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Complexity of Probabilistic Reasoning for Neurosymbolic Classification Techniques (2404.08404v1)
Published 12 Apr 2024 in cs.AI, cs.CC, cs.LG, and cs.SC
Abstract: Neurosymbolic artificial intelligence is a growing field of research aiming to combine neural network learning capabilities with the reasoning abilities of symbolic systems. Informed multi-label classification is a sub-field of neurosymbolic AI which studies how to leverage prior knowledge to improve neural classification systems. A well known family of neurosymbolic techniques for informed classification use probabilistic reasoning to integrate this knowledge during learning, inference or both. Therefore, the asymptotic complexity of probabilistic reasoning is of cardinal importance to assess the scalability of such techniques. However, this topic is rarely tackled in the neurosymbolic literature, which can lead to a poor understanding of the limits of probabilistic neurosymbolic techniques. In this paper, we introduce a formalism for informed supervised classification tasks and techniques. We then build upon this formalism to define three abstract neurosymbolic techniques based on probabilistic reasoning. Finally, we show computational complexity results on several representation languages for prior knowledge commonly found in the neurosymbolic literature.
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International Journal of Computer Vision 115, 211–252 (2015) https://doi.org/10.1007/s11263-015-0816-y Miller [1995] Miller, G.A.: Wordnet. Communications of the ACM 38, 39–41 (1995) https://doi.org/10.1145/219717.219748 [6] Gebru, T., Krause, J., Wang, Y., Chen, D., Deng, J., Fei-Fei, L.: Fine-grained car detection for visual census estimation 31(1) https://doi.org/10.1609/aaai.v31i1.11174 . Number: 1. Accessed 2024-03-31 Van Horn et al. [2018] Van Horn, G., Mac Aodha, O., Song, Y., Cui, Y., Sun, C., Shepard, A., Adam, H., Perona, P., Belongie, S.: The inaturalist species classification and detection dataset. In: 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 8769–8778 (2018). https://doi.org/10.1109/CVPR.2018.00914 van Krieken et al. [2022] Krieken, E., Thanapalasingam, T., Tomczak, J.M., Harmelen, F., Teije, A.: A-NeSI: A Scalable Approximate Method for Probabilistic Neurosymbolic Inference (2022) [9] Maene, J., De Raedt, L.: Soft-unification in deep probabilistic logic 36. Accessed 2024-02-21 Deng et al. [2014] Deng, J., Ding, N., Jia, Y., Frome, A., Murphy, K., Bengio, S., Li, Y., Neven, H., Adam, H.: Large-Scale Object Classification Using Label Relation Graphs. In: Computer Vision – ECCV 2014, pp. 48–64. Springer, ??? (2014) Xu et al. [2018] Xu, J., Zhang, Z., Friedman, T., Liang, Y., Broeck, G.V.D.: A semantic loss function for deep learning with symbolic knowledge. In: 35th International Conference on Machine Learning, ICML 2018, vol. 12, pp. 8752–8760. International Machine Learning Society (IMLS), ??? (2018) Ahmed et al. [2022] Ahmed, K., Teso, S., Chang, K.-W., Broeck, G., Vergari, A.: Semantic Probabilistic Layers for Neuro-Symbolic Learning. In: Advances in Neural Information Processing Systems, vol. 35, pp. 29944–29959. Curran Associates, Inc., ??? (2022) [13] Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. [2021] Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. 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[2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Rueden, L., Mayer, S., Beckh, K., Georgiev, B., Giesselbach, S., Heese, R., Kirsch, B., Pfrommer, J., Pick, A., Ramamurthy, R., Walczak, M., Garcke, J., Bauckhage, C., Schuecker, J.: Informed Machine Learning – A Taxonomy and Survey of Integrating Prior Knowledge into Learning Systems. IEEE Transactions on Knowledge and Data Engineering 35(1), 614–633 (2023) https://doi.org/10.1109/TKDE.2021.3079836 . Conference Name: IEEE Transactions on Knowledge and Data Engineering Russakovsky et al. [2015] Russakovsky, O., Deng, J., Su, H., Krause, J., Satheesh, S., Ma, S., Huang, Z., Karpathy, A., Khosla, A., Bernstein, M., Berg, A.C., Fei-Fei, L.: Imagenet large scale visual recognition challenge. International Journal of Computer Vision 115, 211–252 (2015) https://doi.org/10.1007/s11263-015-0816-y Miller [1995] Miller, G.A.: Wordnet. Communications of the ACM 38, 39–41 (1995) https://doi.org/10.1145/219717.219748 [6] Gebru, T., Krause, J., Wang, Y., Chen, D., Deng, J., Fei-Fei, L.: Fine-grained car detection for visual census estimation 31(1) https://doi.org/10.1609/aaai.v31i1.11174 . Number: 1. Accessed 2024-03-31 Van Horn et al. [2018] Van Horn, G., Mac Aodha, O., Song, Y., Cui, Y., Sun, C., Shepard, A., Adam, H., Perona, P., Belongie, S.: The inaturalist species classification and detection dataset. In: 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 8769–8778 (2018). https://doi.org/10.1109/CVPR.2018.00914 van Krieken et al. [2022] Krieken, E., Thanapalasingam, T., Tomczak, J.M., Harmelen, F., Teije, A.: A-NeSI: A Scalable Approximate Method for Probabilistic Neurosymbolic Inference (2022) [9] Maene, J., De Raedt, L.: Soft-unification in deep probabilistic logic 36. Accessed 2024-02-21 Deng et al. [2014] Deng, J., Ding, N., Jia, Y., Frome, A., Murphy, K., Bengio, S., Li, Y., Neven, H., Adam, H.: Large-Scale Object Classification Using Label Relation Graphs. In: Computer Vision – ECCV 2014, pp. 48–64. Springer, ??? (2014) Xu et al. [2018] Xu, J., Zhang, Z., Friedman, T., Liang, Y., Broeck, G.V.D.: A semantic loss function for deep learning with symbolic knowledge. In: 35th International Conference on Machine Learning, ICML 2018, vol. 12, pp. 8752–8760. International Machine Learning Society (IMLS), ??? (2018) Ahmed et al. [2022] Ahmed, K., Teso, S., Chang, K.-W., Broeck, G., Vergari, A.: Semantic Probabilistic Layers for Neuro-Symbolic Learning. In: Advances in Neural Information Processing Systems, vol. 35, pp. 29944–29959. Curran Associates, Inc., ??? (2022) [13] Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. [2021] Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. [2024] Ledaguenel, A., Hudelot, C., Khouadjia, M.: Improving Neural-based Classification with Logical Background Knowledge (2024). https://arxiv.org/abs/2402.13019 Manhaeve et al. [2021] Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. Artificial Intelligence 298, 103504 (2021) https://doi.org/10.1016/j.artint.2021.103504 . Accessed 2023-08-07 De Raedt et al. [2007] De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence. IJCAI’07, pp. 2468–2473. