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**Published in:** LIPIcs, Volume 305, 27th International Conference on Theory and Applications of Satisfiability Testing (SAT 2024)

It has become standard that, when a SAT solver decides that a CNF Γ is unsatisfiable, it produces a certificate of unsatisfiability in the form of a refutation of Γ in some proof system. The system typically used is DRAT, which is equivalent to extended resolution (ER) - for example, until this year DRAT refutations were required in the annual SAT competition. Recently [Bogaerts et al. 2023] introduced a new proof system, associated with the tool VeriPB, which is at least as strong as DRAT and is further able to handle certain symmetry-breaking techniques. We show that this system simulates the proof system G₁, which allows limited reasoning with QBFs and forms the first level above ER in a natural hierarchy of proof systems. This hierarchy is not known to be strict, but nevertheless this is evidence that the system of [Bogaerts et al. 2023] is plausibly strictly stronger than ER and DRAT. In the other direction, we show that symmetry-breaking for a single symmetry can be handled inside ER.

Leszek Aleksander Kołodziejczyk and Neil Thapen. The Strength of the Dominance Rule. In 27th International Conference on Theory and Applications of Satisfiability Testing (SAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 305, pp. 20:1-20:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kolodziejczyk_et_al:LIPIcs.SAT.2024.20, author = {Ko{\l}odziejczyk, Leszek Aleksander and Thapen, Neil}, title = {{The Strength of the Dominance Rule}}, booktitle = {27th International Conference on Theory and Applications of Satisfiability Testing (SAT 2024)}, pages = {20:1--20:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-334-8}, ISSN = {1868-8969}, year = {2024}, volume = {305}, editor = {Chakraborty, Supratik and Jiang, Jie-Hong Roland}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2024.20}, URN = {urn:nbn:de:0030-drops-205421}, doi = {10.4230/LIPIcs.SAT.2024.20}, annote = {Keywords: proof complexity, DRAT, symmetry breaking, dominance} }

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**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

The complexity class CLS was introduced by Daskalakis and Papadimitriou (SODA 2010) to capture the computational complexity of important TFNP problems solvable by local search over continuous domains and, thus, lying in both PLS and PPAD. It was later shown that, e.g., the problem of computing fixed points guaranteed by Banach’s fixed point theorem is CLS-complete by Daskalakis et al. (STOC 2018). Recently, Fearnley et al. (J. ACM 2023) disproved the plausible conjecture of Daskalakis and Papadimitriou that CLS is a proper subclass of PLS∩PPAD by proving that CLS = PLS∩PPAD.
To study the possibility of other collapses in TFNP, we connect classes formed as the intersection of existing subclasses of TFNP with the phenomenon of feasible disjunction in propositional proof complexity; where a proof system has the feasible disjunction property if, whenever a disjunction F ∨ G has a small proof, and F and G have no variables in common, then either F or G has a small proof. Based on some known and some new results about feasible disjunction, we separate the classes formed by intersecting the classical subclasses PLS, PPA, PPAD, PPADS, PPP and CLS. We also give the first examples of proof systems which have the feasible interpolation property, but not the feasible disjunction property.

Pavel Hubáček, Erfan Khaniki, and Neil Thapen. TFNP Intersections Through the Lens of Feasible Disjunction. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 63:1-63:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hubacek_et_al:LIPIcs.ITCS.2024.63, author = {Hub\'{a}\v{c}ek, Pavel and Khaniki, Erfan and Thapen, Neil}, title = {{TFNP Intersections Through the Lens of Feasible Disjunction}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {63:1--63:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.63}, URN = {urn:nbn:de:0030-drops-195917}, doi = {10.4230/LIPIcs.ITCS.2024.63}, annote = {Keywords: TFNP, feasible disjunction, proof complexity, TFNP intersection classes} }

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**Published in:** LIPIcs, Volume 79, 32nd Computational Complexity Conference (CCC 2017)

We study the random resolution refutation system definedin [Buss et al. 2014]. This attempts to capture the notion of a resolution refutation that may make mistakes but is correct most of the time. By proving the equivalence of several different definitions, we show that this concept is robust. On the other hand, if P does not equal NP, then random resolution cannot be polynomially simulated by any proof system in which correctness of proofs is checkable in polynomial time.
We prove several upper and lower bounds on the width and size of random resolution refutations of explicit and random unsatisfiable CNF formulas. Our main result is a separation between polylogarithmic width random resolution and quasipolynomial size resolution, which solves the problem stated in [Buss et al. 2014]. We also prove exponential size lower bounds on random resolution refutations of the pigeonhole principle CNFs, and of a family of CNFs which have polynomial size refutations in constant depth Frege.

Pavel Pudlák and Neil Thapen. Random Resolution Refutations. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 1:1-1:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{pudlak_et_al:LIPIcs.CCC.2017.1, author = {Pudl\'{a}k, Pavel and Thapen, Neil}, title = {{Random Resolution Refutations}}, booktitle = {32nd Computational Complexity Conference (CCC 2017)}, pages = {1:1--1:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-040-8}, ISSN = {1868-8969}, year = {2017}, volume = {79}, editor = {O'Donnell, Ryan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2017.1}, URN = {urn:nbn:de:0030-drops-75235}, doi = {10.4230/LIPIcs.CCC.2017.1}, annote = {Keywords: proof complexity, random, resolution} }

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**Published in:** LIPIcs, Volume 33, 30th Conference on Computational Complexity (CCC 2015)

We study the space complexity of the cutting planes proof system, in which the lines in a proof are integral linear inequalities. We measure the space used by a refutation as the number of linear inequalities that need to be kept on a blackboard while verifying it. We show that any unsatisfiable set of linear inequalities has a cutting planes refutation in space five. This is in contrast to the weaker resolution proof system, for which the analogous space measure has been well-studied and many optimal linear lower bounds are known.
Motivated by this result we consider a natural restriction of cutting planes, in which all coefficients have size bounded by a constant. We show that there is a CNF which requires super-constant space to refute in this system. The system nevertheless already has an exponential speed-up over resolution with respect to size, and we additionally show that it is stronger than resolution with respect to space, by constructing constant-space cutting planes proofs, with coefficients bounded by two, of the pigeonhole principle.
We also consider variable instance space for cutting planes, where we count the number of instances of variables on the blackboard, and total space, where we count the total number of symbols.

Nicola Galesi, Pavel Pudlák, and Neil Thapen. The Space Complexity of Cutting Planes Refutations. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 433-447, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{galesi_et_al:LIPIcs.CCC.2015.433, author = {Galesi, Nicola and Pudl\'{a}k, Pavel and Thapen, Neil}, title = {{The Space Complexity of Cutting Planes Refutations}}, booktitle = {30th Conference on Computational Complexity (CCC 2015)}, pages = {433--447}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-81-1}, ISSN = {1868-8969}, year = {2015}, volume = {33}, editor = {Zuckerman, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.433}, URN = {urn:nbn:de:0030-drops-50551}, doi = {10.4230/LIPIcs.CCC.2015.433}, annote = {Keywords: Proof Complexity, Cutting Planes, Space Complexity} }

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