A parameterized halting problem

  • Authors:
  • Yijia Chen;Jörg Flum

  • Affiliations:
  • Shanghai Jiaotong University, China;Albert-Ludwigs-Universität Freiburg, Germany

  • Venue:
  • The Multivariate Algorithmic Revolution and Beyond
  • Year:
  • 2012

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Abstract

The parameterized problem $p\textsc{-Halt}$ takes as input a nondeterministic Turing machine $\mathbb{M}$ and a natural number n, the size of $\mathbb{M}$ being the parameter. It asks whether every accepting run of $\mathbb{M}$ 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 $\mathbb{M}$ 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 $p{\rm \textsc{-Halt}\in{XP}_{uni}}$. 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.