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2007) Diligenti et al. [2017] Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. Artificial Intelligence 244, 143–165 (2017) https://doi.org/10.1016/j.artint.2015.08.011 Giannini et al. [2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Russakovsky, O., Deng, J., Su, H., Krause, J., Satheesh, S., Ma, S., Huang, Z., Karpathy, A., Khosla, A., Bernstein, M., Berg, A.C., Fei-Fei, L.: Imagenet large scale visual recognition challenge. International Journal of Computer Vision 115, 211–252 (2015) https://doi.org/10.1007/s11263-015-0816-y Miller [1995] Miller, G.A.: Wordnet. Communications of the ACM 38, 39–41 (1995) https://doi.org/10.1145/219717.219748 [6] Gebru, T., Krause, J., Wang, Y., Chen, D., Deng, J., Fei-Fei, L.: Fine-grained car detection for visual census estimation 31(1) https://doi.org/10.1609/aaai.v31i1.11174 . Number: 1. Accessed 2024-03-31 Van Horn et al. [2018] Van Horn, G., Mac Aodha, O., Song, Y., Cui, Y., Sun, C., Shepard, A., Adam, H., Perona, P., Belongie, S.: The inaturalist species classification and detection dataset. In: 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 8769–8778 (2018). https://doi.org/10.1109/CVPR.2018.00914 van Krieken et al. [2022] Krieken, E., Thanapalasingam, T., Tomczak, J.M., Harmelen, F., Teije, A.: A-NeSI: A Scalable Approximate Method for Probabilistic Neurosymbolic Inference (2022) [9] Maene, J., De Raedt, L.: Soft-unification in deep probabilistic logic 36. Accessed 2024-02-21 Deng et al. [2014] Deng, J., Ding, N., Jia, Y., Frome, A., Murphy, K., Bengio, S., Li, Y., Neven, H., Adam, H.: Large-Scale Object Classification Using Label Relation Graphs. In: Computer Vision – ECCV 2014, pp. 48–64. Springer, ??? (2014) Xu et al. [2018] Xu, J., Zhang, Z., Friedman, T., Liang, Y., Broeck, G.V.D.: A semantic loss function for deep learning with symbolic knowledge. In: 35th International Conference on Machine Learning, ICML 2018, vol. 12, pp. 8752–8760. International Machine Learning Society (IMLS), ??? (2018) Ahmed et al. [2022] Ahmed, K., Teso, S., Chang, K.-W., Broeck, G., Vergari, A.: Semantic Probabilistic Layers for Neuro-Symbolic Learning. In: Advances in Neural Information Processing Systems, vol. 35, pp. 29944–29959. Curran Associates, Inc., ??? (2022) [13] Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. [2021] Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. [2024] Ledaguenel, A., Hudelot, C., Khouadjia, M.: Improving Neural-based Classification with Logical Background Knowledge (2024). https://arxiv.org/abs/2402.13019 Manhaeve et al. [2021] Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. Artificial Intelligence 298, 103504 (2021) https://doi.org/10.1016/j.artint.2021.103504 . Accessed 2023-08-07 De Raedt et al. [2007] De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence. IJCAI’07, pp. 2468–2473. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2007) Diligenti et al. [2017] Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. Artificial Intelligence 244, 143–165 (2017) https://doi.org/10.1016/j.artint.2015.08.011 Giannini et al. [2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Miller, G.A.: Wordnet. Communications of the ACM 38, 39–41 (1995) https://doi.org/10.1145/219717.219748 [6] Gebru, T., Krause, J., Wang, Y., Chen, D., Deng, J., Fei-Fei, L.: Fine-grained car detection for visual census estimation 31(1) https://doi.org/10.1609/aaai.v31i1.11174 . Number: 1. Accessed 2024-03-31 Van Horn et al. [2018] Van Horn, G., Mac Aodha, O., Song, Y., Cui, Y., Sun, C., Shepard, A., Adam, H., Perona, P., Belongie, S.: The inaturalist species classification and detection dataset. In: 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 8769–8778 (2018). https://doi.org/10.1109/CVPR.2018.00914 van Krieken et al. [2022] Krieken, E., Thanapalasingam, T., Tomczak, J.M., Harmelen, F., Teije, A.: A-NeSI: A Scalable Approximate Method for Probabilistic Neurosymbolic Inference (2022) [9] Maene, J., De Raedt, L.: Soft-unification in deep probabilistic logic 36. Accessed 2024-02-21 Deng et al. [2014] Deng, J., Ding, N., Jia, Y., Frome, A., Murphy, K., Bengio, S., Li, Y., Neven, H., Adam, H.: Large-Scale Object Classification Using Label Relation Graphs. In: Computer Vision – ECCV 2014, pp. 48–64. Springer, ??? (2014) Xu et al. [2018] Xu, J., Zhang, Z., Friedman, T., Liang, Y., Broeck, G.V.D.: A semantic loss function for deep learning with symbolic knowledge. In: 35th International Conference on Machine Learning, ICML 2018, vol. 12, pp. 8752–8760. International Machine Learning Society (IMLS), ??? (2018) Ahmed et al. [2022] Ahmed, K., Teso, S., Chang, K.-W., Broeck, G., Vergari, A.: Semantic Probabilistic Layers for Neuro-Symbolic Learning. In: Advances in Neural Information Processing Systems, vol. 35, pp. 29944–29959. Curran Associates, Inc., ??? (2022) [13] Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. [2021] Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. [2024] Ledaguenel, A., Hudelot, C., Khouadjia, M.: Improving Neural-based Classification with Logical Background Knowledge (2024). https://arxiv.org/abs/2402.13019 Manhaeve et al. [2021] Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. Artificial Intelligence 298, 103504 (2021) https://doi.org/10.1016/j.artint.2021.103504 . Accessed 2023-08-07 De Raedt et al. [2007] De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence. IJCAI’07, pp. 2468–2473. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2007) Diligenti et al. [2017] Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. Artificial Intelligence 244, 143–165 (2017) https://doi.org/10.1016/j.artint.2015.08.011 Giannini et al. [2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Van Horn, G., Mac Aodha, O., Song, Y., Cui, Y., Sun, C., Shepard, A., Adam, H., Perona, P., Belongie, S.: The inaturalist species classification and detection dataset. In: 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 8769–8778 (2018). https://doi.org/10.1109/CVPR.2018.00914 van Krieken et al. [2022] Krieken, E., Thanapalasingam, T., Tomczak, J.M., Harmelen, F., Teije, A.: A-NeSI: A Scalable Approximate Method for Probabilistic Neurosymbolic Inference (2022) [9] Maene, J., De Raedt, L.: Soft-unification in deep probabilistic logic 36. Accessed 2024-02-21 Deng et al. [2014] Deng, J., Ding, N., Jia, Y., Frome, A., Murphy, K., Bengio, S., Li, Y., Neven, H., Adam, H.: Large-Scale Object Classification Using Label Relation Graphs. In: Computer Vision – ECCV 2014, pp. 48–64. Springer, ??? (2014) Xu et al. [2018] Xu, J., Zhang, Z., Friedman, T., Liang, Y., Broeck, G.V.D.: A semantic loss function for deep learning with symbolic knowledge. In: 35th International Conference on Machine Learning, ICML 2018, vol. 12, pp. 8752–8760. International Machine Learning Society (IMLS), ??? (2018) Ahmed et al. [2022] Ahmed, K., Teso, S., Chang, K.-W., Broeck, G., Vergari, A.: Semantic Probabilistic Layers for Neuro-Symbolic Learning. In: Advances in Neural Information Processing Systems, vol. 35, pp. 29944–29959. Curran Associates, Inc., ??? (2022) [13] Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. [2021] Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. [2024] Ledaguenel, A., Hudelot, C., Khouadjia, M.: Improving Neural-based Classification with Logical Background Knowledge (2024). https://arxiv.org/abs/2402.13019 Manhaeve et al. [2021] Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. Artificial Intelligence 298, 103504 (2021) https://doi.org/10.1016/j.artint.2021.103504 . Accessed 2023-08-07 De Raedt et al. [2007] De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. 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Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . 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Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . 