Instance Compression for the Polynomial Hierarchy and beyond

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 Σ i 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, thenΣiCIRCUITSAT 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 ([12]), Polynomial Hierarchy (PH) collapses.