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DOI: 10.4230/artifacts.24095
MichaelHartisch/EQuIPS

Authors: Michael Hartisch and Leroy Chew

Abstract
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Michael Hartisch, Leroy Chew. MichaelHartisch/EQuIPS (Software, Source Code). Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@misc{dagstuhl-artifact-24095,
   title = {{MichaelHartisch/EQuIPS}}, 
   author = {Hartisch, Michael and Chew, Leroy},
   note = {Software, DFG-534441421, FWF Project ESP197, swhId: \href{https://archive.softwareheritage.org/swh:1:dir:29f0d2d80f69841007fd3a315a47c2865669f3a6}{\texttt{swh:1:dir:29f0d2d80f69841007fd3a315a47c2865669f3a6}} (visited on 2025-08-08)},
   url = {https://github.com/MichaelHartisch/EQuIPS},
   doi = {10.4230/artifacts.24095},
}
Document
DOI: 10.4230/LIPIcs.CP.2025.12
An Expansion-Based Approach for Quantified Integer Programming

Authors: Michael Hartisch and Leroy Chew

Published in: LIPIcs, Volume 340, 31st International Conference on Principles and Practice of Constraint Programming (CP 2025)

Abstract
Quantified Integer Programming (QIP) bridges multiple domains by extending Quantified Boolean Formulas (QBF) to incorporate general integer variables and linear constraints while also generalizing Integer Programming through variable quantification. As a special case of Quantified Constraint Satisfaction Problems (QCSP), QIP provides a versatile framework for addressing complex decision-making scenarios. Additionally, the inclusion of a linear objective function enables QIP to effectively model multistage robust discrete linear optimization problems, making it a powerful tool for tackling uncertainty in optimization. While two primary solution paradigms exist for QBF - search-based and expansion-based approaches - only search-based methods have been explored for QIP and QCSP. We introduce an expansion-based approach for QIP using Counterexample-Guided Abstraction Refinement (CEGAR), adapting techniques from QBF. We extend this methodology to tackle multistage robust discrete optimization problems with linear constraints and further embed it in an optimization framework, enhancing its applicability. Our experimental results highlight the advantages of this approach, demonstrating superior performance over existing search-based solvers for QIP in specific instances. Furthermore, the ability to model problems using linear constraints enables notable performance gains over state-of-the-art expansion-based solvers for QBF.
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Michael Hartisch and Leroy Chew. An Expansion-Based Approach for Quantified Integer Programming. In 31st International Conference on Principles and Practice of Constraint Programming (CP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 340, pp. 12:1-12:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{hartisch_et_al:LIPIcs.CP.2025.12,
  author =	{Hartisch, Michael and Chew, Leroy},
  title =	{{An Expansion-Based Approach for Quantified Integer Programming}},
  booktitle =	{31st International Conference on Principles and Practice of Constraint Programming (CP 2025)},
  pages =	{12:1--12:26},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-380-5},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{340},
  editor =	{de la Banda, Maria Garcia},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2025.12},
  URN =		{urn:nbn:de:0030-drops-238736},
  doi =		{10.4230/LIPIcs.CP.2025.12},
  annote =	{Keywords: Quantified Integer Programming, Quantified Constraint Satisfaction, Robust Discrete Optimization, Expansion, CEGAR}
}
Document
DOI: 10.4230/LIPIcs.SAT.2025.11
Better Extension Variables in DQBF via Independence

Authors: Leroy Chew and Tomáš Peitl

Published in: LIPIcs, Volume 341, 28th International Conference on Theory and Applications of Satisfiability Testing (SAT 2025)

