{"type":"ScholarlyArticle","title":"A Logic for Reasoning about Aggregate-Combine Graph Neural Networks","authors":[{"name":"Pierre Nunn","slug":"pierre-nunn","given":"Pierre","family":"Nunn","affiliation":"University of Rennes , IRISA , CNRS , France"},{"name":"Marco Sälzer","slug":"marco-salzer","given":"Marco","family":"Sälzer","affiliation":"Theoretical Computer Science / Formal Methods , University of Kassel , Germany"},{"name":"François Schwarzentruber","slug":"francois-schwarzentruber","given":"François","family":"Schwarzentruber","affiliation":"University of Rennes , IRISA , CNRS , France"},{"name":"Nicolas Troquard","slug":"nicolas-troquard","given":"Nicolas","family":"Troquard","affiliation":"Gran Sasso Science Institute , L'Aquila , Italy"}],"abstract":"We propose a modal logic in which counting modalities appear in linear inequalities. We show that each formula can be transformed into an equivalent graph neural network (GNN). We also show that a broad class of GNNs can be transformed efficiently into a\r\nformula, thus significantly improving upon the literature about the logical expressiveness of GNNs. We also show that the satisfiability problem is PSPACE-complete. These results bring together the promise of using standard logical methods for reasoning about GNNs and their properties, particularly in applications such as GNN querying, equivalence checking, etc. We prove that such natural problems can be solved in polynomial space.","doi":"10.24963/ijcai.2024/391","url":"https://ijcai.org/proceedings/2024/391","pdf":"https://ijcai.org/proceedings/2024/0391.pdf","markdown":"https://ijcai.org/proceedings/2024/391.md","bibtex":"https://ijcai.org/proceedings/2024/bibtex/391","volume":{"slug":"2024","title":"Proceedings of the Thirty-Third International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2024","url":"https://ijcai.org/proceedings/2024/","year":2024,"location":"Jeju, Korea","dates":"3-9 August 2024","editors":"Kate Larson","isbn":"978-1-956792-04-1","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2024"},"section":"Knowledge Representation and Reasoning","track":"Main Track","number":391,"pages":{"first":3532,"last":3540},"keywords":["Machine Learning: ML: Explainable/Interpretable machine learning","Machine Learning: ML: Learning theory","Knowledge Representation and Reasoning: KRR: Learning and reasoning"],"videos":[],"references":[{"position":1,"raw":"[Areces et al., 2010] Carlos Areces, Guillaume Hoffmann, and Alexandre Denis. Modal logics with counting. In Anuj Dawar and Ruy J. G. B. de Queiroz, editors, Logic, Lan- guage, Information and Computation, 17th International Workshop, WoLLIC 2010, Brasilia, Brazil, July 6-9, 2010. Proceedings, volume 6188 of Lecture Notes in Computer Science, pages 98-109. Springer, 2010.","title":"Modal logics with counting","authors":"Carlos Areces, Guillaume Hoffmann, Alexandre Denis","year":"2010","venue":"Logic, Language, Information and Computation, 17th International Workshop, WoLLIC 2010","doi":null,"url":null,"internal_url":null},{"position":2,"raw":"[Azzolin et al., 2022] Steve Azzolin, Antonio Longa, Pietro Barbiero, Pietro Liò, and Andrea Passerini. Global ex- plainability of GNNs via logic combination of learned concepts. CoRR, abs/2210.07147, 2022.","title":"Global explainability of GNNs via logic combination of learned concepts","authors":"Steve Azzolin, Antonio Longa, Pietro Barbiero, Pietro Liò, Andrea Passerini","year":"2022","venue":null,"doi":null,"url":null,"internal_url":null},{"position":3,"raw":"[Baader et al., 2020] Franz Baader, Bartosz Bednarczyk, and Sebastian Rudolph. Satisfiability and query answering in description logics with global and local cardinality con- straints. In 24th European Conference on Artificial Intel- ligence (ECAI 2020), volume 325 of Frontiers in Artificial Intelligence and Applications, pages 616-623. IOS Press, 2020.","title":"Satisfiability