I'm a CS undergrad so my math/CS knowledge is not that deep so please correct me if my premise is flawed or I have made some incorrect assumptions.
So I was thinking, much in the way that some primality testers are probabilistic(they give you yes or no but have a chance to be wrong). Would it be possible to build a probabilistic halting problem solver? One that reports within a certain degree of error, whether a problem halts or not?
It depends what you mean by "probabilistic". There are at least two interpretations. First, the algorithm has some probability of success for every input. Second, the algorithm succeeds on a certain fraction of inputs.
For the first interpretation, it is easy to rule out such an algorithm: probabilistic computation can be simulated (inefficiently but effectively) using deterministic computation, by trying all possible coin tosses.
For the second interpretation, we will have to work a bit harder. Suppose that your algorithm is guaranteed to work with an asymptotic success probability of $2/3$. That means that the fraction of inputs in $[1,N]$ for which it gives the correct answer is some $p_N \to 2/3$. Now suppose you're interested in a certain program $P$. It seems that under a reasonable encoding of programs, you would be able to come up with a long stretch $[M,10M]$ (say) of programs equivalent to $P$. By taking $M$ large enough, it should be the case that by taking the majority vote on the answer of a "probabilistic" algorithm on all of these equivalent programs, you will be able to ascertain whether $P$ halts or not. This rules out even this interpretation of a "probabilistic" algorithm.
