6 Search Results for "Spachmann, Luc Nicolas"

Document
DOI: 10.4230/LIPIcs.SAT.2026.3
Proof Systems for QBF Synthesis: Extracting Skolem and Herbrand Functions

Authors: S. Akshay, Olaf Beyersdorff, Supratik Chakraborty, Lea Kasche, Meena Mahajan, and Luc Nicolas Spachmann

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

Abstract
Strategy extraction in QBF proof systems usually attempts to extract winning strategies from valid proofs. However, an alternative (and arguably more powerful) view is to extract Skolem/Herbrand functions, or equivalently synthesis of the game values at all intermediate points. In this paper, we investigate the existence and properties of such proof systems from which one can extract Skolem and Herbrand functions. We propose such a proof system for QBF, which we show is sound and complete, and from which extraction of Skolem/Herbrand functions can be performed, and game values computed, in polynomial time. We also show that this system is optimal among all proof systems that allow efficient extraction of Skolem/Herbrand functions. We provide conditional lower bound results for our new proof system and compare it to several existing/standard proof systems for QBF that have been studied in the literature, showing interesting orthogonality results. Finally, we provide a compilation algorithm that takes an arbitrary QBF and synthesizes a proof in our system, from which Skolem and Herbrand functions can be easily computed.
Cite as

S. Akshay, Olaf Beyersdorff, Supratik Chakraborty, Lea Kasche, Meena Mahajan, and Luc Nicolas Spachmann. Proof Systems for QBF Synthesis: Extracting Skolem and Herbrand Functions. In 29th International Conference on Theory and Applications of Satisfiability Testing (SAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 377, pp. 3:1-3:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{akshay_et_al:LIPIcs.SAT.2026.3,
  author =	{Akshay, S. and Beyersdorff, Olaf and Chakraborty, Supratik and Kasche, Lea and Mahajan, Meena and Spachmann, Luc Nicolas},
  title =	{{Proof Systems for QBF Synthesis: Extracting Skolem and Herbrand Functions}},
  booktitle =	{29th International Conference on Theory and Applications of Satisfiability Testing (SAT 2026)},
  pages =	{3:1--3:20},
  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.2026.3},
  URN =		{urn:nbn:de:0030-drops-263096},
  doi =		{10.4230/LIPIcs.SAT.2026.3},
  annote =	{Keywords: Quantified Boolean Formulas, Skolem and Herbrand functions, Automated synthesis, Knowledge compilation, Proof systems and complexity}
}
Document
DOI: 10.4230/LIPIcs.SAT.2025.5
Semi-Algebraic Proof Systems for QBF

Authors: Olaf Beyersdorff, Ilario Bonacina, Kaspar Kasche, Meena Mahajan, and Luc Nicolas Spachmann

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

Abstract
We introduce new semi-algebraic proof systems for Quantified Boolean Formulas (QBF) analogous to the propositional systems Nullstellensatz, Sherali-Adams and Sum-of-Squares. We transfer to this setting techniques both from the QBF literature (strategy extraction) and from propositional proof complexity (size-degree relations and pseudo-expectation). We obtain a number of strong QBF lower bounds and separations between these systems, even when disregarding propositional hardness.
Cite as

Olaf Beyersdorff, Ilario Bonacina, Kaspar Kasche, Meena Mahajan, and Luc Nicolas Spachmann. Semi-Algebraic Proof Systems for QBF. In 28th International Conference on Theory and Applications of Satisfiability Testing (SAT 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 341, pp. 5:1-5:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{beyersdorff_et_al:LIPIcs.SAT.2025.5,
  author =	{Beyersdorff, Olaf and Bonacina, Ilario and Kasche, Kaspar and Mahajan, Meena and Spachmann, Luc Nicolas},
  title =	{{Semi-Algebraic Proof Systems for QBF}},
  booktitle =	{28th International Conference on Theory and Applications of Satisfiability Testing (SAT 2025)},
  pages =	{5:1--5:19},
  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.2025.5},
  URN =		{urn:nbn:de:0030-drops-237394},
  doi =		{10.4230/LIPIcs.SAT.2025.5},
  annote =	{Keywords: QBF, Proof Complexity, Sums-of-Squares, Nullstellensatz, Sherali-Adams, Semi-Algebraic Proof Systems}
}
Document
DOI: 10.4230/LIPIcs.SAT.2025.8
Certifying Projected Knowledge Compilation

