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**Published in:** Dagstuhl Reports, Volume 12, Issue 1 (2022)

Finite and algorithmic model theory (FAMT) studies the expressive power of logical languages on finite structures or, more generally, structures that can be finitely presented. These are the structures that serve as input to computation, and for this reason the study of FAMT is intimately connected with computer science. Over the last four decades, the subject has developed through a close interaction between theoretical computer science and related areas of mathematics, including logic and combinatorics. This report documents the program and the outcomes of Dagstuhl Seminar 22051 "Finite and Algorithmic Model Theory".

Albert Atserias, Christoph Berkholz, Kousha Etessami, and Joanna Ochremiak. Finite and Algorithmic Model Theory (Dagstuhl Seminar 22051). In Dagstuhl Reports, Volume 12, Issue 1, pp. 101-118, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@Article{atserias_et_al:DagRep.12.1.101, author = {Atserias, Albert and Berkholz, Christoph and Etessami, Kousha and Ochremiak, Joanna}, title = {{Finite and Algorithmic Model Theory (Dagstuhl Seminar 22051)}}, pages = {101--118}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2022}, volume = {12}, number = {1}, editor = {Atserias, Albert and Berkholz, Christoph and Etessami, Kousha and Ochremiak, Joanna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.12.1.101}, URN = {urn:nbn:de:0030-drops-169232}, doi = {10.4230/DagRep.12.1.101}, annote = {Keywords: automata and game theory, database theory, descriptive complexity, finite model theory, homomorphism counts, Query enumeration} }

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Invited Talk

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

A possibly unexpected by-product of the mathematical study of the lengths of proofs, as is done in the field of propositional proof complexity, is, I claim, that it may lead to new polynomial-time algorithms. To explain this, I will first recall the origins of proof complexity as a field, and then explain why some of the recent progress in it could lead to some new algorithms.

Albert Atserias. Towards a Theory of Algorithmic Proof Complexity (Invited Talk). In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 1:1-1:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{atserias:LIPIcs.ICALP.2022.1, author = {Atserias, Albert}, title = {{Towards a Theory of Algorithmic Proof Complexity}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {1:1--1:2}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.1}, URN = {urn:nbn:de:0030-drops-163423}, doi = {10.4230/LIPIcs.ICALP.2022.1}, annote = {Keywords: proof complexity, logic, computational complexity} }

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Invited Talk

**Published in:** LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

Let P be a sound proof system for propositional logic. For each CNF formula F, let SAT(F) be a CNF formula whose satisfying assignments are in 1-to-1 correspondence with those of F (e.g., F itself). For each integer s, let REF(F,s) be a CNF formula whose satisfying assignments are in 1-to-1 correspondence with the P-refutations of F of length s. Since P is sound, it is obvious that the conjunction formula SAT(F) & REF(F,s) is unsatisfiable for any choice of F and s. It has been long known that, for many natural proof systems P and for the most natural formalizations of the formulas SAT and REF, the unsatisfiability of SAT(F) & REF(F,s) can be established by a short refutation. In addition, for many P, these short refutations live in the proof system P itself. This is the case for all Frege proof systems. In contrast it was known since the early 2000’s that Resolution proofs of Resolution’s soundness statements must have superpolynomial length. In this talk I will explain how the soundness formulas for a proof system P relate to the computational complexity of the proof search problem for P. In particular, I will explain how such formulas are used in the recent proof that the problem of approximating the minimum proof-length for Resolution is NP-hard (Atserias-Müller 2019). Besides playing a key role in this hardness of approximation result, the renewed interest in the soundness formulas led to a complete answer to the question whether Resolution has subexponential-length proofs of its own soundness statements (Garlík 2019).

