We earlier mentioned that it is required that the finite mapping be precomputed. If it is for arbitrary Turing machines, including those that don’t halt, we need infinite time, so the claim that we can map to arbitrary Turing machines fails. If we restrict it to those which halt, we need to check that before providing the map, which requires solving the halting problem to provide the map.
Edit to add: I’m confused why this is getting “disagree” votes—can someone explain why or how this is an incorrect logical step, or
This isn’t arguing other side, just a request for clarification:
There are some Turing machines that we don’t know and can’t prove if they halt. There are some that are sufficiently simple that we can show that they halt.
But why dors the discussion encompass the question of all possible Turing machines and their maps? Why isn’t it sufficient to discuss the subset which are tractable within some compute threshold?
Wait, how does mapping Turing machine states to integers require us to solve the halting problem?
We earlier mentioned that it is required that the finite mapping be precomputed. If it is for arbitrary Turing machines, including those that don’t halt, we need infinite time, so the claim that we can map to arbitrary Turing machines fails. If we restrict it to those which halt, we need to check that before providing the map, which requires solving the halting problem to provide the map.
Edit to add: I’m confused why this is getting “disagree” votes—can someone explain why or how this is an incorrect logical step, or
This isn’t arguing other side, just a request for clarification:
There are some Turing machines that we don’t know and can’t prove if they halt. There are some that are sufficiently simple that we can show that they halt.
But why dors the discussion encompass the question of all possible Turing machines and their maps? Why isn’t it sufficient to discuss the subset which are tractable within some compute threshold?