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Documents authored by Blinkhorn, Joshua

Document
DOI: 10.4230/LIPIcs.FSTTCS.2020.12
Hard QBFs for Merge Resolution

Authors: Olaf Beyersdorff, Joshua Blinkhorn, Meena Mahajan, Tomáš Peitl, and Gaurav Sood

Published in: LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

Abstract
We prove the first proof size lower bounds for the proof system Merge Resolution (MRes [Olaf Beyersdorff et al., 2020]), 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 [Olaf Beyersdorff et al., 2020], 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 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.
Cite as

Olaf Beyersdorff, Joshua Blinkhorn, Meena Mahajan, Tomáš Peitl, and Gaurav Sood. Hard QBFs for Merge Resolution. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{beyersdorff_et_al:LIPIcs.FSTTCS.2020.12,
  author =	{Beyersdorff, Olaf and Blinkhorn, Joshua and Mahajan, Meena and Peitl, Tom\'{a}\v{s} and Sood, Gaurav},
  title =	{{Hard QBFs for Merge Resolution}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{12:1--12:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Saxena, Nitin and Simon, Sunil},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.12},
  URN =		{urn:nbn:de:0030-drops-132530},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.12},
  annote =	{Keywords: QBF, resolution, proof complexity, lower bounds}
}
Document
DOI: 10.4230/LIPIcs.STACS.2019.14
Building Strategies into QBF Proofs

Authors: Olaf Beyersdorff, Joshua Blinkhorn, and Meena Mahajan

Published in: LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

Abstract
Strategy extraction is of paramount importance for quantified Boolean formulas (QBF), both in solving and proof complexity. It extracts (counter)models for a QBF from a run of the solver resp. the proof of the QBF, thereby allowing to certify the solver’s answer resp. establish soundness of the system. So far in the QBF literature, strategy extraction has been algorithmically performed from proofs. Here we devise the first QBF system where (partial) strategies are built into the proof and are piecewise constructed by simple operations along with the derivation. This has several advantages: (1) lines of our calculus have a clear semantic meaning as they are accompanied by semantic objects; (2) partial strategies are represented succinctly (in contrast to some previous approaches); (3) our calculus has strategy extraction by design; and (4) the partial strategies allow new sound inference steps which are disallowed in previous central QBF calculi such as Q-Resolution and long-distance Q-Resolution. The last item (4) allows us to show an exponential separation between our new system and the previously studied reductionless long-distance resolution calculus, introduced to model QCDCL solving. Our approach also naturally lifts to dependency QBFs (DQBF), where it yields the first sound and complete CDCL-type calculus for DQBF, thus opening future avenues into DQBF CDCL solving.
Cite as

Olaf Beyersdorff, Joshua Blinkhorn, and Meena Mahajan. Building Strategies into QBF Proofs. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{beyersdorff_et_al:LIPIcs.STACS.2019.14,
  author =	{Beyersdorff, Olaf and Blinkhorn, Joshua and Mahajan, Meena},
  title =	{{Building Strategies into QBF Proofs}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{14:1--14:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-100-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{126},
  editor =	{Niedermeier, Rolf and Paul, Christophe},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.14},
  URN =		{urn:nbn:de:0030-drops-102538},
  doi =		{10.4230/LIPIcs.STACS.2019.14},
  annote =	{Keywords: QBF, DQBF, resolution, proof complexity}
}
Document
DOI: 10.4230/LIPIcs.STACS.2018.12
Genuine Lower Bounds for QBF Expansion

Authors: Olaf Beyersdorff and Joshua Blinkhorn

Published in: LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

Abstract
We propose the first general technique for proving genuine lower bounds in expansion-based QBF proof systems. We present the technique 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 a result with manifest practical import: in the absence of propositional hardness, formulas separating the two major QBF expansion systems must have unbounded quantifier alternations.
Cite as

Olaf Beyersdorff and Joshua Blinkhorn. Genuine Lower Bounds for QBF Expansion. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{beyersdorff_et_al:LIPIcs.STACS.2018.12,
  author =	{Beyersdorff, Olaf and Blinkhorn, Joshua},
  title =	{{Genuine Lower Bounds for QBF Expansion}},
  booktitle =	{35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
  pages =	{12:1--12:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-062-0},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{96},
  editor =	{Niedermeier, Rolf and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.12},
  URN =		{urn:nbn:de:0030-drops-85174},
  doi =		{10.4230/LIPIcs.STACS.2018.12},
  annote =	{Keywords: QBF, proof complexity, lower-bound techniques, resolution}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2018.9
Size, Cost and Capacity: A Semantic Technique for Hard Random QBFs

Authors: Olaf Beyersdorff, Joshua Blinkhorn, and Luke Hinde

Published in: LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

Abstract
As a natural extension of the SAT problem, an array of proof systems for quantified Boolean formulas (QBF) have been proposed, many of which extend a propositional proof system to handle universal quantification. By formalising the construction of the QBF proof system obtained from a propositional proof system by adding universal reduction (Beyersdorff, Bonacina & Chew, ITCS'16), we present a new technique for proving proof-size lower bounds in these systems. The technique relies only on two semantic measures: the cost of a QBF, and the capacity of a proof. By examining the capacity of proofs in several QBF systems, we are able to use the technique to obtain lower bounds based on cost alone. As applications of the technique, we first prove exponential lower bounds for a new family of simple QBFs representing equality. The main application is in proving exponential lower bounds with high probability for a class of randomly generated QBFs, the first 'genuine' lower bounds of this kind, which apply to the QBF analogues of resolution, Cutting Planes, and Polynomial Calculus. Finally, we employ the technique to give a simple proof of hardness for a prominent family of QBFs.
Cite as

Olaf Beyersdorff, Joshua Blinkhorn, and Luke Hinde. Size, Cost and Capacity: A Semantic Technique for Hard Random QBFs. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 9:1-9:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{beyersdorff_et_al:LIPIcs.ITCS.2018.9,
  author =	{Beyersdorff, Olaf and Blinkhorn, Joshua and Hinde, Luke},
  title =	{{Size, Cost and Capacity: A Semantic Technique for Hard Random QBFs}},
  booktitle =	{9th Innovations in Theoretical Computer Science Conference (ITCS 2018)},
  pages =	{9:1--9:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-060-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{94},
  editor =	{Karlin, Anna R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.9},
  URN =		{urn:nbn:de:0030-drops-83228},
  doi =		{10.4230/LIPIcs.ITCS.2018.9},
  annote =	{Keywords: quantified Boolean formulas, proof complexity, lower bounds}
}
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