Albert Atserias. Proofs of Soundness and Proof Search (Invited Talk). In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, p. 2:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{atserias:LIPIcs.FSTTCS.2020.2, author = {Atserias, Albert}, title = {{Proofs of Soundness and Proof Search}}, booktitle = {40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)}, pages = {2:1--2:1}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-174-0}, ISSN = {1868-8969}, year = {2020}, volume = {182}, editor = {Saxena, Nitin and Simon, Sunil}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.2}, URN = {urn:nbn:de:0030-drops-132439}, doi = {10.4230/LIPIcs.FSTTCS.2020.2}, annote = {Keywords: Proof complexity, automatability, Resolution, proof search, consistency statements, lower bounds, reflection principle, satisfiability} }

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

We show that if a system of degree-k polynomial constraints on n Boolean variables has a Sums-of-Squares (SOS) proof of unsatisfiability with at most s many monomials, then it also has one whose degree is of the order of the square root of n log s plus k. A similar statement holds for the more general Positivstellensatz (PS) proofs. This establishes size-degree trade-offs for SOS and PS that match their analogues for weaker proof systems such as Resolution, Polynomial Calculus, and the proof systems for the LP and SDP hierarchies of Lovász and Schrijver. As a corollary to this, and to the known degree lower bounds, we get optimal integrality gaps for exponential size SOS proofs for sparse random instances of the standard NP-hard constraint optimization problems. We also get exponential size SOS lower bounds for Tseitin and Knapsack formulas. The proof of our main result relies on a zero-gap duality theorem for pre-ordered vector spaces that admit an order unit, whose specialization to PS and SOS may be of independent interest.

Albert Atserias and Tuomas Hakoniemi. Size-Degree Trade-Offs for Sums-of-Squares and Positivstellensatz Proofs. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 24:1-24:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{atserias_et_al:LIPIcs.CCC.2019.24, author = {Atserias, Albert and Hakoniemi, Tuomas}, title = {{Size-Degree Trade-Offs for Sums-of-Squares and Positivstellensatz Proofs}}, booktitle = {34th Computational Complexity Conference (CCC 2019)}, pages = {24:1--24:20}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.24}, URN = {urn:nbn:de:0030-drops-108464}, doi = {10.4230/LIPIcs.CCC.2019.24}, annote = {Keywords: Proof complexity, semialgebraic proof systems, Sums-of-Squares, Positivstellensatz, trade-offs, lower bounds, monomial size, degree} }

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**Published in:** LIPIcs, Volume 119, 27th EACSL Annual Conference on Computer Science Logic (CSL 2018)

We consider the hardness of approximation of optimization problems from the point of view of definability. For many NP-hard optimization problems it is known that, unless P = NP, no polynomial-time algorithm can give an approximate solution guaranteed to be within a fixed constant factor of the optimum. We show, in several such instances and without any complexity theoretic assumption, that no algorithm that is expressible in fixed-point logic with counting (FPC) can compute an approximate solution. Since important algorithmic techniques for approximation algorithms (such as linear or semidefinite programming) are expressible in FPC, this yields lower bounds on what can be achieved by such methods. The results are established by showing lower bounds on the number of variables required in first-order logic with counting to separate instances with a high optimum from those with a low optimum for fixed-size instances.

Albert Atserias and Anuj Dawar. Definable Inapproximability: New Challenges for Duplicator. In 27th EACSL Annual Conference on Computer Science Logic (CSL 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 119, pp. 7:1-7:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{atserias_et_al:LIPIcs.CSL.2018.7, author = {Atserias, Albert and Dawar, Anuj}, title = {{Definable Inapproximability: New Challenges for Duplicator}}, booktitle = {27th EACSL Annual Conference on Computer Science Logic (CSL 2018)}, pages = {7:1--7:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-088-0}, ISSN = {1868-8969}, year = {2018}, volume = {119}, editor = {Ghica, Dan R. and Jung, Achim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2018.7}, URN = {urn:nbn:de:0030-drops-96742}, doi = {10.4230/LIPIcs.CSL.2018.7}, annote = {Keywords: Descriptive Compleixty, Hardness of Approximation, MAX SAT, Vertex Cover, Fixed-point logic with counting} }

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**Published in:** Dagstuhl Reports, Volume 8, Issue 1 (2018)

