would open the trap door
What actually happened was...:
the trap door did not open
Steps to Reproduce:
1. put down a slab level with the top of a block
2. place a pressure plate on the slab
3. place a trap door on the same level, on the slab or appropriate block
4. walk on to the pressure plate
5. an open trapdoor will close, a closed one will do nothing.
Linked issues
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Comments

Since the top-slab is not a whole block, I would think it shouldn't transmit redstone signal at all. So, what I would expect to happen would be that the trap door doesn't move at all.
I think the bug here is that an open one is closing.
Main issue: any redstone update close to a door/trapdoor/fence gates causes it to update.
Same with wooden doors.
I just did some tests with other "block updates", since in a certain way it does make sense that a (trap)door which has been put into a "bad" state by a player would change to the "right" state when receiving a block update.
There are some inconsistencies. When I open an unpowered (trap)door (so by a player) it will close again if there is a redstone block update, but not if there are other block updates like placing/removing a block. But when it is the other way around, so when I close a powered (trap)door it will open again with any block update.
Confirmed in 13w19a.
Very annoying behavior. Not very usable in redstone contraptions. Should be fixed ASAP!
Device to replicate this:
! = Door/Trapdoor/Fence Gate
O = Redstone
+ = Block
8 = Button
!
+
OOOOOOO+8
Open the door/trapdoor/fence gate and press the button.
Can this issue please be updated to include 1.6.2 and Survival mode as well?
Fixed only for doors in 14w11b.

Affects 18w46a (at least the trapdoor thing does - why are there like three bug reports crammed into one here?)

Can confirm fixed in 1.14 for trapdoors/doors/fence gates
The report is only about trapdoors, so fixed?

I just tested it: Upper slab, pressure plate on top, trap door next to the pressure plate, open trap door manually, step on pressure plate, step off pressure plate. Result: Trap door closes.
The case in the screenshots does indeed not apply anymore, can someone remove them, please?
@unknown What do you mean with three reports in one?

Can I have my ownership of this ticket revoked? Don't want to deal with it anymore.

The issue described in this ticket is actually the trapdoor closing to the pressure plate, as it not opening through the slab is intended. This was finally fixed in a 1.13 snapshot, I presume due to new blockstates.
Can confirm.