Instance compression for the polynomial hierarchy and beyond

  • Authors:

  • Chiranjit Chakraborty;Rahul Santhanam

  • Affiliations:

  • School of Informatics, University of Edinburgh, UK;School of Informatics, University of Edinburgh, UK

  • Venue:
  • IPEC'12 Proceedings of the 7th international conference on Parameterized and Exact Computation
  • Year:
  • 2012

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Abstract

We define instance compressibility ([5,7]) for parametric problems in PH and PSPACE. We observe that the problem ΣiCircuitSAT 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 If CircuitSAT is non-uniformly compressible within NP, then ΣiCircuitSAT is non-uniformly compressible within NP, for any i≥1. 2 If QBCSAT is non-uniformly compressible (or even if satisfiability of quantified Boolean CNF formulae is non-uniformly compressible), then PSPACE ⊆ NP/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.