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**Published in:** LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

The Tensor Isomorphism problem (TI) has recently emerged as having connections to multiple areas of research within complexity and beyond, but the current best upper bound is essentially the brute force algorithm. Being an algebraic problem, TI (or rather, proving that two tensors are non-isomorphic) lends itself very naturally to algebraic and semi-algebraic proof systems, such as the Polynomial Calculus (PC) and Sum of Squares (SoS). For its combinatorial cousin Graph Isomorphism, essentially optimal lower bounds are known for approaches based on PC and SoS (Berkholz & Grohe, SODA '17). Our main results are an Ω(n) lower bound on PC degree or SoS degree for Tensor Isomorphism, and a nontrivial upper bound for testing isomorphism of tensors of bounded rank.
We also show that PC cannot perform basic linear algebra in sub-linear degree, such as comparing the rank of two matrices (which is essentially the same as 2-TI), or deriving BA = I from AB = I. As linear algebra is a key tool for understanding tensors, we introduce a strictly stronger proof system, PC+Inv, which allows as derivation rules all substitution instances of the implication AB = I → BA = I. We conjecture that even PC+Inv cannot solve TI in polynomial time either, but leave open getting lower bounds on PC+Inv for any system of equations, let alone those for TI. We also highlight many other open questions about proof complexity approaches to TI.

Nicola Galesi, Joshua A. Grochow, Toniann Pitassi, and Adrian She. On the Algebraic Proof Complexity of Tensor Isomorphism. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 4:1-4:40, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{galesi_et_al:LIPIcs.CCC.2023.4, author = {Galesi, Nicola and Grochow, Joshua A. and Pitassi, Toniann and She, Adrian}, title = {{On the Algebraic Proof Complexity of Tensor Isomorphism}}, booktitle = {38th Computational Complexity Conference (CCC 2023)}, pages = {4:1--4:40}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-282-2}, ISSN = {1868-8969}, year = {2023}, volume = {264}, editor = {Ta-Shma, Amnon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.4}, URN = {urn:nbn:de:0030-drops-182748}, doi = {10.4230/LIPIcs.CCC.2023.4}, annote = {Keywords: Algebraic proof complexity, Tensor Isomorphism, Graph Isomorphism, Polynomial Calculus, Sum-of-Squares, reductions, lower bounds, proof complexity of linear algebra} }

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**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

Vanishing sums of roots of unity can be seen as a natural generalization of knapsack from Boolean variables to variables taking values over the roots of unity. We show that these sums are hard to prove for polynomial calculus and for sum-of-squares, both in terms of degree and size.

Ilario Bonacina, Nicola Galesi, and Massimo Lauria. On Vanishing Sums of Roots of Unity in Polynomial Calculus and Sum-Of-Squares. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bonacina_et_al:LIPIcs.MFCS.2022.23, author = {Bonacina, Ilario and Galesi, Nicola and Lauria, Massimo}, title = {{On Vanishing Sums of Roots of Unity in Polynomial Calculus and Sum-Of-Squares}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {23:1--23:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.23}, URN = {urn:nbn:de:0030-drops-168211}, doi = {10.4230/LIPIcs.MFCS.2022.23}, annote = {Keywords: polynomial calculus, sum-of-squares, roots of unity, knapsack} }

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

Stabbing Planes is a proof system introduced very recently which, informally speaking, extends the DPLL method by branching on integer linear inequalities instead of single variables. The techniques known so far to prove size and depth lower bounds for Stabbing Planes are generalizations of those used for the Cutting Planes proof system established via communication complexity arguments. Rank lower bounds for Cutting Planes are also obtained by geometric arguments called protection lemmas.
In this work we introduce two new geometric approaches to prove size/depth lower bounds in Stabbing Planes working for any formula: (1) the antichain method, relying on Sperner’s Theorem and (2) the covering method which uses results on essential coverings of the boolean cube by linear polynomials, which in turn relies on Alon’s combinatorial Nullenstellensatz.
We demonstrate their use on classes of combinatorial principles such as the Pigeonhole principle, the Tseitin contradictions and the Linear Ordering Principle. By the first method we prove almost linear size lower bounds and optimal logarithmic depth lower bounds for the Pigeonhole principle and analogous lower bounds for the Tseitin contradictions over the complete graph and for the Linear Ordering Principle. By the covering method we obtain a superlinear size lower bound and a logarithmic depth lower bound for Stabbing Planes proof of Tseitin contradictions over a grid graph.

