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DOI: 10.4230/LIPIcs.SAT.2026.32

On Proof Systems for #QBF (Short Paper)

Authors: Sravanthi Chede, Leroy Chew, Vaibhav Krishan, and Anil Shukla

Published in: LIPIcs, Volume 377, 29th International Conference on Theory and Applications of Satisfiability Testing (SAT 2026)

Abstract

For a quantified Boolean formula (QBF), the problem of computing the number of winning strategies is known as the #QBF problem. This problem is considered harder than the analogous #SAT problem. Recently, important proof systems for QBFs and #SAT have been studied. By extending the ideas from both fields, we show that it is possible to design proof systems for #QBF. Such proof systems are important not only for advancing the theory of #QBF but also for certifying and designing better #QBF solvers, an area that is still in its early stages. In this paper, we explore #QBF proof systems to count the number of Skolem functions. In addition to a naive system, we study #QBF systems based on the ∀-expansion rule of QBFs. We observe that these systems have inherent structural weaknesses that lead to lower bounds. As an alternative, we propose a #QBF proof system that we call Q-MICE, which consists of sound inference rules for computing and certifying the #QBF solution, similar to the line-based #SAT proof system MICE. To demonstrate the strength of Q-MICE, we present various upper bounds, such as the quantified version of the propositional XOR-PAIRS formula, which are known to be hard for MICE. Consequently, we also separate Q-MICE from ∀-expansion based #QBF proof systems.

Cite as

Sravanthi Chede, Leroy Chew, Vaibhav Krishan, and Anil Shukla. On Proof Systems for #QBF (Short Paper). In 29th International Conference on Theory and Applications of Satisfiability Testing (SAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 377, pp. 32:1-32:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{chede_et_al:LIPIcs.SAT.2026.32,
  author =	{Chede, Sravanthi and Chew, Leroy and Krishan, Vaibhav and Shukla, Anil},
  title =	{{On Proof Systems for #QBF}},
  booktitle =	{29th International Conference on Theory and Applications of Satisfiability Testing (SAT 2026)},
  pages =	{32:1--32:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-431-4},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{377},
  editor =	{Ignatiev, Alexey and Szeider, Stefan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2026.32},
  URN =		{urn:nbn:de:0030-drops-263380},
  doi =		{10.4230/LIPIcs.SAT.2026.32},
  annote =	{Keywords: QBF, Model Counting, Proof Systems, #QBF}
}

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.STACS.2023.21

Extending Merge Resolution to a Family of QBF-Proof Systems

Authors: Sravanthi Chede and Anil Shukla

Published in: LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

Abstract

Merge Resolution (MRes [Olaf Beyersdorff et al., 2021]) is a recently introduced proof system for false QBFs. Unlike other known QBF proof systems, it builds winning strategies for the universal player (countermodels) within the proofs as merge maps. Merge maps are deterministic branching programs in which isomorphism checking is efficient, as a result MRes is a polynomial time verifiable proof system. In this paper, we introduce a family of proof systems MRes-ℛ in which the information of countermodels are stored in any pre-fixed complete representation ℛ. Hence, corresponding to each possible complete representation ℛ, we have a sound and refutationally complete QBF-proof system in MRes-ℛ. To handle these arbitrary representations, we introduce consistency checking rules in MRes-ℛ instead of the isomorphism checking in MRes. As a result these proof systems are not polynomial time verifiable (Non-P). Consequently, the paper shows that using merge maps is too restrictive and with a slight change in rules, it can be replaced with arbitrary representations leading to several interesting proof systems. We relate these new systems with the implicit proof system from the algorithm in [Joshua Blinkhorn et al., 2021], which was designed to solve DQBFs (Dependency QBFs) using clause-strategy pairs like MRes. We use the OBDD (Ordered Binary Decision Diagrams) representation suggested in [Joshua Blinkhorn et al., 2021] and deduce that "Ordered" versions of the proof systems in MRes-ℛ are indeed polynomial time verifiable. On the lower bound side, we lift the lower bound result of regular MRes ([Olaf Beyersdorff et al., 2020]) by showing that the completion principle formulas (CR_n) from [Mikolás Janota and João Marques-Silva, 2015] which are shown to be hard for regular MRes in [Olaf Beyersdorff et al., 2020], are also hard for any regular proof system in MRes-ℛ. Thereby, the paper lifts the lower bound of regular MRes to an entire class of proof systems, which use various complete representations, including those undiscovered, instead of only merge maps. Thereby proving that the hardness of CR_n formulas is intact even after changing the weak isomorphism checking in MRes to the stronger consistency checking in MRes-ℛ.

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Sravanthi Chede and Anil Shukla. Extending Merge Resolution to a Family of QBF-Proof Systems. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 21:1-21:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chede_et_al:LIPIcs.STACS.2023.21,
  author =	{Chede, Sravanthi and Shukla, Anil},
  title =	{{Extending Merge Resolution to a Family of QBF-Proof Systems}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{21:1--21:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.21},
  URN =		{urn:nbn:de:0030-drops-176737},
  doi =		{10.4230/LIPIcs.STACS.2023.21},
  annote =	{Keywords: Proof complexity, QBFs, Merge Resolution, Simulation, Lower Bound}
}

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Documents authored by Chede, Sravanthi

On Proof Systems for #QBF (Short Paper)

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Circuits, Proofs and Propositional Model Counting

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Extending Merge Resolution to a Family of QBF-Proof Systems

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