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Instance Compression for the Polynomial Hierarchy and beyond

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 7535)

Abstract

We define instance compressibility ([5,7]) for parametric problems in PH and PSPACE. We observe that the problem Σ i CircuitSAT of deciding satisfiability of a quantified Boolean circuit with i − 1 alternations of quantifiers starting with an existential quantifier is complete for parametric problems in the class \(\Sigma_{i}^{p}\) with respect to W-reductions, and that analogously the problem QBCSAT (Quantified Boolean Circuit Satisfiability) is complete for parametric problems in PSPACE with respect to W-reductions. We show the following results about these problems:

  1. 1

    If CircuitSAT is non-uniformly compressible within NP, then Σ i CircuitSAT is non-uniformly compressible within NP, for any i ≥ 1.

  2. 2

    If QBCSAT is non-uniformly compressible (or even if satisfiability of quantified Boolean CNF formulae is non-uniformly compressible), then PSPACENP/poly and PH collapses to the third level.

Next, we define Succinct Interactive Proof (Succinct IP) and by adapting the proof of IP = PSPACE ([4,2]), we show that QBFormulaSAT (Quantified Boolean Formula Satisfiability) is in Succinct IP. On the contrary if QBFormulaSAT has Succinct PCPs ([11]), Polynomial Hierarchy (PH) collapses.

Keywords

  • Polynomial Time
  • Parametric Problem
  • Boolean Formula
  • Input String
  • Compression Function

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Chakraborty, C., Santhanam, R. (2012). Instance Compression for the Polynomial Hierarchy and beyond. In: Thilikos, D.M., Woeginger, G.J. (eds) Parameterized and Exact Computation. IPEC 2012. Lecture Notes in Computer Science, vol 7535. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33293-7_13

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  • DOI: https://doi.org/10.1007/978-3-642-33293-7_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-33292-0

  • Online ISBN: 978-3-642-33293-7

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