How strong is an oracle that avoid don't-halt

Question

Consider such an oracle:

Given a turing machine[1], return the halting state it falls on, or arbitary result(but don't stuck in) if the TM doesn't halt.

1. How strong is a TM with the oracle?
2. Can the oracle exist(or does the question always have an answer) if change [1] into a TM with the oracle?

Some results I get:

1. 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.
2. 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.

(Moved from here)

Answer 1

This same question appears on cs.stackexchange; see my answer to it there for details. Since we can't close as a duplicate of a question on another site, I've made this answer community wiki to avoid double-dipping reputation.

These are exactly the PA degrees - the Turing degrees which compute some complete consistent extension of (first-order) Peano arithmetic. The class of PA degrees is incredibly important in computability theory and has a number of equivalent characterizations, with possibly the most useful being:

$X$ has PA degree iff for every computable infinite binary tree $T$, there is an infinite path through $T$ computable relative to $X$.

Although always non-computable (this is Rosser's improvement of Godel's incompleteness theorem), PA degrees can be very weak - in particular, they can be low. There are also PA degrees which are Turing incomparable with the halting problem.