The complexity of recursive constraint satisfaction problems

We investigate the complexity of finding solutions to infinite recursive constraint satisfaction problems. We show that, in general, the problem of finding a solution to an infinite recursive constraint satisfaction problem is equivalent to the problem of finding an infinite path through a recursive tree. We also identify natural classes of infinite recursive constraint problems where the problem of finding a solution to the infinite recursive constraint problem is equivalent of the problem of finding an infinite path through a recursive finitely branching recursive trees or a recursive binary tree. There are a large number of results in the literature on the complexity of the problem of finding an infinite path through a recursive tree. Our main result allows us to automatically transfer such results to to give equivalent results about the complexity of the problem of finding a solution to a recursive CSP problem.