Abstract
We examine the tree-like versions of QBF Frege and extended Frege systems. While in the propositional setting, tree-like and dag-like Frege are equivalent, we show that this is not the case for QBF Frege, where tree-like systems are exponentially weaker. This applies to the version of QBF Frege where the universal reduction rule substitutes universal variables by 0/1 constants.To show lower bounds for tree-like QBF Frege we devise a general technique that provides lower bounds for all tree-like QBF systems of the form P+∀red, where P is a propositional system. The lower bound is based on the semantic measure of strategy size corresponding to the size of countermodels for false QBFs.We also obtain a full characterisation of hardness for tree-like QBF Frege. Lower bounds for this system either arise from a lower bound to propositional Frege, from a circuit lower bound, or from a lower bound to strategy size.
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