Abstract
We show that extension variables in (D)QBF can be generalised by conditioning on universal assignments. The benefit of this is that the dependency sets of such conditioned extension variables can be made smaller to allow easier refutations. This simple modification instantly solves many challenges in p-simulating the QBF expansion rule, which cannot be p-simulated in proof systems that have strategy extraction [Leroy Chew and Judith Clymo, 2020]. Simulating expansion is even more crucial in DQBF, where other methods are incomplete. In this paper we provide an overview of the strength of this new independent extension rule. We find that a new version of Extended Frege called IndExtFrege + ∀red can p-simulate a multitude of difficult QBF and DQBF techniques, even techniques that are difficult to approach with eFrege + ∀red. We show five p-simulations, that IndExtFrege + ∀red p-simulates QRAT, DQBF-IR-calc, IR(𝒟^rrs)-calc, Fork-Resolution and DQRAT which together underpin most DQBF solving and preprocessing techniques. The p-simulations work despite these systems using complicated rules and our new extension rule being relatively simple. Moreover, unlike recent p-simulations by eFrege + ∀red we can simulate the proof rules line by line, which allows us to mix QBF rules more easily with other inference steps.
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Leroy Chew and Tomáš Peitl. Better Extension Variables in DQBF via Independence. In 28th International Conference on Theory and Applications of Satisfiability Testing (SAT 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 341, pp. 11:1-11:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chew_et_al:LIPIcs.SAT.2025.11,
  author =	{Chew, Leroy and Peitl, Tom\'{a}\v{s}},
  title =	{{Better Extension Variables in DQBF via Independence}},
  booktitle =	{28th International Conference on Theory and Applications of Satisfiability Testing (SAT 2025)},
  pages =	{11:1--11:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-381-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{341},
  editor =	{Berg, Jeremias and Nordstr\"{o}m, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2025.11},
  URN =		{urn:nbn:de:0030-drops-237453},
  doi =		{10.4230/LIPIcs.SAT.2025.11},
  annote =	{Keywords: DQBF, QBF, Proof Systems, Dependency Schemes, RAT, Extended Frege, Skolem functions}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2024.18
Circuits, Proofs and Propositional Model Counting

Authors: Sravanthi Chede, Leroy Chew, and Anil Shukla

Published in: LIPIcs, Volume 323, 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024)

Abstract
In this paper we present a new proof system framework CLIP (Circuit Linear Induction Proposition) for propositional model counting (#SAT). A CLIP proof firstly involves a Boolean circuit, calculating the cumulative function (or running count) of models counted up to a point, and secondly a propositional proof arguing for the correctness of the circuit. This concept is remarkably simple and CLIP is modular so it allows us to use existing checking formats from propositional logic, especially strong proof systems. CLIP has polynomial-size proofs for XOR-pairs which are known to require exponential-size proofs in MICE [Fichte et al., 2022]. The existence of a strong proof system that can tackle these hard problems was posed as an open problem in Beyersdorff et al. [Olaf Beyersdorff et al., 2023]. In addition, CLIP systems can p-simulate all other existing #SAT proofs systems (KCPS(#SAT) [Florent Capelli, 2019], CPOG [Bryant et al., 2023], MICE). Furthermore, CLIP has a theoretical advantage over the other #SAT proof systems in the sense that CLIP only has lower bounds from its propositional proof system or if 𝖯^#𝖯 is not contained in P/poly, which is a major open problem in circuit complexity. CLIP uses unrestricted circuits in its proof as compared to restricted structures used by the existing #SAT proof systems. In this way, CLIP avoids hardness or limitations due to circuit restrictions.
Cite as

Sravanthi Chede, Leroy Chew, and Anil Shukla. Circuits, Proofs and Propositional Model Counting. In 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 323, pp. 18:1-18:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chede_et_al:LIPIcs.FSTTCS.2024.18,
  author =	{Chede, Sravanthi and Chew, Leroy and Shukla, Anil},
  title =	{{Circuits, Proofs and Propositional Model Counting}},
  booktitle =	{44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024)},
  pages =	{18:1--18:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-355-3},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{323},
  editor =	{Barman, Siddharth and Lasota, S{\l}awomir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2024.18},
  URN =		{urn:nbn:de:0030-drops-222079},
  doi =		{10.4230/LIPIcs.FSTTCS.2024.18},
  annote =	{Keywords: Propositional model counting, Boolean circuits, #SAT, Proof Systems, Certified Partition Operation Graph (CPOG)}
}
Document
DOI: 10.4230/LIPIcs.SAT.2022.10
Relating Existing Powerful Proof Systems for QBF

Authors: Leroy Chew and Marijn J. H. Heule

Published in: LIPIcs, Volume 236, 25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022)