The study of proof complexity was initiated in [Cook and Reckhow 1979] as a way to attack the P vs.NP problem, and in the ensuing decades many powerful techniques have been discovered for analyzing different proof systems. Proof complexity also gives a way of studying subsystems of Peano Arithmetic where the power of mathematical reasoning is restricted, and to quantify how complex different mathematical theorems are measured in terms of the strength of the methods of reasoning required to establish their validity. Moreover, it allows to analyse the power and limitations of satisfiability algorithms (SAT solvers) used in industrial applications with formulas containing up to millions of variables.
During the last 10--15 years the area of proof complexity has seen a
revival with many exciting results, and new connections have also been revealed with other areas such as, e.g., cryptography, algebraic complexity theory, communication complexity, and combinatorial optimization. While many longstanding open problems from the 1980s and 1990s still remain unsolved, recent progress gives hope that the area may be ripe for decisive breakthroughs. This workshop, gathering researchers from different strands of the proof complexity community, gave opportunities to take stock of
where we stand and discuss the way ahead.

Albert Atserias, Jakob Nordström, Pavel Pudlák, and Rahul Santhanam. Proof Complexity (Dagstuhl Seminar 18051). In Dagstuhl Reports, Volume 8, Issue 1, pp. 124-157, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@Article{atserias_et_al:DagRep.8.1.124, author = {Atserias, Albert and Nordstr\"{o}m, Jakob and Pudl\'{a}k, Pavel and Santhanam, Rahul}, title = {{Proof Complexity (Dagstuhl Seminar 18051)}}, pages = {124--157}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2018}, volume = {8}, number = {1}, editor = {Atserias, Albert and Nordstr\"{o}m, Jakob and Pudl\'{a}k, Pavel and Santhanam, Rahul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.8.1.124}, URN = {urn:nbn:de:0030-drops-92864}, doi = {10.4230/DagRep.8.1.124}, annote = {Keywords: bounded arithmetic, computational complexity, logic, proof complexity, satisfiability algorithms} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We show that for no surface except for the plane does monadic second-order logic (MSO) have a zero-one-law - and not even a convergence law - on the class of (connected) graphs embeddable on the surface. In addition we show that every rational in [0,1] is the limiting probability of some MSO formula. This strongly refutes a conjecture by Heinig et al. (2014) who proved a convergence law for planar graphs, and a zero-one law for connected planar graphs, and also identified the so-called gaps of [0,1]: the subintervals that are not limiting probabilities of any MSO formula. The proof relies on a combination of methods from structural graph theory, especially large face-width embeddings of graphs on surfaces, analytic combinatorics, and finite model theory, and several parts of the proof may be of independent interest. In particular, we identify precisely the properties that make the zero-one law work on planar graphs but fail for every other surface.

Albert Atserias, Stephan Kreutzer, and Marc Noy. On Zero-One and Convergence Laws for Graphs Embeddable on a Fixed Surface. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 116:1-116:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{atserias_et_al:LIPIcs.ICALP.2018.116, author = {Atserias, Albert and Kreutzer, Stephan and Noy, Marc}, title = {{On Zero-One and Convergence Laws for Graphs Embeddable on a Fixed Surface}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {116:1--116:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.116}, URN = {urn:nbn:de:0030-drops-91206}, doi = {10.4230/LIPIcs.ICALP.2018.116}, annote = {Keywords: topological graph theory, monadic second-order logic, random graphs, zero-one law, convergence law} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

We analyse how the standard reductions between constraint satisfaction problems affect their proof complexity. We show that, for the most studied propositional and semi-algebraic proof systems, the classical constructions of pp-interpretability, homomorphic equivalence and addition of constants to a core preserve the proof complexity of the CSP. As a result, for those proof systems, the classes of constraint languages for which small unsatisfiability certificates exist can be characterised algebraically. We illustrate our results by a gap theorem saying that a constraint language either has resolution refutations of bounded width, or does not have bounded-depth Frege refutations of subexponential size. The former holds exactly for the widely studied class of constraint languages of bounded width. This class is also known to coincide with the class of languages with Sums-of-Squares refutations of sublinear degree, a fact for which we provide an alternative proof. We hence ask for the existence of a natural proof system with good behaviour with respect to reductions and simultaneously small size refutations beyond bounded width. We give an example of such a proof system by showing that bounded-degree Lovasz-Schrijver satisfies both requirements.

