The Riis Complexity Gap for QBF Resolution

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We give an analogue of the Riis Complexity Gap Theorem in Resolution for Quantified Boolean Formulas (QBFs). Every first-order sentence ϕ without finite models gives rise to a sequence of QBFs whose minimal refutations in tree-like QBF Resolution systems are either of polynomial size (if ϕ has no models) or at least exponential in size (if ϕ has some infinite model). However, we show that this gap theorem is sensitive to the translation and different translations are needed for different QBF resolution systems. For tree-like Q-Resolution, the translation to QBF must be given additional structure in order for the polynomial upper bound to hold. This extra structure is not needed in the system tree-like ∀Exp+Res, where we see the complexity gap on a natural translation to QBF.