and query answering in description logics with global and local cardinality constraints","authors":"Baader","year":"2020","venue":"24th European Conference on Artificial Intelligence (ECAI 2020)","doi":null,"url":null,"internal_url":null},{"position":4,"raw":"[Barceló et al., 2020] Pablo Barceló, Egor V Kostylev, Mikaël Monet, Jorge Pérez, Juan Reutter, and Juan-Pablo Silva. The logical expressiveness of graph neural net- works. In 8th International Conference on Learning Rep- resentations (ICLR 2020), Virtual conference, Ethiopia, April 2020.","title":"The logical expressiveness of graph neural networks","authors":"Barceló","year":"2020","venue":"8th International Conference on Learning Representations (ICLR 2020), Virtual conference","doi":null,"url":null,"internal_url":null},{"position":5,"raw":"[Blackburn et al., 2001] Patrick Blackburn, Maarten de Ri- jke, and Yde Venema. Modal Logic, volume 53 of Cam- bridge Tracts in Theoretical Computer Science. Cam- bridge University Press, 2001.","title":"Cambridge Tracts in Theoretical Computer Science","authors":"Patrick Blackburn, Maarten De Rijke, Yde Venema","year":"2001","venue":null,"doi":null,"url":null,"internal_url":null},{"position":6,"raw":"[Cucala et al., 2023] David Tena Cucala, Bernardo Cuenca Grau, Boris Motik, and Egor V. Kostylev. On the corre- spondence between monotonic max-sum GNNs and Dat- alog. In Pierre Marquis, Tran Cao Son, and Gabriele Kern-Isberner, editors, Proceedings of the 20th Interna- tional Conference on Principles of Knowledge Represen- tation and Reasoning, KR 2023, Rhodes, Greece, Septem- ber 2-8, 2023, pages 658-667, 2023.","title":"On the correspondence between monotonic max-sum GNNs and Datalog","authors":"David Tena Cucala, Bernardo Cuenca Grau, Boris Motik, Egor V Kostylev","year":"2023","venue":"Proceedings of the 20th International Conference on Principles of Knowledge Representation and Reasoning, KR 2023","doi":null,"url":null,"internal_url":null},{"position":7,"raw":"[de Rijke, 2000] Maarten de Rijke. A note on graded modal logic. Studia Logica, 64(2):271-283, 2000.","title":"Maarten de Rijke. A note on graded modal logic","authors":"De Rijke","year":"2000","venue":"Studia Logica","doi":null,"url":null,"internal_url":null},{"position":8,"raw":"[Demri and Lugiez, 2010] Stéphane Demri and Denis Lugiez. Complexity of modal logics with Presburger constraints. J. Appl. Log., 8(3):233-252, 2010.","title":"Complexity of modal logics with Presburger constraints","authors":"Stéphane Demri, Denis Lugiez","year":"2010","venue":"J. Appl. Log","doi":null,"url":null,"internal_url":null},{"position":9,"raw":"[Fattorosi-Barnaba and Caro, 1985] Maurizio Fattorosi- Barnaba and F. Caro. Graded modalities. I. Studia Logica, 44(2):197-221, 1985.","title":"Graded modalities","authors":"Maurizio Fattorosi-Barnaba, F Caro","year":"1985","venue":"I. Studia Logica","doi":null,"url":null,"internal_url":null},{"position":10,"raw":"[Galliani et al., 2023] Pietro Galliani, Oliver Kutz, and Nico- las Troquard. Succinctness and complexity of ALC with counting perceptrons. In Pierre Marquis, Tran Cao Son, and Gabriele Kern-Isberner, editors, Proceedings of the 20th International Conference on Principles of Knowledge Representation and Reasoning, KR 2023, Rhodes, Greece, September 2-8, 2023, pages 291-300, 2023.","title":"Succinctness and complexity of ALC with counting perceptrons","authors":"Galliani","year":"2023","venue":"Proceedings of the 20th International Conference on Principles of Knowledge Representation and Reasoning, KR 2023","doi":null,"url":null,"internal_url":null},{"position":11,"raw":"[Gilmer et al., 2017] Justin Gilmer, Samuel S. Schoenholz, Patrick F. Riley, Oriol Vinyals, and George E. Dahl. Neural message passing for quantum chemistry. In Doina Precup and Yee Whye Teh, editors, Proceedings of the 34th Inter- national Conference on Machine Learning, ICML 