Abstract
We advance the theory of QBF proof systems by showing the first simulation of the universal checking format QRAT by a theory-friendly system. We show that the sequent system G fully p-simulates QRAT, including the Extended Universal Reduction (EUR) rule which was recently used to show QRAT does not have strategy extraction. Because EUR heavily uses resolution paths our technique also brings resolution path dependency and sequent systems closer together. While we do not recommend G for practical applications this work can potentially show what features are needed for a new QBF checking format stronger than QRAT.
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Leroy Chew and Marijn J. H. Heule. Relating Existing Powerful Proof Systems for QBF. In 25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 236, pp. 10:1-10:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chew_et_al:LIPIcs.SAT.2022.10,
  author =	{Chew, Leroy and Heule, Marijn J. H.},
  title =	{{Relating Existing Powerful Proof Systems for QBF}},
  booktitle =	{25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022)},
  pages =	{10:1--10:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-242-6},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{236},
  editor =	{Meel, Kuldeep S. and Strichman, Ofer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2022.10},
  URN =		{urn:nbn:de:0030-drops-166845},
  doi =		{10.4230/LIPIcs.SAT.2022.10},
  annote =	{Keywords: QBF, Proof Complexity, Verification, Strategy Extraction, Sequent Calculus}
}
Document
DOI: 10.4230/LIPIcs.STACS.2022.22
Towards Uniform Certification in QBF

Authors: Leroy Chew and Friedrich Slivovsky

Published in: LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Abstract
We pioneer a new technique that allows us to prove a multitude of previously open simulations in QBF proof complexity. In particular, we show that extended QBF Frege p-simulates clausal proof systems such as IR-Calculus, IRM-Calculus, Long-Distance Q-Resolution, and Merge Resolution. These results are obtained by taking a technique of Beyersdorff et al. (JACM 2020) that turns strategy extraction into simulation and combining it with new local strategy extraction arguments. This approach leads to simulations that are carried out mainly in propositional logic, with minimal use of the QBF rules. Our proofs therefore provide a new, largely propositional interpretation of the simulated systems. We argue that these results strengthen the case for uniform certification in QBF solving, since many QBF proof systems now fall into place underneath extended QBF Frege.
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Leroy Chew and Friedrich Slivovsky. Towards Uniform Certification in QBF. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 22:1-22:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chew_et_al:LIPIcs.STACS.2022.22,
  author =	{Chew, Leroy and Slivovsky, Friedrich},
  title =	{{Towards Uniform Certification in QBF}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{22:1--22:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.22},
  URN =		{urn:nbn:de:0030-drops-158326},
  doi =		{10.4230/LIPIcs.STACS.2022.22},
  annote =	{Keywords: QBF, Proof Complexity, Verification, Frege, Extended Frege, Strategy Extraction}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2016.40
Understanding Cutting Planes for QBFs

Authors: Olaf Beyersdorff, Leroy Chew, Meena Mahajan, and Anil Shukla

Published in: LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

Abstract
We define a cutting planes system CP+ForallRed for quantified Boolean formulas (QBF) and analyse the proof-theoretic strength of this new calculus. While in the propositional case, Cutting Planes is of intermediate strength between resolution and Frege, our findings here show that the situation in QBF is slightly more complex: while CP+ForallRed is again weaker than QBF Frege and stronger than the CDCL-based QBF resolution systems Q-Res and QU-Res, it turns out to be incomparable to even the weakest expansion-based QBF resolution system ForallExp+Res. Technically, our results establish the effectiveness of two lower bound techniques for CP+ForallRed: via strategy extraction and via monotone feasible interpolation.
Cite as

Olaf Beyersdorff, Leroy Chew, Meena Mahajan, and Anil Shukla. Understanding Cutting Planes for QBFs. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 40:1-40:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{beyersdorff_et_al:LIPIcs.FSTTCS.2016.40,
  author =	{Beyersdorff, Olaf and Chew, Leroy and Mahajan, Meena and Shukla, Anil},
  title =	{{Understanding Cutting Planes for QBFs}},
  booktitle =	{36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)},
  pages =	{40:1--40:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-027-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{65},
  editor =	{Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.40},
  URN =		{urn:nbn:de:0030-drops-68758},
  doi =		{10.4230/LIPIcs.FSTTCS.2016.40},
  annote =	{Keywords: proof complexity, QBF, cutting planes, resolution, simulations}
}
Document
DOI: 10.4230/LIPIcs.STACS.2016.15
Are Short Proofs Narrow? QBF Resolution is not Simple