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(2018) Ahmed et al. [2022] Ahmed, K., Teso, S., Chang, K.-W., Broeck, G., Vergari, A.: Semantic Probabilistic Layers for Neuro-Symbolic Learning. In: Advances in Neural Information Processing Systems, vol. 35, pp. 29944–29959. Curran Associates, Inc., ??? (2022) [13] Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. [2021] Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. 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In: Advances in Neural Information Processing Systems, vol. 35, pp. 29944–29959. Curran Associates, Inc., ??? (2022) [13] Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. [2021] Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. 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Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . 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Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence. IJCAI’07, pp. 2468–2473. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2007) Diligenti et al. [2017] Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. Artificial Intelligence 244, 143–165 (2017) https://doi.org/10.1016/j.artint.2015.08.011 Giannini et al. [2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. 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Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . 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Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . 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In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. 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In: Advances in Neural Information Processing Systems, vol. 35, pp. 29944–29959. Curran Associates, Inc., ??? (2022) [13] Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. [2021] Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. 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International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. [2021] Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. [2024] Ledaguenel, A., Hudelot, C., Khouadjia, M.: Improving Neural-based Classification with Logical Background Knowledge (2024). https://arxiv.org/abs/2402.13019 Manhaeve et al. [2021] Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. Artificial Intelligence 298, 103504 (2021) https://doi.org/10.1016/j.artint.2021.103504 . Accessed 2023-08-07 De Raedt et al. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. 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In: Computer Vision – ECCV 2014, pp. 48–64. Springer, ??? (2014) Xu et al. [2018] Xu, J., Zhang, Z., Friedman, T., Liang, Y., Broeck, G.V.D.: A semantic loss function for deep learning with symbolic knowledge. In: 35th International Conference on Machine Learning, ICML 2018, vol. 12, pp. 8752–8760. International Machine Learning Society (IMLS), ??? (2018) Ahmed et al. [2022] Ahmed, K., Teso, S., Chang, K.-W., Broeck, G., Vergari, A.: Semantic Probabilistic Layers for Neuro-Symbolic Learning. In: Advances in Neural Information Processing Systems, vol. 35, pp. 29944–29959. Curran Associates, Inc., ??? (2022) [13] Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. [2021] Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. [2024] Ledaguenel, A., Hudelot, C., Khouadjia, M.: Improving Neural-based Classification with Logical Background Knowledge (2024). https://arxiv.org/abs/2402.13019 Manhaeve et al. [2021] Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. Artificial Intelligence 298, 103504 (2021) https://doi.org/10.1016/j.artint.2021.103504 . Accessed 2023-08-07 De Raedt et al. [2007] De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. 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Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . 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[2018] Xu, J., Zhang, Z., Friedman, T., Liang, Y., Broeck, G.V.D.: A semantic loss function for deep learning with symbolic knowledge. In: 35th International Conference on Machine Learning, ICML 2018, vol. 12, pp. 8752–8760. International Machine Learning Society (IMLS), ??? (2018) Ahmed et al. [2022] Ahmed, K., Teso, S., Chang, K.-W., Broeck, G., Vergari, A.: Semantic Probabilistic Layers for Neuro-Symbolic Learning. In: Advances in Neural Information Processing Systems, vol. 35, pp. 29944–29959. Curran Associates, Inc., ??? (2022) [13] Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. 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Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . 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(2018) Ahmed et al. [2022] Ahmed, K., Teso, S., Chang, K.-W., Broeck, G., Vergari, A.: Semantic Probabilistic Layers for Neuro-Symbolic Learning. In: Advances in Neural Information Processing Systems, vol. 35, pp. 29944–29959. Curran Associates, Inc., ??? (2022) [13] Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. [2021] Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. 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[2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. 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In: Advances in Neural Information Processing Systems, vol. 35, pp. 29944–29959. Curran Associates, Inc., ??? (2022) [13] Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. [2021] Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. 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Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . 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Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Ledaguenel, A., Hudelot, C., Khouadjia, M.: Improving Neural-based Classification with Logical Background Knowledge (2024). https://arxiv.org/abs/2402.13019 Manhaeve et al. [2021] Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. 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Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence. IJCAI’07, pp. 2468–2473. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2007) Diligenti et al. [2017] Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. Artificial Intelligence 244, 143–165 (2017) https://doi.org/10.1016/j.artint.2015.08.011 Giannini et al. [2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. 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Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . 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Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . 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In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. 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[2018] Xu, J., Zhang, Z., Friedman, T., Liang, Y., Broeck, G.V.D.: A semantic loss function for deep learning with symbolic knowledge. In: 35th International Conference on Machine Learning, ICML 2018, vol. 12, pp. 8752–8760. International Machine Learning Society (IMLS), ??? (2018) Ahmed et al. [2022] Ahmed, K., Teso, S., Chang, K.-W., Broeck, G., Vergari, A.: Semantic Probabilistic Layers for Neuro-Symbolic Learning. In: Advances in Neural Information Processing Systems, vol. 35, pp. 29944–29959. Curran Associates, Inc., ??? (2022) [13] Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. 