Genuine Lower Bounds for QBF Expansion

We propose the first general technique for proving genuine lower bounds in the expansion-based QBF proof systems ∀Exp+Res and IR-calc. We present the technique in a framework centred on natural properties of winning strategies in the ‘evaluation game’ interpretation of QBF semantics. As applications, we prove an exponential lower bound on IR-calc proof size for a whole class of formula families, and demonstrate the power of our approach over existing methods by providing alternative short proofs of two known hardness results. We also use our technique to deduce a result with manifest practical import: in the absence of propositional hardness, formulas separating IR-calc from ∀Exp+Res must have unbounded quantifier alternations.