Albert Atserias and Joanna Ochremiak. Proof Complexity Meets Algebra. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 110:1-110:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{atserias_et_al:LIPIcs.ICALP.2017.110, author = {Atserias, Albert and Ochremiak, Joanna}, title = {{Proof Complexity Meets Algebra}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {110:1--110:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.110}, URN = {urn:nbn:de:0030-drops-74956}, doi = {10.4230/LIPIcs.ICALP.2017.110}, annote = {Keywords: Constraint Satisfaction Problem, Proof Complexity, Reductions, Gap Theorems} }

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**Published in:** LIPIcs, Volume 62, 25th EACSL Annual Conference on Computer Science Logic (CSL 2016)

Homogenization is a powerful way of taming a class of finite structures with several interesting applications in different areas, from Ramsey theory in combinatorics to constraint satisfaction problems (CSPs) in computer science, through (finite) model theory. A few sufficient conditions for a class of finite structures to allow homogenization are known, and here we provide a necessary condition. This lets us show that certain natural classes are not homogenizable: 1) the class of locally consistent systems of linear equations over the two-element field or any finite Abelian group, and 2) the class of finite structures that forbid homomorphisms from a specific MSO-definable class of structures of treewidth two. In combination with known results, the first example shows that, up to pp-interpretability, the CSPs that are solvable by local consistency methods are distinguished from the rest by the fact that their classes of locally consistent instances are homogenizable. The second example shows that, for MSO-definable classes of forbidden patterns, treewidth one versus two is the dividing line to homogenizability.

Albert Atserias and Szymon Torunczyk. Non-Homogenizable Classes of Finite Structures. In 25th EACSL Annual Conference on Computer Science Logic (CSL 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 62, pp. 16:1-16:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{atserias_et_al:LIPIcs.CSL.2016.16, author = {Atserias, Albert and Torunczyk, Szymon}, title = {{Non-Homogenizable Classes of Finite Structures}}, booktitle = {25th EACSL Annual Conference on Computer Science Logic (CSL 2016)}, pages = {16:1--16:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-022-4}, ISSN = {1868-8969}, year = {2016}, volume = {62}, editor = {Talbot, Jean-Marc and Regnier, Laurent}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2016.16}, URN = {urn:nbn:de:0030-drops-65563}, doi = {10.4230/LIPIcs.CSL.2016.16}, annote = {Keywords: Fra\"{i}ss\'{e} class, amalgmation class, reduct, Constraint Satisfaction Problem, bounded width} }

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**Published in:** LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

Tree-width is a well-studied parameter of structures that measures their similarity to a tree. Many important NP-complete problems, such as Boolean satisfiability (SAT), are tractable on bounded tree-width instances. In this paper we focus on the canonical PSPACE-complete problem QBF, the fully-quantified version of SAT. It was shown by Pan and Vardi [LICS 2006] that this problem is PSPACE-complete even for formulas whose tree-width grows extremely slowly. Vardi also posed the question of whether the problem is tractable when restricted to instances of bounded tree-width. We answer this question by showing that QBF on instances with constant tree-width is PSPACE-complete.

Albert Atserias and Sergi Oliva. Bounded-width QBF is PSPACE-complete. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 44-54, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{atserias_et_al:LIPIcs.STACS.2013.44, author = {Atserias, Albert and Oliva, Sergi}, title = {{Bounded-width QBF is PSPACE-complete}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {44--54}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-50-7}, ISSN = {1868-8969}, year = {2013}, volume = {20}, editor = {Portier, Natacha and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.44}, URN = {urn:nbn:de:0030-drops-39217}, doi = {10.4230/LIPIcs.STACS.2013.44}, annote = {Keywords: Tree-width, QBF, PSPACE-complete} }

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