Stefan Dantchev, Nicola Galesi, Abdul Ghani, and Barnaby Martin. Depth Lower Bounds in Stabbing Planes for Combinatorial Principles. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 24:1-24:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dantchev_et_al:LIPIcs.STACS.2022.24, author = {Dantchev, Stefan and Galesi, Nicola and Ghani, Abdul and Martin, Barnaby}, title = {{Depth Lower Bounds in Stabbing Planes for Combinatorial Principles}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {24:1--24:18}, 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.2022.24}, URN = {urn:nbn:de:0030-drops-158349}, doi = {10.4230/LIPIcs.STACS.2022.24}, annote = {Keywords: proof complexity, computational complexity, lower bounds, cutting planes, stabbing planes} }

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**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

We prove that there is a constant K such that Tseitin formulas for an undirected graph G requires proofs of size 2^{tw(G)^{Omega(1/d)}} in depth-d Frege systems for d<(K log n)/(log log n), where tw(G) is the treewidth of G. This extends Håstad recent lower bound for the grid graph to any graph. Furthermore, we prove tightness of our bound up to a multiplicative constant in the top exponent. Namely, we show that if a Tseitin formula for a graph G has size s, then for all large enough d, it has a depth-d Frege proof of size 2^{tw(G)^{O(1/d)}} poly(s). Through this result we settle the question posed by M. Alekhnovich and A. Razborov of showing that the class of Tseitin formulas is quasi-automatizable for resolution.

Nicola Galesi, Dmitry Itsykson, Artur Riazanov, and Anastasia Sofronova. Bounded-Depth Frege Complexity of Tseitin Formulas for All Graphs. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 49:1-49:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{galesi_et_al:LIPIcs.MFCS.2019.49, author = {Galesi, Nicola and Itsykson, Dmitry and Riazanov, Artur and Sofronova, Anastasia}, title = {{Bounded-Depth Frege Complexity of Tseitin Formulas for All Graphs}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {49:1--49:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.49}, URN = {urn:nbn:de:0030-drops-109932}, doi = {10.4230/LIPIcs.MFCS.2019.49}, annote = {Keywords: Tseitin formula, treewidth, AC0-Frege} }

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**Published in:** LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

Res(s) is an extension of Resolution working on s-DNFs. We prove tight n^{Omega(k)} lower bounds for the size of refutations of the binary version of the k-Clique Principle in Res(o(log log n)). Our result improves that of Lauria, Pudlák et al. [Massimo Lauria et al., 2017] who proved the lower bound for Res(1), i.e. Resolution. The exact complexity of the (unary) k-Clique Principle in Resolution is unknown. To prove the lower bound we do not use any form of the Switching Lemma [Nathan Segerlind et al., 2004], instead we apply a recursive argument specific for binary encodings. Since for the k-Clique and other principles lower bounds in Resolution for the unary version follow from lower bounds in Res(log n) for their binary version we start a systematic study of the complexity of proofs in Resolution-based systems for families of contradictions given in the binary encoding.
We go on to consider the binary version of the weak Pigeonhole Principle Bin-PHP^m_n for m>n. Using the the same recursive approach we prove the new result that for any delta>0, Bin-PHP^m_n requires proofs of size 2^{n^{1-delta}} in Res(s) for s=o(log^{1/2}n). Our lower bound is almost optimal since for m >= 2^{sqrt{n log n}} there are quasipolynomial size proofs of Bin-PHP^m_n in Res(log n).
Finally we propose a general theory in which to compare the complexity of refuting the binary and unary versions of large classes of combinatorial principles, namely those expressible as first order formulae in Pi_2-form and with no finite model.

