Instance Compression for the Polynomial Hierarchy and beyond

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 If CircuitSAT is non-uniformly compressible within NP, then Σ _i CircuitSAT 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.