Hard Instances of Algorithms and Proof Systems

  • Yijia Chen
  • Jörg Flum
  • Moritz Müller
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7318)


If the class Taut of tautologies of propositional logic has no almost optimal algorithm, then every algorithm \(\mathbb{A}\) deciding Taut has a polynomial time computable sequence witnessing that \(\mathbb{A}\) is not almost optimal. We show that this result extends to every \(\Pi_t^p\)-complete problem with t ≥ 1; however, assuming the Measure Hypothesis, there is a problem which has no almost optimal algorithm but is decided by an algorithm without such a hard sequence. Assuming that a problem Q has an almost optimal algorithm, we analyze whether every algorithm deciding Q, which is not almost optimal algorithm, has a hard sequence.


Optimal Algorithm Polynomial Time Turing Machine Polynomial Time Algorithm Propositional Logic 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Yijia Chen
    • 1
  • Jörg Flum
    • 2
  • Moritz Müller
    • 3
  1. 1.Department of Computer Science and EngineeringShanghai Jiao Tong UniversityShanghaiChina
  2. 2.Abteilung für mathematische LogikAlbert-Ludwigs-Universität FreiburgFreiburgGermany
  3. 3.Kurt Gödel Research Center for Mathematical LogicWienAustria

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