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Proof Complexity of Fragments of Long-Distance Q-Resolution

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Theory and Applications of Satisfiability Testing – SAT 2019 (SAT 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11628))

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Abstract

Q-resolution is perhaps the most well-studied proof system for Quantified Boolean Formulas (QBFs). Its proof complexity is by now well understood, and several general proof size lower bound techniques have been developed. The situation is quite different for long-distance Q-resolution (LDQ-resolution). While lower bounds on LDQ-resolution proof size have been established for specific families of formulas, we lack semantically grounded lower bound techniques for LDQ-resolution.

In this work, we study restrictions of LDQ-resolution. We show that a specific lower bound technique based on bounded-depth strategy extraction does not work even for reductionless Q-resolution by presenting short proofs of the QParity formulas. Reductionless Q-resolution is a variant of LDQ-resolution that admits merging but no universal reduction. We also prove a lower bound on the proof size of the completion principle formulas in reductionless Q-resolution. This shows that two natural fragments of LDQ-resolution are incomparable: Q-resolution, which allows universal reductions but no merging, and reductionless Q-resolution, which allows merging but no universal reductions. Finally, we develop semantically grounded lower bound techniques for fragments of LDQ-resolution, specifically tree-like LDQ-resolution and regular reductionless Q-resolution.

This research was partially supported by FWF grants P27721 and W1255-N23.

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Notes

  1. 1.

    At least without techniques like dependency learning [25].

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Authors and Affiliations

  1. Algorithms and Complexity Group, TU Wien, Vienna, Austria

    Tomáš Peitl, Friedrich Slivovsky & Stefan Szeider

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  1. Tomáš Peitl
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  2. Friedrich Slivovsky
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  3. Stefan Szeider
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Correspondence to Friedrich Slivovsky .

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  1. University of Lisbon, Lisbon, Portugal

    Mikoláš Janota

  2. University of Lisbon, Lisbon, Portugal

    Inês Lynce

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Peitl, T., Slivovsky, F., Szeider, S. (2019). Proof Complexity of Fragments of Long-Distance Q-Resolution. In: Janota, M., Lynce, I. (eds) Theory and Applications of Satisfiability Testing – SAT 2019. SAT 2019. Lecture Notes in Computer Science(), vol 11628. Springer, Cham. https://doi.org/10.1007/978-3-030-24258-9_23

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  • DOI: https://doi.org/10.1007/978-3-030-24258-9_23

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