Consider such an oracle:
Given a turing machine, return the halting state it falls on, or arbitary result(but don't stuck in) if the TM doesn't halt.
- How strong is a TM with the oracle?
- Can the oracle exist(or does the question always have an answer) if change  into a TM with the oracle?
Some results I get:
- It's as strong as a TM with oracle that compare running time of two programs, or arbitary returned value if both don't halt.
- If the oracle returns integer the TM returns(may need to define a way it outputs integer), or arbitary integer if it doesn't halt, we can solve the halting problem by getting the runtime and check. However, currently I can't output from the oracle a string of any finite length if the length can be arbitary long, promising it's finite.