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In: Proceedings of the 20th International Joint Conference on Artifical Intelligence. IJCAI’07, pp. 2468–2473. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2007) Diligenti et al. [2017] Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. Artificial Intelligence 244, 143–165 (2017) https://doi.org/10.1016/j.artint.2015.08.011 Giannini et al. [2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Maene, J., De Raedt, L.: Soft-unification in deep probabilistic logic 36. Accessed 2024-02-21 Deng et al. [2014] Deng, J., Ding, N., Jia, Y., Frome, A., Murphy, K., Bengio, S., Li, Y., Neven, H., Adam, H.: Large-Scale Object Classification Using Label Relation Graphs. In: Computer Vision – ECCV 2014, pp. 48–64. Springer, ??? (2014) Xu et al. [2018] Xu, J., Zhang, Z., Friedman, T., Liang, Y., Broeck, G.V.D.: A semantic loss function for deep learning with symbolic knowledge. In: 35th International Conference on Machine Learning, ICML 2018, vol. 12, pp. 8752–8760. International Machine Learning Society (IMLS), ??? (2018) Ahmed et al. [2022] Ahmed, K., Teso, S., Chang, K.-W., Broeck, G., Vergari, A.: Semantic Probabilistic Layers for Neuro-Symbolic Learning. In: Advances in Neural Information Processing Systems, vol. 35, pp. 29944–29959. Curran Associates, Inc., ??? (2022) [13] Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. [2021] Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. [2024] Ledaguenel, A., Hudelot, C., Khouadjia, M.: Improving Neural-based Classification with Logical Background Knowledge (2024). https://arxiv.org/abs/2402.13019 Manhaeve et al. [2021] Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. Artificial Intelligence 298, 103504 (2021) https://doi.org/10.1016/j.artint.2021.103504 . Accessed 2023-08-07 De Raedt et al. [2007] De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence. IJCAI’07, pp. 2468–2473. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2007) Diligenti et al. [2017] Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. Artificial Intelligence 244, 143–165 (2017) https://doi.org/10.1016/j.artint.2015.08.011 Giannini et al. [2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Deng, J., Ding, N., Jia, Y., Frome, A., Murphy, K., Bengio, S., Li, Y., Neven, H., Adam, H.: Large-Scale Object Classification Using Label Relation Graphs. In: Computer Vision – ECCV 2014, pp. 48–64. Springer, ??? (2014) Xu et al. [2018] Xu, J., Zhang, Z., Friedman, T., Liang, Y., Broeck, G.V.D.: A semantic loss function for deep learning with symbolic knowledge. In: 35th International Conference on Machine Learning, ICML 2018, vol. 12, pp. 8752–8760. International Machine Learning Society (IMLS), ??? (2018) Ahmed et al. [2022] Ahmed, K., Teso, S., Chang, K.-W., Broeck, G., Vergari, A.: Semantic Probabilistic Layers for Neuro-Symbolic Learning. In: Advances in Neural Information Processing Systems, vol. 35, pp. 29944–29959. Curran Associates, Inc., ??? (2022) [13] Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. [2021] Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. [2024] Ledaguenel, A., Hudelot, C., Khouadjia, M.: Improving Neural-based Classification with Logical Background Knowledge (2024). https://arxiv.org/abs/2402.13019 Manhaeve et al. [2021] Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. Artificial Intelligence 298, 103504 (2021) https://doi.org/10.1016/j.artint.2021.103504 . Accessed 2023-08-07 De Raedt et al. [2007] De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence. IJCAI’07, pp. 2468–2473. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2007) Diligenti et al. [2017] Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. Artificial Intelligence 244, 143–165 (2017) https://doi.org/10.1016/j.artint.2015.08.011 Giannini et al. [2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Xu, J., Zhang, Z., Friedman, T., Liang, Y., Broeck, G.V.D.: A semantic loss function for deep learning with symbolic knowledge. In: 35th International Conference on Machine Learning, ICML 2018, vol. 12, pp. 8752–8760. International Machine Learning Society (IMLS), ??? (2018) Ahmed et al. [2022] Ahmed, K., Teso, S., Chang, K.-W., Broeck, G., Vergari, A.: Semantic Probabilistic Layers for Neuro-Symbolic Learning. In: Advances in Neural Information Processing Systems, vol. 35, pp. 29944–29959. Curran Associates, Inc., ??? (2022) [13] Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. [2021] Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. [2024] Ledaguenel, A., Hudelot, C., Khouadjia, M.: Improving Neural-based Classification with Logical Background Knowledge (2024). https://arxiv.org/abs/2402.13019 Manhaeve et al. [2021] Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. Artificial Intelligence 298, 103504 (2021) https://doi.org/10.1016/j.artint.2021.103504 . Accessed 2023-08-07 De Raedt et al. [2007] De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence. IJCAI’07, pp. 2468–2473. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2007) Diligenti et al. [2017] Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. Artificial Intelligence 244, 143–165 (2017) https://doi.org/10.1016/j.artint.2015.08.011 Giannini et al. [2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Ahmed, K., Teso, S., Chang, K.-W., Broeck, G., Vergari, A.: Semantic Probabilistic Layers for Neuro-Symbolic Learning. In: Advances in Neural Information Processing Systems, vol. 35, pp. 29944–29959. Curran Associates, Inc., ??? (2022) [13] Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. [2021] Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. [2024] Ledaguenel, A., Hudelot, C., Khouadjia, M.: Improving Neural-based Classification with Logical Background Knowledge (2024). https://arxiv.org/abs/2402.13019 Manhaeve et al. [2021] Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. Artificial Intelligence 298, 103504 (2021) https://doi.org/10.1016/j.artint.2021.103504 . Accessed 2023-08-07 De Raedt et al. [2007] De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence. IJCAI’07, pp. 2468–2473. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2007) Diligenti et al. [2017] Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. Artificial Intelligence 244, 143–165 (2017) https://doi.org/10.1016/j.artint.2015.08.011 Giannini et al. 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[2018] Xu, J., Zhang, Z., Friedman, T., Liang, Y., Broeck, G.V.D.: A semantic loss function for deep learning with symbolic knowledge. In: 35th International Conference on Machine Learning, ICML 2018, vol. 12, pp. 8752–8760. International Machine Learning Society (IMLS), ??? (2018) Ahmed et al. [2022] Ahmed, K., Teso, S., Chang, K.-W., Broeck, G., Vergari, A.: Semantic Probabilistic Layers for Neuro-Symbolic Learning. In: Advances in Neural Information Processing Systems, vol. 35, pp. 29944–29959. Curran Associates, Inc., ??? (2022) [13] Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. [2021] Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. [2024] Ledaguenel, A., Hudelot, C., Khouadjia, M.: Improving Neural-based Classification with Logical Background Knowledge (2024). https://arxiv.org/abs/2402.13019 Manhaeve et al. [2021] Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. Artificial Intelligence 298, 103504 (2021) https://doi.org/10.1016/j.artint.2021.103504 . Accessed 2023-08-07 De Raedt et al. [2007] De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence. IJCAI’07, pp. 2468–2473. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2007) Diligenti et al. [2017] Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. Artificial Intelligence 244, 143–165 (2017) https://doi.org/10.1016/j.artint.2015.08.011 Giannini et al. [2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. 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Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Maene, J., De Raedt, L.: Soft-unification in deep probabilistic logic 36. Accessed 2024-02-21 Deng et al. [2014] Deng, J., Ding, N., Jia, Y., Frome, A., Murphy, K., Bengio, S., Li, Y., Neven, H., Adam, H.: Large-Scale Object Classification Using Label Relation Graphs. In: Computer Vision – ECCV 2014, pp. 48–64. Springer, ??? (2014) Xu et al. [2018] Xu, J., Zhang, Z., Friedman, T., Liang, Y., Broeck, G.V.D.: A semantic loss function for deep learning with symbolic knowledge. In: 35th International Conference on Machine Learning, ICML 2018, vol. 12, pp. 8752–8760. International Machine Learning Society (IMLS), ??? (2018) Ahmed et al. [2022] Ahmed, K., Teso, S., Chang, K.-W., Broeck, G., Vergari, A.: Semantic Probabilistic Layers for Neuro-Symbolic Learning. In: Advances in Neural Information Processing Systems, vol. 35, pp. 29944–29959. Curran Associates, Inc., ??? (2022) [13] Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. [2021] Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. 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[2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Deng, J., Ding, N., Jia, Y., Frome, A., Murphy, K., Bengio, S., Li, Y., Neven, H., Adam, H.: Large-Scale Object Classification Using Label Relation Graphs. In: Computer Vision – ECCV 2014, pp. 48–64. Springer, ??? (2014) Xu et al. [2018] Xu, J., Zhang, Z., Friedman, T., Liang, Y., Broeck, G.V.D.: A semantic loss function for deep learning with symbolic knowledge. In: 35th International Conference on Machine Learning, ICML 2018, vol. 12, pp. 8752–8760. International Machine Learning Society (IMLS), ??? (2018) Ahmed et al. [2022] Ahmed, K., Teso, S., Chang, K.-W., Broeck, G., Vergari, A.: Semantic Probabilistic Layers for Neuro-Symbolic Learning. In: Advances in Neural Information Processing Systems, vol. 35, pp. 29944–29959. Curran Associates, Inc., ??? (2022) [13] Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. [2021] Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. [2024] Ledaguenel, A., Hudelot, C., Khouadjia, M.: Improving Neural-based Classification with Logical Background Knowledge (2024). https://arxiv.org/abs/2402.13019 Manhaeve et al. [2021] Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. Artificial Intelligence 298, 103504 (2021) https://doi.org/10.1016/j.artint.2021.103504 . Accessed 2023-08-07 De Raedt et al. [2007] De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence. IJCAI’07, pp. 2468–2473. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2007) Diligenti et al. [2017] Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. Artificial Intelligence 244, 143–165 (2017) https://doi.org/10.1016/j.artint.2015.08.011 Giannini et al. [2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . 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[2022] Ahmed, K., Teso, S., Chang, K.-W., Broeck, G., Vergari, A.: Semantic Probabilistic Layers for Neuro-Symbolic Learning. In: Advances in Neural Information Processing Systems, vol. 35, pp. 29944–29959. Curran Associates, Inc., ??? (2022) [13] Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. [2021] Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. [2024] Ledaguenel, A., Hudelot, C., Khouadjia, M.: Improving Neural-based Classification with Logical Background Knowledge (2024). https://arxiv.org/abs/2402.13019 Manhaeve et al. [2021] Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. Artificial Intelligence 298, 103504 (2021) https://doi.org/10.1016/j.artint.2021.103504 . Accessed 2023-08-07 De Raedt et al. [2007] De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence. IJCAI’07, pp. 2468–2473. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2007) Diligenti et al. [2017] Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. Artificial Intelligence 244, 143–165 (2017) https://doi.org/10.1016/j.artint.2015.08.011 Giannini et al. [2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Ahmed, K., Teso, S., Chang, K.-W., Broeck, G., Vergari, A.: Semantic Probabilistic Layers for Neuro-Symbolic Learning. In: Advances in Neural Information Processing Systems, vol. 35, pp. 29944–29959. Curran Associates, Inc., ??? (2022) [13] Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. [2021] Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. [2024] Ledaguenel, A., Hudelot, C., Khouadjia, M.: Improving Neural-based Classification with Logical Background Knowledge (2024). https://arxiv.org/abs/2402.13019 Manhaeve et al. [2021] Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. Artificial Intelligence 298, 103504 (2021) https://doi.org/10.1016/j.artint.2021.103504 . Accessed 2023-08-07 De Raedt et al. [2007] De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence. IJCAI’07, pp. 2468–2473. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2007) Diligenti et al. [2017] Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. Artificial Intelligence 244, 143–165 (2017) https://doi.org/10.1016/j.artint.2015.08.011 Giannini et al. [2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. [2021] Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. [2024] Ledaguenel, A., Hudelot, C., Khouadjia, M.: Improving Neural-based Classification with Logical Background Knowledge (2024). https://arxiv.org/abs/2402.13019 Manhaeve et al. [2021] Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. Artificial Intelligence 298, 103504 (2021) https://doi.org/10.1016/j.artint.2021.103504 . Accessed 2023-08-07 De Raedt et al. [2007] De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence. IJCAI’07, pp. 2468–2473. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2007) Diligenti et al. [2017] Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. Artificial Intelligence 244, 143–165 (2017) https://doi.org/10.1016/j.artint.2015.08.011 Giannini et al. [2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. [2024] Ledaguenel, A., Hudelot, C., Khouadjia, M.: Improving Neural-based Classification with Logical Background Knowledge (2024). https://arxiv.org/abs/2402.13019 Manhaeve et al. [2021] Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. Artificial Intelligence 298, 103504 (2021) https://doi.org/10.1016/j.artint.2021.103504 . Accessed 2023-08-07 De Raedt et al. [2007] De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence. IJCAI’07, pp. 2468–2473. