Abstract
We aim to understand inherent reasons for lower bounds for QBF proof systems and revisit and compare two previous approaches in this direction. The first of these relates size lower bounds for strong QBF Frege systems to circuit lower bounds via strategy extraction (Beyersdorff and Pich, LICS’16). Here, we show a refined version of strategy extraction and thereby for any QBF proof system obtain a trichotomy for hardness: (1) via circuit lower bounds, (2) via propositional Resolution lower bounds, or (3) “genuine” QBF lower bounds. The second approach tries to explain QBF lower bounds through quantifier alternations in a system called relaxing QU-Res (Chen, ACM TOCT 2017). We prove a strong lower bound for relaxing QU-Res, which at the same time exhibits significant shortcomings of that model. Prompted by this, we introduce a hierarchy of new systems that improve Chen’s model and prove a strict separation for the complexity of proofs in this hierarchy. We show that lower bounds in our new model correspond to the trichotomy obtained via strategy extraction.
Highlights
Proof complexity studies the question of how difficult it is to prove theorems in different formal proof systems
There are circuits Ci ∈ C witnessing ψn so that the witnessed formulas have polynomial-size Resolution refutations, but for all such circuits Ci it is hard to derive i Ci(x1, . . . , xi, y1, . . . , yi−1) = yi from ¬φn in P. This means that any quantified Boolean formula (QBF) lower bound on P is either a circuit lower bound, a propositional proof complexity lower bound, or a ‘genuine’ QBF proof complexity lower bound in the sense that P cannot derive efficiently some circuits witnessing the existential quantifiers in the original formula and whenever it can do that for some other witnessing circuits, the witnessed formula is hard for Resolution
The lower bounds we show on the size of QU-Res proofs of these formulas are clearly due to a lower bound on Resolution proofs, rather than alternation of quantifiers, or any other ‘genuine’ QBF reasons
Summary
Proof complexity studies the question of how difficult it is to prove theorems in different formal proof systems. This leads to the rather disturbing fact that lower bounds for e.g. Resolution trivially lift to any of the studied QBF Resolution systems Motivated by this observation, Chen [12] introduced the new notions of relaxing QU-Res and a proof system ensemble, with the aim of distinguishing ‘genuine’ QU-Res lower bounds, arising from the alternation of quantifiers, from those lifted from propositional Resolution. The exponential lower bound for relaxing QU-Res given in [12] applies only to quantified Boolean circuits with no small CNF representations (Appendix ??) As this is a somewhat atypical feature in proof complexity, we improve this by presenting QBFs with CNF matrices that require exponential-size relaxing QU-Res proofs (Theorem 9).
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