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I have a pretty good handle on what recursive and recursively enume table languages mean with respect to Turing machines and how they relate to one another through my algorithms class. What I don't understand is how these languages relate to the computability of problems, and whether these languages correlate to problems or something else. I'm missing the bridge between the theory and the practical application of theory so abstract, could somebody bridge the gap?
In particular, what does the recursive nature of a language tell us about the problem being considered? Recursive nature being recursive or r.e.