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2007) Diligenti et al. [2017] Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. Artificial Intelligence 244, 143–165 (2017) https://doi.org/10.1016/j.artint.2015.08.011 Giannini et al. [2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Ledaguenel, A., Hudelot, C., Khouadjia, M.: Improving Neural-based Classification with Logical Background Knowledge (2024). https://arxiv.org/abs/2402.13019 Manhaeve et al. [2021] Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. Artificial Intelligence 298, 103504 (2021) https://doi.org/10.1016/j.artint.2021.103504 . Accessed 2023-08-07 De Raedt et al. [2007] De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence. IJCAI’07, pp. 2468–2473. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2007) Diligenti et al. [2017] Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. Artificial Intelligence 244, 143–165 (2017) https://doi.org/10.1016/j.artint.2015.08.011 Giannini et al. [2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. Artificial Intelligence 298, 103504 (2021) https://doi.org/10.1016/j.artint.2021.103504 . Accessed 2023-08-07 De Raedt et al. [2007] De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence. IJCAI’07, pp. 2468–2473. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2007) Diligenti et al. [2017] Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. Artificial Intelligence 244, 143–165 (2017) https://doi.org/10.1016/j.artint.2015.08.011 Giannini et al. [2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. 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[2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Xu, J., Zhang, Z., Friedman, T., Liang, Y., Broeck, G.V.D.: A semantic loss function for deep learning with symbolic knowledge. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Ahmed, K., Teso, S., Chang, K.-W., Broeck, G., Vergari, A.: Semantic Probabilistic Layers for Neuro-Symbolic Learning. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. [2021] Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. [2024] Ledaguenel, A., Hudelot, C., Khouadjia, M.: Improving Neural-based Classification with Logical Background Knowledge (2024). https://arxiv.org/abs/2402.13019 Manhaeve et al. [2021] Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. 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Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . 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[2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Ledaguenel, A., Hudelot, C., Khouadjia, M.: Improving Neural-based Classification with Logical Background Knowledge (2024). https://arxiv.org/abs/2402.13019 Manhaeve et al. [2021] Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. 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Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. Artificial Intelligence 298, 103504 (2021) https://doi.org/10.1016/j.artint.2021.103504 . Accessed 2023-08-07 De Raedt et al. [2007] De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence. IJCAI’07, pp. 2468–2473. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2007) Diligenti et al. [2017] Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. Artificial Intelligence 244, 143–165 (2017) https://doi.org/10.1016/j.artint.2015.08.011 Giannini et al. [2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. 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[2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Maene, J., De Raedt, L.: Soft-unification in deep probabilistic logic 36. Accessed 2024-02-21 Deng et al. [2014] Deng, J., Ding, N., Jia, Y., Frome, A., Murphy, K., Bengio, S., Li, Y., Neven, H., Adam, H.: Large-Scale Object Classification Using Label Relation Graphs. In: Computer Vision – ECCV 2014, pp. 48–64. Springer, ??? (2014) Xu et al. [2018] Xu, J., Zhang, Z., Friedman, T., Liang, Y., Broeck, G.V.D.: A semantic loss function for deep learning with symbolic knowledge. In: 35th International Conference on Machine Learning, ICML 2018, vol. 12, pp. 8752–8760. International Machine Learning Society (IMLS), ??? (2018) Ahmed et al. [2022] Ahmed, K., Teso, S., Chang, K.-W., Broeck, G., Vergari, A.: Semantic Probabilistic Layers for Neuro-Symbolic Learning. In: Advances in Neural Information Processing Systems, vol. 35, pp. 29944–29959. Curran Associates, Inc., ??? (2022) [13] Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. [2021] Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. [2024] Ledaguenel, A., Hudelot, C., Khouadjia, M.: Improving Neural-based Classification with Logical Background Knowledge (2024). https://arxiv.org/abs/2402.13019 Manhaeve et al. [2021] Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. Artificial Intelligence 298, 103504 (2021) https://doi.org/10.1016/j.artint.2021.103504 . Accessed 2023-08-07 De Raedt et al. [2007] De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence. IJCAI’07, pp. 2468–2473. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2007) Diligenti et al. [2017] Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. Artificial Intelligence 244, 143–165 (2017) https://doi.org/10.1016/j.artint.2015.08.011 Giannini et al. [2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Deng, J., Ding, N., Jia, Y., Frome, A., Murphy, K., Bengio, S., Li, Y., Neven, H., Adam, H.: Large-Scale Object Classification Using Label Relation Graphs. In: Computer Vision – ECCV 2014, pp. 48–64. Springer, ??? (2014) Xu et al. [2018] Xu, J., Zhang, Z., Friedman, T., Liang, Y., Broeck, G.V.D.: A semantic loss function for deep learning with symbolic knowledge. In: 35th International Conference on Machine Learning, ICML 2018, vol. 12, pp. 8752–8760. International Machine Learning Society (IMLS), ??? (2018) Ahmed et al. [2022] Ahmed, K., Teso, S., Chang, K.-W., Broeck, G., Vergari, A.: Semantic Probabilistic Layers for Neuro-Symbolic Learning. In: Advances in Neural Information Processing Systems, vol. 35, pp. 29944–29959. Curran Associates, Inc., ??? (2022) [13] Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. [2021] Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. [2024] Ledaguenel, A., Hudelot, C., Khouadjia, M.: Improving Neural-based Classification with Logical Background Knowledge (2024). https://arxiv.org/abs/2402.13019 Manhaeve et al. [2021] Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. Artificial Intelligence 298, 103504 (2021) https://doi.org/10.1016/j.artint.2021.103504 . Accessed 2023-08-07 De Raedt et al. [2007] De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence. IJCAI’07, pp. 2468–2473. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2007) Diligenti et al. [2017] Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. Artificial Intelligence 244, 143–165 (2017) https://doi.org/10.1016/j.artint.2015.08.011 Giannini et al. [2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Xu, J., Zhang, Z., Friedman, T., Liang, Y., Broeck, G.V.D.: A semantic loss function for deep learning with symbolic knowledge. In: 35th International Conference on Machine Learning, ICML 2018, vol. 12, pp. 8752–8760. International Machine Learning Society (IMLS), ??? (2018) Ahmed et al. [2022] Ahmed, K., Teso, S., Chang, K.