The description of this game is already exists in this link
I am quoting from this link the description of the game to ease the reading of this question:
In a game, where there are m active bombs, each one with a number above zero, and n standing levers, where each can be either pushed forward or pulled backward, and each lever is connected to some bombs with green wires/cables and connected to some bombs with red wires/cables.
If any lever is pushed forward then all bombs that are connected to this lever via green wire/cable are defused and all bombs that are connected to this lever via red wire/cable, their number is decremented by 1.
If any lever is pulled backward then all bombs that are connected to this lever via red wire/cable are defused and all bombs that are connected to this lever via green wire/cable, their number is decremented by 1.
A defused bomb cannot explode, but any active bomb that it's number reaches zero explodes immediately.
Only standing levers can be touched (either pushed forward or pulled backward). You shouldn't touch already touched (either pushed forward or pulled backward) levers.
To win/beat the game the purpose is to defuse all the active bombs without exploding any of them.
My greedy algorithm described in this question suppose to run in polynomial time, but it's not proven if it is correct.
Even though proving it's correctness is very difficult for me I am attempting to prove it, but this question isn't about this greedy algorithm at all.
Now this question is more general:
Can deterministic Turing machine beats/wins the game described in this question in polynomial time?
If yes, then there should exist polynomial time algorithm that returns all steps that should be followed to beat/win the given instance of this game (if it is possible).
The deterministic Turing machine or the algorithm doesn't have to be greedy necessarily.