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Lower Bound Techniques for QBF Expansion

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Abstract

We propose some general techniques for proving lower bounds in expansion-based QBF proof systems. We present them in a framework centred on natural properties of winning strategies in the ‘evaluation game’ interpretation of QBF semantics. As applications, we prove an exponential proof-size lower bound 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 an interesting result: in the absence of propositional hardness, formulas separating the two major QBF expansion systems must have unbounded quantifier alternations.

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Notes

  1. Proof system P1 simulates proof system P2 whenever P1-proofs and be transformed into P2-proofs with at most polynomial increase in proof size.

  2. This is our notation; in [30], the formulas are referred to as ‘(2)’.

  3. That such a clause and its subderivation remain is proved as part of Proposition 2. We note that this subderivation may include weakening steps – the addition of arbitrary literals to a clause – but such steps are easily erased from a refutation.

  4. In fact, the authors separated Q-Res from ∀Exp+Res; since IR-calcp-simulates Q-Res [9], the result stated in the text is an immediate corollary.

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Acknowledgments

We thank Meena Mahajan and Anil Shukla for helpful discussions on this work.

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  1. Institut für Informatik, Friedrich Schiller Universität Jena, Jena, Germany

    Olaf Beyersdorff & Joshua Blinkhorn

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  1. Olaf Beyersdorff
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Correspondence to Joshua Blinkhorn.

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This article is part of the Topical Collection on Special Issue on Theoretical Aspects of Computer Science (2018)

An extended abstract of this article appeared in the proceedings of the conference STACS’18 [6]. Research was supported by grants from the John Templeton Foundation (grant no. 60842), the Carl-Zeiss Foundation, and the EU (Marie Curie IRSES project CORCON).

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Beyersdorff, O., Blinkhorn, J. Lower Bound Techniques for QBF Expansion. Theory Comput Syst 64, 400–421 (2020). https://doi.org/10.1007/s00224-019-09940-0

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