2017, Sydney, NSW, Australia, 6-11 August 2017, volume 70 of Proceedings of Machine Learning Research, pages 1263- 1272. PMLR, 2017.","title":"Neural message passing for quantum chemistry","authors":"Justin Gilmer, Samuel S Schoenholz, Patrick F Riley, Oriol Vinyals, George E Dahl","year":"2017","venue":"Proceedings of the 34th International Conference on Machine Learning, ICML 2017","doi":null,"url":null,"internal_url":null},{"position":12,"raw":"[Grohe, 2021] Martin Grohe. The logic of graph neural net- works. In 36th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2021, Rome, Italy, June 29 -July 2, 2021, pages 1-17. IEEE, 2021.","title":"The logic of graph neural networks","authors":"Martin Grohe","year":"2021","venue":"36th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2021","doi":null,"url":null,"internal_url":null},{"position":13,"raw":"[Hampson, 2016] Christopher Hampson. Decidable first- order modal logics with counting quantifiers. In Lev D. Beklemishev, Stéphane Demri, and András Maté, editors, Advances in Modal Logic 11, proceedings of the 11th con- ference on \"Advances in Modal Logic,\" held in Budapest, Hungary, August 30 -September 2, 2016, pages 382-400. College Publications, 2016.","title":"Decidable firstorder modal logics with counting quantifiers","authors":"Christopher Hampson","year":"2016","venue":"Advances in Modal Logic 11, proceedings of the 11th conference on \"Advances in Modal Logic","doi":null,"url":null,"internal_url":null},{"position":14,"raw":"[Huang et al., 2020] Xiaowei Huang, Daniel Kroening, Wenjie Ruan, James Sharp, Youcheng Sun, Emese Thamo, Min Wu, and Xinping Yi. A survey of safety and trustwor- thiness of deep neural networks: Verification, testing, ad- versarial attack and defence, and interpretability. Comput. Sci. Rev., 37:100270, 2020.","title":"A survey of safety and trustworthiness of deep neural networks: Verification, testing, adversarial attack and defence, and interpretability","authors":"Xiaowei Huang, Daniel Kroening, Wenjie Ruan, James Sharp, Youcheng Sun, Emese Thamo, Min Wu, Xinping Yi","year":"2020","venue":"Comput. Sci. Rev","doi":null,"url":null,"internal_url":null},{"position":15,"raw":"[Katz et al., 2017] Guy Katz, Clark W. Barrett, David L. Dill, Kyle Julian, and Mykel J. Kochenderfer. Reluplex: An efficient SMT solver for verifying deep neural net- works. In Rupak Majumdar and Viktor Kuncak, editors, Computer Aided Verification -29th International Confer- ence, CAV 2017, Heidelberg, Germany, July 24-28, 2017, Proceedings, Part I, volume 10426 of Lecture Notes in Computer Science, pages 97-117. Springer, 2017.","title":"Reluplex: An efficient SMT solver for verifying deep neural networks","authors":"Guy Katz, Clark W Barrett, David L Dill, Kyle Julian, Mykel J Kochenderfer","year":"2017","venue":"Computer Aided Verification -29th International Conference, CAV 2017","doi":null,"url":null,"internal_url":null},{"position":16,"raw":"[Ladner, 1977] Richard E. Ladner. The computational com- plexity of provability in systems of modal propositional logic. SIAM Journal on Computing, 6(3):467-480, 1977.","title":"The computational complexity of provability in systems of modal propositional logic","authors":"Richard E Ladner","year":"1977","venue":"SIAM Journal on Computing","doi":null,"url":null,"internal_url":null},{"position":17,"raw":"[Marques-Silva and Ignatiev, 2022] João Marques-Silva and Alexey Ignatiev. Delivering trustworthy AI through for- mal XAI. In Thirty-Sixth AAAI Conference on Artificial Intelligence, AAAI 2022, Thirty-Fourth Conference on In- novative Applications of Artificial Intelligence, IAAI 2022, The Twelveth Symposium on Educational Advances in Ar- tificial Intelligence, EAAI 2022 Virtual Event, February 22 -March 1, 2022, pages 12342-12350. AAAI Press, 2022.","title":"João Marques-Silva and Alexey Ignatiev. Delivering trustworthy AI through formal