Abstract
We provide a characterisation for the size of proofs in tree-like Q-Resolution and tree-like QU-Resolution by a Prover–Delayer game, which is inspired by a similar characterisation for the proof size in classical tree-like Resolution. This gives one of the first successful transfers of one of the lower bound techniques for classical proof systems to QBF proof systems. We apply our technique to show the hardness of three classes of formulas for tree-like Q-Resolution. In particular, we give a proof of the hardness of the parity formulas from Beyersdorff et al. (2015) [10] for tree-like Q-Resolution and of the formulas of Kleine Büning et al. (1995) [29] for tree-like QU-Resolution.
Highlights
Proof complexity is a well established field that has rich connections to fundamental problems in computational complexity and logic [21,30]
As most modern SAT solvers employ methods based on conflict-driven clause learning (CDCL) [34], they correspond to Resolution
As for SAT solvers, each execution trace of a quantified Boolean formulas (QBF) solver can be interpreted as a witness for the truth of the QBF or respectively as a proof of its unsatisfiability, and there has been great interest in trying to understand which formal systems would correspond to the solvers
Summary
Proof complexity is a well established field that has rich connections to fundamental problems in computational complexity (the separation of complexity classes) and logic (the separation of theories of bounded arithmetic) [21,30]. They apply only to special classes of formulas, expressing principles for which we have circuit lower bounds (which are embarrassingly few) For this reason, all present lower bounds for QBF proof systems — except for recent results proven by the new strategy extraction method [8,10] and feasible interpolation [11] — are either shown ad hoc or are obtained by lifting known classical lower bounds or previous QBF bounds We use our new technique to show that these formulas require exponential-size proofs in tree-like QU-Resolution, which in contrast to the previous two examples provides a new hardness result.
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