Hard QBFs for Merge Resolution

Beyersdorff, Olaf; Blinkhorn, Joshua; Mahajan, Meena; Peitl, Tomáš; Sood, Gaurav

doi:10.4230/lipics.fsttcs.2020.12

Cited by 2 publications

(8 citation statements)

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“…that there are formula families having short refutations in M-Res but requiring exponentialsize refutations in LQU + -Res and IRM -the most powerful resolution-based QBF proof systems. Combining this with the results in [7], we conclude that M-Res is incomparable with LQU-Res and LQU + -Res; see Theorems 3.6 and 3.12.…”

Section: Introductionsupporting

confidence: 67%

“…The power of M-Res is shown using modifications of two well-known formula families: KBKF-lq [3] which is hard for M-Res [7], and QUParity [9] which we believe is also hard. The main observation is that the reason making these formulas hard for M-Res is the mismatch of partial strategies at some point in the refutation.…”

Section: Introductionmentioning

confidence: 94%

“…Its definition is a bit technical and can be found in [6]. We give an informal description, see [7] for a slightly longer but still informal description.…”

Section: Merge Resolutionmentioning

confidence: 99%

“…The variant KBKF-lq was defined in [3] and used to show that LD-Q-Res does not simulate QU-Res. In [7], KBKF-lq was also shown to require exponentially large M-Res refutations. We reproduce the definition of this family below and then define two further variants that will be useful for our purpose.…”

Section: Advantage Over Irmmentioning

confidence: 99%

“…They only showed advantage over a restricted version of LD-Q-Res, the system reductionless LD-Q-Res. In a subsequent paper [7], limitations of M-Res were shown -there are formula families which have polynomial-size refutations in QU-Res, LQU-Res, LQU + -Res and CP + ∀red, but require exponential size refutations in M-Res. This, combined with the results from [6], showed that M-Res is incomparable with QU-Res and CP + ∀red.…”

Section: Introductionmentioning

confidence: 99%

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QBF Merge Resolution is powerful but unnatural

Mahajan¹,

Sood²

2022

Preprint

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The Merge Resolution proof system (M-Res) for QBFs, proposed by Beyersdorff et al. in 2019, explicitly builds partial strategies inside refutations. The original motivation for this approach was to overcome the limitations encountered in long-distance Q-Resolution proof system (LD-Q-Res), where the syntactic side-conditions, while prohibiting all unsound resolutions, also end up prohibiting some sound resolutions. However, while the advantage of M-Res over many other resolution-based QBF proof systems was already demonstrated, a comparison with LD-Q-Res itself had remained open. In this paper, we settle this question. We show that M-Res has an exponential advantage over not only LD-Q-Res, but even over LQU + -Res and IRM, the most powerful among currently known resolution-based QBF proof systems. Combining this with results from Beyersdorff et al. 2020, we conclude that M-Res is incomparable with LQU-Res and LQU + -Res.Our proof method reveals two additional and curious features about MRes: (i) M-Res is not closed under restrictions, and is hence not a natural proof system, and (ii) weakening axiom clauses with existential variables provably yields an exponential advantage over MRes without weakening. We further show that in the context of regular derivations, weakening axiom clauses with universal variables provably yields an exponential advantage over M-Res without weakening. These results suggest that M-Res is better used with weakening, though whether M-Res with weakening is closed under restrictions remains open. We note that even with weakening, M-Res continues to be simulated by eFrege + ∀red (the simulation of ordinary M-Res was shown recently by Chew and Slivovsky).

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Section: Introductionsupporting

confidence: 67%

Section: Introductionmentioning

confidence: 94%

“…Its definition is a bit technical and can be found in [6]. We give an informal description, see [7] for a slightly longer but still informal description.…”

Section: Merge Resolutionmentioning

confidence: 99%

Section: Advantage Over Irmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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QBF Merge Resolution is powerful but unnatural

Mahajan¹,

Sood²

2022

Preprint

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Hard QBFs for Merge Resolution

Beyersdorff,

Blinkhorn,

Mahajan

et al. 2024

ACM Trans. Comput. Theory

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We prove the first genuine QBF proof size lower bounds for the proof system Merge Resolution (MRes [7]), a refutational proof system for prenex quantified Boolean formulas (QBF) with a CNF matrix. Unlike most QBF resolution systems in the literature, proofs in MRes consist of resolution steps together with information on countermodels, which are syntactically stored in the proofs as merge maps. As demonstrated in [7], this makes MRes quite powerful: it has strategy extraction by design and allows short proofs for formulas which are hard for classical QBF resolution systems. Here we show the first genuine QBF exponential lower bounds for MRes , thereby uncovering limitations of MRes. Technically, the results are either transferred from bounds from circuit complexity (for restricted versions of MRes) or directly obtained by combinatorial arguments (for full MRes). Our results imply that the MRes approach is largely orthogonal to other QBF resolution models such as the QCDCL resolution systems QRes and QURes and the expansion systems ∀Exp + Res and IR.

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Hard QBFs for Merge Resolution

Cited by 2 publications

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QBF Merge Resolution is powerful but unnatural

QBF Merge Resolution is powerful but unnatural

Hard QBFs for Merge Resolution

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