XAI","authors":"Marques-Silva, Ignatiev","year":"2022","venue":"Thirty-Sixth AAAI Conference on Artificial Intelligence, AAAI 2022, Thirty-Fourth Conference on Innovative Applications of Artificial Intelligence, IAAI 2022, The Twelveth Symposium on Educational Advances in Artificial Intelligence, EAAI 2022 Virtual Event","doi":null,"url":null,"internal_url":null},{"position":18,"raw":"[Müller et al., 2022] Peter Müller, Lukas Faber, Karolis Martinkus, and Roger Wattenhofer. DT+GNN: A fully explainable graph neural network using decision trees. CoRR, abs/2205.13234, 2022.","title":"DT+GNN: A fully explainable graph neural network using decision trees","authors":"Müller","year":"2022","venue":null,"doi":null,"url":null,"internal_url":null},{"position":19,"raw":"[Nunn et al., 2024] Pierre Nunn, Marco Sälzer, François Schwarzentruber, and Nicolas Troquard. A Logic for Reasoning About Aggregate-Combine Graph Neural Net- works. CoRR, abs/2405.00205, 2024.","title":"A Logic for Reasoning About Aggregate-Combine Graph Neural Networks","authors":"Nunn","year":"2024","venue":null,"doi":null,"url":null,"internal_url":null},{"position":20,"raw":"[Pinchinat et al., 2022] Sophie Pinchinat, Sasha Rubin, and François Schwarzentruber. Formula synthesis in proposi- tional dynamic logic with shuffle. In Thirty-Sixth AAAI Conference on Artificial Intelligence, AAAI 2022, Thirty- Fourth Conference on Innovative Applications of Artificial Intelligence, IAAI 2022, The Twelveth Symposium on Ed- ucational Advances in Artificial Intelligence, EAAI 2022 Virtual Event, February 22 -March 1, 2022, pages 9902- 9909. AAAI Press, 2022.","title":"Formula synthesis in propositional dynamic logic with shuffle","authors":"Sophie Pinchinat, Sasha Rubin, François Schwarzentruber","year":"2022","venue":"Thirty-Sixth AAAI Conference on Artificial Intelligence, AAAI 2022, Thirty-Fourth Conference on Innovative Applications of Artificial Intelligence, IAAI 2022, The Twelveth Symposium on Educational Advances in Artificial Intelligence, EAAI 2022 Virtual Event","doi":null,"url":null,"internal_url":null},{"position":21,"raw":"[Reiser et al., 2022] Patrick Reiser, Marlen Neubert, André Eberhard, Luca Torresi, Chen Zhou, Chen Shao, Houssam Metni, Clint van Hoesel, Henrik Schopmans, Timo Som- mer, and Pascal Friederich. Graph neural networks for ma- terials science and chemistry. Communications Materials, 3(93), 2022.","title":"Graph neural networks for materials science and chemistry","authors":"Patrick Reiser, Marlen Neubert, André Eberhard, Luca Torresi, Chen Zhou, Chen Shao, Houssam Metni, Clint Van Hoesel, Henrik Schopmans, Timo Sommer, Pascal Friederich","year":"2022","venue":"Communications Materials","doi":null,"url":null,"internal_url":null},{"position":22,"raw":"[Rudin, 2019] Cynthia Rudin. Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead. Nat. Mach. Intell., 1(5):206- 215, 2019.","title":"Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead","authors":"Cynthia Rudin","year":"2019","venue":"Nat. Mach. Intell","doi":null,"url":null,"internal_url":null},{"position":23,"raw":"[Salamat et al., 2021] Amirreza Salamat, Xiao Luo, and Ali Jafari. Heterographrec: A heterogeneous graph-based neu- ral networks for social recommendations. Knowl. Based Syst., 217:106817, 2021.","title":"Heterographrec: A heterogeneous graph-based neural networks for social recommendations","authors":"Salamat","year":"2021","venue":"Knowl. Based Syst","doi":null,"url":null,"internal_url":null},{"position":24,"raw":"[Sälzer and Lange, 2022] Marco Sälzer and Martin Lange. We cannot guarantee safety: The undecidability of graph neural network verification. CoRR, abs/2206.05070, 2022.","title":"We cannot guarantee safety: The undecidability of graph neural network verification","authors":"Lange Sälzer, Marco Sälzer, Martin Lange","year":"2022","venue":null,"doi":null,"url":null,"internal_url":null},{"position":25,"raw":"[Shirazi