A parameterized halting problem

Abstract

The parameterized problem 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 XP uni, 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-H ALT XP uni. 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. © 2012 Springer-Verlag Berlin Heidelberg.