-W., Broeck, G., Vergari, A.: Semantic Probabilistic Layers for Neuro-Symbolic Learning. In: Advances in Neural Information Processing Systems, vol. 35, pp. 29944–29959. Curran Associates, Inc., ??? (2022) [13] Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. [2021] Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. [2024] Ledaguenel, A., Hudelot, C., Khouadjia, M.: Improving Neural-based Classification with Logical Background Knowledge (2024). https://arxiv.org/abs/2402.13019 Manhaeve et al. [2021] Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. Artificial Intelligence 298, 103504 (2021) https://doi.org/10.1016/j.artint.2021.103504 . Accessed 2023-08-07 De Raedt et al. [2007] De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence. IJCAI’07, pp. 2468–2473. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2007) Diligenti et al. [2017] Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. Artificial Intelligence 244, 143–165 (2017) https://doi.org/10.1016/j.artint.2015.08.011 Giannini et al. [2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Ahmed, K., Teso, S., Chang, K.-W., Broeck, G., Vergari, A.: Semantic Probabilistic Layers for Neuro-Symbolic Learning. In: Advances in Neural Information Processing Systems, vol. 35, pp. 29944–29959. Curran Associates, Inc., ??? (2022) [13] Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. [2021] Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. [2024] Ledaguenel, A., Hudelot, C., Khouadjia, M.: Improving Neural-based Classification with Logical Background Knowledge (2024). https://arxiv.org/abs/2402.13019 Manhaeve et al. [2021] Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. Artificial Intelligence 298, 103504 (2021) https://doi.org/10.1016/j.artint.2021.103504 . Accessed 2023-08-07 De Raedt et al. [2007] De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence. IJCAI’07, pp. 2468–2473. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2007) Diligenti et al. [2017] Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. Artificial Intelligence 244, 143–165 (2017) https://doi.org/10.1016/j.artint.2015.08.011 Giannini et al. [2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. 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Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Xu, J., Zhang, Z., Friedman, T., Liang, Y., Broeck, G.V.D.: A semantic loss function for deep learning with symbolic knowledge. 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(2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. [2024] Ledaguenel, A., Hudelot, C., Khouadjia, M.: Improving Neural-based Classification with Logical Background Knowledge (2024). https://arxiv.org/abs/2402.13019 Manhaeve et al. [2021] Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. Artificial Intelligence 298, 103504 (2021) https://doi.org/10.1016/j.artint.2021.103504 . Accessed 2023-08-07 De Raedt et al. [2007] De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence. IJCAI’07, pp. 2468–2473. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2007) Diligenti et al. [2017] Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Ahmed, K., Teso, S., Chang, K.-W., Broeck, G., Vergari, A.: Semantic Probabilistic Layers for Neuro-Symbolic Learning. In: Advances in Neural Information Processing Systems, vol. 35, pp. 29944–29959. Curran Associates, Inc., ??? (2022) [13] Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. [2021] Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. 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[2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. In: Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pp. 1755–1762. International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2020/243 Niepert et al. [2021] Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. [2024] Ledaguenel, A., Hudelot, C., Khouadjia, M.: Improving Neural-based Classification with Logical Background Knowledge (2024). https://arxiv.org/abs/2402.13019 Manhaeve et al. [2021] Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. Artificial Intelligence 298, 103504 (2021) https://doi.org/10.1016/j.artint.2021.103504 . Accessed 2023-08-07 De Raedt et al. [2007] De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence. IJCAI’07, pp. 2468–2473. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2007) Diligenti et al. [2017] Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. Artificial Intelligence 244, 143–165 (2017) https://doi.org/10.1016/j.artint.2015.08.011 Giannini et al. [2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Niepert, M., Minervini, P., Franceschi, L.: Implicit MLE: Backpropagating through discrete exponential family distributions. In: Advances in Neural Information Processing Systems, vol. 34, pp. 14567–14579. Curran Associates, Inc., ??? (2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. [2024] Ledaguenel, A., Hudelot, C., Khouadjia, M.: Improving Neural-based Classification with Logical Background Knowledge (2024). https://arxiv.org/abs/2402.13019 Manhaeve et al. [2021] Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. Artificial Intelligence 298, 103504 (2021) https://doi.org/10.1016/j.artint.2021.103504 . Accessed 2023-08-07 De Raedt et al. [2007] De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence. IJCAI’07, pp. 2468–2473. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2007) Diligenti et al. [2017] Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. Artificial Intelligence 244, 143–165 (2017) https://doi.org/10.1016/j.artint.2015.08.011 Giannini et al. [2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Ledaguenel, A., Hudelot, C., Khouadjia, M.: Improving Neural-based Classification with Logical Background Knowledge (2024). https://arxiv.org/abs/2402.13019 Manhaeve et al. [2021] Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. Artificial Intelligence 298, 103504 (2021) https://doi.org/10.1016/j.artint.2021.103504 . Accessed 2023-08-07 De Raedt et al. [2007] De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence. IJCAI’07, pp. 2468–2473. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2007) Diligenti et al. [2017] Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. Artificial Intelligence 244, 143–165 (2017) https://doi.org/10.1016/j.artint.2015.08.011 Giannini et al. [2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. 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Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. 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In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. 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[2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. 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Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. 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In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. 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Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . 