A Parameterized Halting Problem

The parameterized problem p-Halt takes as input a nondeterministic Turing machine M and a natural number n, the size of M being the parameter. It asks whether every accepting run of M on empty input tape takes more than n steps. This problem is in the class XPuni, the class “uniform XP,” if there is an algorithm deciding it, which for fixed machine M runs in time polynomial in n. It turns out that various open problems of different areas of theoretical computer science are related or even equivalent to p-Halt ∈ XPuni. Thus this statement forms a bridge which allows to derive equivalences between statements of different areas (proof theory, complexity theory, descriptive complexity, . . . ) which at first glance seem to be unrelated. As our presentation shows, various of these equivalences may be obtained by the same method.