Authors: Randal E. Bryant, Yong Kiam Tan, and Marijn J. H. Heule

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

Abstract
Knowledge compilers convert Boolean formulas, given in conjunctive normal form (CNF), into representations that enable efficient evaluation of unweighted and weighted model counts, as well as a variety of other useful properties. With projected knowledge compilation, the generated representation describes the restriction of the formula to a designated set of data variables, with the remaining ones eliminated by existential quantification. Projected knowledge compilation has applications in a variety of domains, including formal verification and synthesis. This paper describes a formally verified proof framework for certifying the output of a projected knowledge compiler. It builds on an earlier clausal proof framework for certifying the output of a standard knowledge compiler. Extending the framework to projected compilation requires a method to represent Skolem assignments, describing how the quantified variables can be assigned, given an assignment for the data variables. We do so by extending the representation generated by the knowledge compiler to also encode Skolem assignments. We also refine the earlier framework, moving beyond purely clausal proofs to enable scaling certification to larger formulas. We present experimental results obtained by making small modifications to the D4 projected knowledge compiler and extensions of our earlier proof generator. We detail a soundness argument stating that a compiler output that passes our certifier is logically equivalent to the quantified input formula; the soundness argument has been formally validated using the HOL4 proof assistant. The checker also ensures that the compiler output satisfies the properties required for efficient unweighted and weighted model counting. We have developed two proof checkers for the certification framework: one written in C and designed for high performance and one written in CakeML and formally verified in HOL4.
Cite as

Randal E. Bryant, Yong Kiam Tan, and Marijn J. H. Heule. Certifying Projected Knowledge Compilation. In 28th International Conference on Theory and Applications of Satisfiability Testing (SAT 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 341, pp. 8:1-8:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bryant_et_al:LIPIcs.SAT.2025.8,
  author =	{Bryant, Randal E. and Tan, Yong Kiam and Heule, Marijn J. H.},
  title =	{{Certifying Projected Knowledge Compilation}},
  booktitle =	{28th International Conference on Theory and Applications of Satisfiability Testing (SAT 2025)},
  pages =	{8:1--8:22},
  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.2025.8},
  URN =		{urn:nbn:de:0030-drops-237422},
  doi =		{10.4230/LIPIcs.SAT.2025.8},
  annote =	{Keywords: Knowledge Compilation, Propositional model counting, Proof checking}
}
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.MFCS.2024.27
Polynomial Calculus for Quantified Boolean Logic: Lower Bounds Through Circuits and Degree

Authors: Olaf Beyersdorff, Tim Hoffmann, Kaspar Kasche, and Luc Nicolas Spachmann

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
We initiate an in-depth proof-complexity analysis of polynomial calculus (𝒬-PC) for Quantified Boolean Formulas (QBF). In the course of this we establish a tight proof-size characterisation of 𝒬-PC in terms of a suitable circuit model (polynomial decision lists). Using this correspondence we show a size-degree relation for 𝒬-PC, similar in spirit, yet different from the classic size-degree formula for propositional PC by Impagliazzo, Pudlák and Sgall (1999). We use the circuit characterisation together with the size-degree relation to obtain various new lower bounds on proof size in 𝒬-PC. This leads to incomparability results for 𝒬-PC systems over different fields.
Cite as

Olaf Beyersdorff, Tim Hoffmann, Kaspar Kasche, and Luc Nicolas Spachmann. Polynomial Calculus for Quantified Boolean Logic: Lower Bounds Through Circuits and Degree. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 27:1-27:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{beyersdorff_et_al:LIPIcs.MFCS.2024.27,
  author =	{Beyersdorff, Olaf and Hoffmann, Tim and Kasche, Kaspar and Spachmann, Luc Nicolas},
  title =	{{Polynomial Calculus for Quantified Boolean Logic: Lower Bounds Through Circuits and Degree}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{27:1--27:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.27},
  URN =		{urn:nbn:de:0030-drops-205834},
  doi =		{10.4230/LIPIcs.MFCS.2024.27},
  annote =	{Keywords: proof complexity, QBF, polynomial calculus, circuits, lower bounds}
}
Document
DOI: 10.4230/LIPIcs.SAT.2023.2
Proof Complexity of Propositional Model Counting

Authors: Olaf Beyersdorff, Tim Hoffmann, and Luc Nicolas Spachmann

Published in: LIPIcs, Volume 271, 26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023)

Abstract
Recently, the proof system MICE for the model counting problem #SAT was introduced by Fichte, Hecher and Roland (SAT'22). As demonstrated by Fichte et al., the system MICE can be used for proof logging for state-of-the-art #SAT solvers. We perform a proof-complexity study of MICE. For this we first simplify the rules of MICE and obtain a calculus MICE' that is polynomially equivalent to MICE. Our main result establishes an exponential lower bound for the number of proof steps in MICE' (and hence also in MICE) for a specific family of CNFs.
Cite as

Olaf Beyersdorff, Tim Hoffmann, and Luc Nicolas Spachmann. Proof Complexity of Propositional Model Counting. In 26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 271, pp. 2:1-2:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{beyersdorff_et_al:LIPIcs.SAT.2023.2,
  author =	{Beyersdorff, Olaf and Hoffmann, Tim and Spachmann, Luc Nicolas},
  title =	{{Proof Complexity of Propositional Model Counting}},
  booktitle =	{26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023)},
  pages =	{2:1--2:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-286-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{271},
  editor =	{Mahajan, Meena and Slivovsky, Friedrich},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2023.2},
  URN =		{urn:nbn:de:0030-drops-184647},
  doi =		{10.4230/LIPIcs.SAT.2023.2},
  annote =	{Keywords: model counting, #SAT, proof complexity, proof systems, lower bounds}
}
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