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(2021). https://proceedings.neurips.cc/paper_files/paper/2021/hash/7a430339c10c642c4b2251756fd1b484-Abstract.html Accessed 2024-01-17 Ledaguenel et al. [2024] Ledaguenel, A., Hudelot, C., Khouadjia, M.: Improving Neural-based Classification with Logical Background Knowledge (2024). https://arxiv.org/abs/2402.13019 Manhaeve et al. [2021] Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. Artificial Intelligence 298, 103504 (2021) https://doi.org/10.1016/j.artint.2021.103504 . Accessed 2023-08-07 De Raedt et al. [2007] De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence. IJCAI’07, pp. 2468–2473. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2007) Diligenti et al. [2017] Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. 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(2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. 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In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Toda, S.: PP is as hard as the polynomial-time hierarchy. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Yang, Z., Ishay, A., Lee, J.: NeurASP: Embracing neural networks into answer set programming. 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Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 De Raedt, L., Kimmig, A., Toivonen, H.: ProbLog: a probabilistic prolog and its application in link discovery. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence. IJCAI’07, pp. 2468–2473. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2007) Diligenti et al. [2017] Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. Artificial Intelligence 244, 143–165 (2017) https://doi.org/10.1016/j.artint.2015.08.011 Giannini et al. [2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. 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Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . 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Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . 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In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. 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In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Toda, S.: PP is as hard as the polynomial-time hierarchy. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. 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Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. 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(2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . 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Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11
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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., De Raedt, L.: Neural probabilistic logic programming in DeepProbLog. 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Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. 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Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. 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In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. 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(2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. 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Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11
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Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. 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In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. 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(2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. 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Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . 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Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . 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In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11
- Diligenti, M., Gori, M., Saccà, C.: Semantic-based regularization for learning and inference. Artificial Intelligence 244, 143–165 (2017) https://doi.org/10.1016/j.artint.2015.08.011 Giannini et al. [2023] Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. 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Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11
- Giannini, F., Diligenti, M., Maggini, M., Gori, M., Marra, G.: T-norms driven loss functions for machine learning. Applied Intelligence 53(15), 18775–18789 (2023) https://doi.org/10.1007/s10489-022-04383-6 . Accessed 2023-08-07 Badreddine et al. [2022] Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Badreddine, S., Garcez, A.d., Serafini, L., Spranger, M.: Logic Tensor Networks. Artificial Intelligence 303, 103649 (2022) https://doi.org/10.1016/j.artint.2021.103649 [21] Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. [2019] Pogančić, M.V., Paulus, A., Musil, V., Martius, G., Rolinek, M.: Differentiation of Blackbox Combinatorial Solvers. (2019). https://openreview.net/forum?id=BkevoJSYPB Accessed 2023-10-27 Ahmed et al. [2022] Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence. Proceedings of Machine Learning Research, vol. 180, pp. 43–53. PMLR, ??? (2022). https://proceedings.mlr.press/v180/ahmed22a.html [26] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Darwiche, A., Marquis, P.: A knowledge compilation map 17(1), 229–264 Darwiche [2011] Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (2011) Toda [1991] Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991) https://doi.org/10.1137/0220053 . Publisher: Society for Industrial and Applied Mathematics. Accessed 2024-01-17 Pogančić et al. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Darwiche, A.: SDD: A New Canonical Representation of Propositional Knowledge Bases. 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Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. 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Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20–22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. 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Watson Research Center, Yorktown Heights, New York, and Sponsored by the Office of Naval Research, Mathematics Program, IBM World Trade Corporation, and the IBM Research Mathematical Sciences Department. The IBM Research Symposia Series, pp. 85–103. Springer. https://doi.org/10.1007/978-1-4684-2001-2_9 . https://doi.org/10.1007/978-1-4684-2001-2_9 Accessed 2024-03-11 [27] Valiant, L.G.: The complexity of enumeration and reliability problems 8(3), 410–421 https://doi.org/10.1137/0208032 . Accessed 2024-02-21 [28] Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices 69B(1), https://doi.org/10.6028/jres.069b.013 . Number: 1 and 2. Accessed 2024-03-11 [29] Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11 Ahmed, K., Wang, E., Chang, K.-W., Broeck, G.: Neuro-symbolic entropy regularization. In: Cussens, J., Zhang, K. (eds.) 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- Amarilli, A., Monet, M.: Weighted Counting of Matchings in Unbounded-Treewidth Graph Families. arXiv. https://doi.org/10.48550/arXiv.2205.00851 . http://arxiv.org/abs/2205.00851 Accessed 2024-03-11
- Arthur Ledaguenel (3 papers)
- Céline Hudelot (50 papers)
- Mostepha Khouadjia (3 papers)