Stefan Dantchev, Nicola Galesi, and Barnaby Martin. Resolution and the Binary Encoding of Combinatorial Principles. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 6:1-6:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dantchev_et_al:LIPIcs.CCC.2019.6, author = {Dantchev, Stefan and Galesi, Nicola and Martin, Barnaby}, title = {{Resolution and the Binary Encoding of Combinatorial Principles}}, booktitle = {34th Computational Complexity Conference (CCC 2019)}, pages = {6:1--6:25}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-116-0}, ISSN = {1868-8969}, year = {2019}, volume = {137}, editor = {Shpilka, Amir}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.6}, URN = {urn:nbn:de:0030-drops-108287}, doi = {10.4230/LIPIcs.CCC.2019.6}, annote = {Keywords: Proof complexity, k-DNF resolution, binary encodings, Clique and Pigeonhole principle} }

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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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**Published in:** Dagstuhl Reports, Volume 2, Issue 11 (2013)

This report documents the programme and outcomes of Dagstuhl Seminar 12471 "SAT Interactions". The seminar brought together researchers from different areas from theoretical computer science as well as the area of SAT solvers. A key objective of the seminar has been to initiate or consolidate discussions among the different groups for a fresh attack on one of the most important problems in theoretical computer science and mathematics.

Nadia Creignou, Nicola Galesi, Oliver Kullmann, and Heribert Vollmer. SAT Interactions (Dagstuhl Seminar 12471). In Dagstuhl Reports, Volume 2, Issue 11, pp. 87-101, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@Article{creignou_et_al:DagRep.2.11.87, author = {Creignou, Nadia and Galesi, Nicola and Kullmann, Oliver and Vollmer, Heribert}, title = {{SAT Interactions (Dagstuhl Seminar 12471)}}, pages = {87--101}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2013}, volume = {2}, number = {11}, editor = {Creignou, Nadia and Galesi, Nicola and Kullmann, Oliver and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.2.11.87}, URN = {urn:nbn:de:0030-drops-39786}, doi = {10.4230/DagRep.2.11.87}, annote = {Keywords: satisfiability problem, computational complexity, P-NP question, proof complexity, combinatorics, SAT-solvers, quantified Boolean formulas} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 10061, Circuits, Logic, and Games (2010)

Parameterized Resolution and, moreover, a general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider (FOCS'07). In that paper, Dantchev et al. show a complexity gap in tree-like Parameterized Resolution for propositional formulas arising from translations of first-order principles.
We broadly investigate Parameterized Resolution obtaining the following main results:
1) We introduce a purely combinatorial approach to obtain lower bounds to the proof size in tree-like Parameterized Resolution. For this we devise a new asymmetric Prover-Delayer game which characterizes proofs in (parameterized) tree-like Resolution.
By exhibiting good Delayer strategies we then show lower bounds for the pigeonhole principle as well as the order principle.
2) Interpreting a well-known FPT algorithm for vertex cover as a DPLL procedure for Parameterized Resolution, we devise a proof search algorithm for Parameterized Resolution and show that tree-like Parameterized Resolution allows short refutations of all parameterized contradictions given as bounded-width CNF's.
3) We answer a question posed by Dantchev, Martin, and Szeider showing that dag-like Parameterized Resolution is not fpt-bounded. We obtain this result by proving that the pigeonhole principle requires proofs of size $n^{Omega(k)}$ in dag-like Parameterized Resolution.
For this lower bound we use a different Prover-Delayer game which was developed for Resolution by Pudlák.

Olaf Beyersdorff, Nicola Galesi, and Massimo Lauria. Hardness of Parameterized Resolution. In Circuits, Logic, and Games. Dagstuhl Seminar Proceedings, Volume 10061, pp. 1-28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{beyersdorff_et_al:DagSemProc.10061.4, author = {Beyersdorff, Olaf and Galesi, Nicola and Lauria, Massimo}, title = {{Hardness of Parameterized Resolution}}, booktitle = {Circuits, Logic, and Games}, pages = {1--28}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2010}, volume = {10061}, editor = {Benjamin Rossman and Thomas Schwentick and Denis Th\'{e}rien and Heribert Vollmer}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.10061.4}, URN = {urn:nbn:de:0030-drops-25254}, doi = {10.4230/DagSemProc.10061.4}, annote = {Keywords: Proof complexity, parameterized complexity, Resolution, Prover-Delayer Games} }

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