and Amir, 2007] Afsaneh Shirazi and Eyal Amir. Probabilistic modal logic. In Proceedings of the Twenty- Second AAAI Conference on Artificial Intelligence, July 22-26, 2007, Vancouver, British Columbia, Canada, pages 489-495. AAAI Press, 2007.","title":"Probabilistic modal logic","authors":"Afsaneh Shirazi, Eyal Amir","year":"2007","venue":"Proceedings of the Twenty-Second AAAI Conference on Artificial Intelligence","doi":null,"url":null,"internal_url":null},{"position":26,"raw":"[Sistla and Clarke, 1985] A. Prasad Sistla and Edmund M. Clarke. The complexity of propositional linear temporal logics. J. ACM, 32(3):733-749, 1985.","title":"The complexity of propositional linear temporal logics","authors":"A, Prasad Sistla, Edmund M Clarke","year":"1985","venue":"J. ACM","doi":null,"url":null,"internal_url":null},{"position":27,"raw":"[Snell et al., 2012] William Snell, Dirk Pattinson, and Flo- rian Widmann. Solving graded/probabilistic modal logic via linear inequalities (system description). In Nikolaj S. Bjørner and Andrei Voronkov, editors, Logic for Program- ming, Artificial Intelligence, and Reasoning -18th Inter- national Conference, LPAR-18, Mérida, Venezuela, March 11-15, 2012. Proceedings, volume 7180 of Lecture Notes in Computer Science, pages 383-390. Springer, 2012.","title":"Solving graded/probabilistic modal logic via linear inequalities (system description)","authors":"William Snell, Dirk Pattinson, Florian Widmann","year":"2012","venue":"Logic for Programming, Artificial Intelligence, and Reasoning -18th International Conference, LPAR-18","doi":null,"url":null,"internal_url":null},{"position":28,"raw":"[Tseitin, 1983] Grigori S. Tseitin. On the complexity of derivation in propositional calculus. Automation of rea- soning: 2: Classical papers on computational logic 1967- 1970, pages 466-483, 1983.","title":"On the complexity of derivation in propositional calculus","authors":"S Grigori, Tseitin","year":"1983","venue":"Automation of reasoning: 2: Classical papers on computational logic 1967-1970","doi":null,"url":null,"internal_url":null},{"position":29,"raw":"[Xiong et al., 2021] Jiacheng Xiong, Zhaoping Xiong, Kaix- ian Chen, Hualiang Jiang, and Mingyue Zheng. Graph neural networks for automated de novo drug design. Drug Discovery Today, 26(6):1382-1393, 2021.","title":"Graph neural networks for automated de novo drug design","authors":"Xiong","year":"2021","venue":"Drug Discovery Today","doi":null,"url":null,"internal_url":null},{"position":30,"raw":"[Ye et al., 2022] Zi Ye, Yogan Jaya Kumar, Goh Ong Sing, Fengyan Song, and Junsong Wang. A comprehensive sur- vey of graph neural networks for knowledge graphs. IEEE Access, 10:75729-75741, 2022.","title":"A comprehensive survey of graph neural networks for knowledge graphs","authors":"Ye","year":"2022","venue":"IEEE Access","doi":null,"url":null,"internal_url":null},{"position":31,"raw":"[Yuan et al., 2023] Hao Yuan, Haiyang Yu, Shurui Gui, and Shuiwang Ji. Explainability in graph neural networks: A taxonomic survey. IEEE Trans. Pattern Anal. Mach. In- tell., 45(5):5782-5799, 2023.","title":"Explainability in graph neural networks: A taxonomic survey","authors":"Haiyang Hao Yuan, Shurui Yu, Shuiwang Gui, Ji","year":"2023","venue":"IEEE Trans. Pattern Anal. Mach. Intell","doi":null,"url":null,"internal_url":null},{"position":32,"raw":"[Zhou et al., 2020] Jie Zhou, Ganqu Cui, Shengding Hu, Zhengyan Zhang, Cheng Yang, Zhiyuan Liu, Lifeng Wang, Changcheng Li, and Maosong Sun. Graph neural networks: A review of methods and applications. AI Open, 1:57-81, 2020.","title":"Graph neural networks: A review of methods and applications","authors":"Jie Zhou, Ganqu Cui, Shengding Hu, Zhengyan Zhang, Cheng Yang, Zhiyuan Liu, Lifeng Wang, Changcheng Li, Maosong Sun","year":"2020","venue":"AI Open","doi":null,"url":null,"internal_url":null}],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}