Authors: Olaf Beyersdorff, Leroy Chew, Meena Mahajan, and Anil Shukla

Published in: LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

Abstract
The groundbreaking paper 'Short proofs are narrow - resolution made simple' by Ben-Sasson and Wigderson (J. ACM 2001) introduces what is today arguably the main technique to obtain resolution lower bounds: to show a lower bound for the width of proofs. Another important measure for resolution is space, and in their fundamental work, Atserias and Dalmau (J. Comput. Syst. Sci. 2008) show that space lower bounds again can be obtained via width lower bounds. Here we assess whether similar techniques are effective for resolution calculi for quantified Boolean formulas (QBF). A mixed picture emerges. Our main results show that both the relations between size and width as well as between space and width drastically fail in Q-resolution, even in its weaker tree-like version. On the other hand, we obtain positive results for the expansion-based resolution systems Forall-Exp+Res and IR-calc, however only in the weak tree-like models. Technically, our negative results rely on showing width lower bounds together with simultaneous upper bounds for size and space. For our positive results we exhibit space and width-preserving simulations between QBF resolution calculi.
Cite as

Olaf Beyersdorff, Leroy Chew, Meena Mahajan, and Anil Shukla. Are Short Proofs Narrow? QBF Resolution is not Simple. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 15:1-15:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{beyersdorff_et_al:LIPIcs.STACS.2016.15,
  author =	{Beyersdorff, Olaf and Chew, Leroy and Mahajan, Meena and Shukla, Anil},
  title =	{{Are Short Proofs Narrow? QBF Resolution is not Simple}},
  booktitle =	{33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)},
  pages =	{15:1--15:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-001-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{47},
  editor =	{Ollinger, Nicolas and Vollmer, Heribert},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.15},
  URN =		{urn:nbn:de:0030-drops-57164},
  doi =		{10.4230/LIPIcs.STACS.2016.15},
  annote =	{Keywords: proof complexity, QBF, resolution, lower bound techniques, simulations}
}
Document
DOI: 10.4230/LIPIcs.STACS.2015.76
Proof Complexity of Resolution-based QBF Calculi

Authors: Olaf Beyersdorff, Leroy Chew, and Mikolás Janota

Published in: LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

Abstract
Proof systems for quantified Boolean formulas (QBFs) provide a theoretical underpinning for the performance of important QBF solvers. However, the proof complexity of these proof systems is currently not well understood and in particular lower bound techniques are missing. In this paper we exhibit a new and elegant proof technique for showing lower bounds in QBF proof systems based on strategy extraction. This technique provides a direct transfer of circuit lower bounds to lengths of proofs lower bounds. We use our method to show the hardness of a natural class of parity formulas for Q-resolution and universal Q-resolution. Variants of the formulas are hard for even stronger systems as long-distance Q-resolution and extensions. With a completely different lower bound argument we show the hardness of the prominent formulas of Kleine Büning et al. [34] for the strong expansion-based calculus IR-calc. Our lower bounds imply new exponential separations between two different types of resolution-based QBF calculi: proof systems for CDCL-based solvers (Q-resolution, long-distance Q-resolution) and proof systems for expansion-based solvers (forallExp+Res and its generalizations IR-calc and IRM-calc). The relations between proof systems from the two different classes were not known before.
Cite as

Olaf Beyersdorff, Leroy Chew, and Mikolás Janota. Proof Complexity of Resolution-based QBF Calculi. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 76-89, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{beyersdorff_et_al:LIPIcs.STACS.2015.76,
  author =	{Beyersdorff, Olaf and Chew, Leroy and Janota, Mikol\'{a}s},
  title =	{{Proof Complexity of Resolution-based QBF Calculi}},
  booktitle =	{32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)},
  pages =	{76--89},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-78-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{30},
  editor =	{Mayr, Ernst W. and Ollinger, Nicolas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.76},
  URN =		{urn:nbn:de:0030-drops-49057},
  doi =		{10.4230/LIPIcs.STACS.2015.76},
  annote =	{Keywords: proof complexity, QBF, lower bound techniques, separations}
}
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