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References

📍 Where we are: The evidence base for the whole volume, gathered in one place.

Every inline marker like [1] in a chapter resolves here. References are grouped by chapter, and the numbering is local to each chapter — each chapter's list restarts at [1], so find the chapter heading first, then the number. A citation appears inline as [k], where the fragment names the chapter and k is the entry's number within that chapter's list. Each entry gives the author, title, venue, and year, is annotated with what it supports, and closes with a bracketed evidence-tier tag.

  1. Nickel, M., Murphy, K., Tresp, V., and Gabrilovich, E. "A Review of Relational Machine Learning for Knowledge Graphs." Proceedings of the IEEE, 2016. — The canonical survey of knowledge-graph completion. [Evidence tier: peer-reviewed]
  2. Bordes, A., Usunier, N., Garcia-Durán, A., Weston, J., and Yakhnenko, O. "Translating Embeddings for Modeling Multi-relational Data." NeurIPS, 2013. — The TransE paper; introduced the ranking protocol used here. [Evidence tier: peer-reviewed]
  3. Dettmers, T., Minervini, P., Stenetorp, P., and Riedel, S. "Convolutional 2D Knowledge Graph Embeddings." AAAI, 2018. — Introduced the WN18RR benchmark by removing WN18's inverse-relation leakage; source of the benchmark scale quoted here. [Evidence tier: peer-reviewed]
  4. Toutanova, K., and Chen, D. "Observed versus Latent Features for Knowledge Base and Text Inference." Workshop on Continuous Vector Space Models and their Compositionality, 2015. — Test leakage through inverse relations; why filtered evaluation and careful splits matter. [Evidence tier: peer-reviewed]
  5. Ruffinelli, D., Broscheit, S., and Gemulla, R. "You CAN Teach an Old Dog New Tricks! On Training Knowledge Graph Embeddings." ICLR, 2020. — Evaluation methodology and the outsized role of training protocol. [Evidence tier: peer-reviewed]

Translational Models: TransE and Its Family​

  1. Bordes, A., Usunier, N., Garcia-Durán, A., Weston, J., and Yakhnenko, O. "Translating Embeddings for Modeling Multi-relational Data." NeurIPS, 2013. — TransE: relations as translations, margin ranking loss. [Evidence tier: peer-reviewed]
  2. Wang, Z., Zhang, J., Feng, J., and Chen, Z. "Knowledge Graph Embedding by Translating on Hyperplanes." AAAI, 2014. — TransH: per-relation hyperplane projection for 1-to-many relations. [Evidence tier: peer-reviewed]
  3. Lin, Y., Liu, Z., Sun, M., Liu, Y., and Zhu, X. "Learning Entity and Relation Embeddings for Knowledge Graph Completion." AAAI, 2015. — TransR: separate entity and relation spaces. [Evidence tier: peer-reviewed]
  4. Sun, Z., Deng, Z.-H., Nie, J.-Y., and Tang, J. "RotatE: Knowledge Graph Embedding by Relational Rotation in Complex Space." ICLR, 2019. — Relations as rotations; relation-pattern analysis (symmetry, antisymmetry, composition). [Evidence tier: peer-reviewed]

Bilinear Models: DistMult and ComplEx​

  1. Nickel, M., Tresp, V., and Kriegel, H.-P. "A Three-Way Model for Collective Learning on Multi-Relational Data." ICML, 2011. — RESCAL: the full bilinear tensor factorization. [Evidence tier: peer-reviewed]
  2. Yang, B., Yih, W., He, X., Gao, J., and Deng, L. "Embedding Entities and Relations for Learning and Inference in Knowledge Bases." ICLR, 2015. — DistMult: the diagonal bilinear model. [Evidence tier: peer-reviewed]
  3. Trouillon, T., Welbl, J., Riedel, S., Gaussier, É., and Bouchard, G. "Complex Embeddings for Simple Link Prediction." ICML, 2016. — ComplEx: complex-valued embeddings with a conjugated tail; the single-relation expressiveness result. [Evidence tier: peer-reviewed]
  4. Trouillon, T., Dance, C. R., Gaussier, É., Welbl, J., Riedel, S., and Bouchard, G. "Knowledge Graph Completion via Complex Tensor Factorization." JMLR, 2017. — The multi-relational full-expressiveness theorem for ComplEx, with an explicit bound on the embedding dimension; source of the capacity guarantee. [Evidence tier: peer-reviewed]
  5. Kazemi, S. M., and Poole, D. "SimplE Embedding for Link Prediction in Knowledge Graphs." NeurIPS, 2018. — Another fully expressive bilinear repair; context for the design space. [Evidence tier: peer-reviewed]

Balls and Cones: Concepts as Regions​

  1. Erk, K. "Representing Words as Regions in Vector Space." CoNLL, 2009. — The early case for regions over points in lexical semantics. [Evidence tier: peer-reviewed]
  2. Vendrov, I., Kiros, R., Fidler, S., and Urtasun, R. "Order-Embeddings of Images and Language." ICLR, 2016. — The reversed-product-order (cone) reading of entailment. [Evidence tier: peer-reviewed]
  3. Ganea, O.-E., Bécigneul, G., and Hofmann, T. "Hyperbolic Entailment Cones for Learning Hierarchical Embeddings." ICML, 2018. — Entailment cones; geodesically convex regions for is-a. [Evidence tier: peer-reviewed]
  4. Vilnis, L., and McCallum, A. "Word Representations via Gaussian Embedding." ICLR, 2015. — Densities as regions; asymmetric entailment by KL divergence. [Evidence tier: peer-reviewed]

Box Embeddings: Query2Box Geometry​

  1. Ren, H., Hu, W., and Leskovec, J. "Query2box: Reasoning over Knowledge Graphs in Vector Space Using Box Embeddings." ICLR, 2020. — The box query calculus this chapter reimplements. [Evidence tier: peer-reviewed]
  2. Hamilton, W. L., Bajaj, P., Zitnik, M., Jurafsky, D., and Leskovec, J. "Embedding Logical Queries on Knowledge Graphs." NeurIPS, 2018. — GQE: the first end-to-end embedded conjunctive queries. [Evidence tier: peer-reviewed]
  3. Arakelyan, E., Daza, D., Minervini, P., and Cochez, M. "Complex Query Answering with Neural Link Predictors." ICLR, 2021. — CQD: answering the same queries by fuzzy composition of 1-hop predictors; the main alternative paradigm. [Evidence tier: peer-reviewed]
  4. Vilnis, L., Li, X., Murty, S., and McCallum, A. "Probabilistic Embedding of Knowledge Graphs with Box Lattice Measures." ACL, 2018. — Boxes as a probabilistic lattice; the box calculus’ measure-theoretic side. [Evidence tier: peer-reviewed]

Beta and Probabilistic Embeddings: Buying Negation with Densities​

  1. Ren, H., and Leskovec, J. "Beta Embeddings for Multi-Hop Logical Reasoning in Knowledge Graphs." NeurIPS, 2020. — BetaE: probabilistic embeddings with closed-form negation. [Evidence tier: peer-reviewed]
  2. Ren, H., Galkin, M., Cochez, M., Zhu, Z., and Leskovec, J. "Neural Graph Reasoning: Complex Logical Query Answering Meets Graph Databases." arXiv:2303.14617, 2023. — The survey mapping this whole query-embedding landscape. [Evidence tier: preprint]
  3. Zhang, Z., Wang, J., Chen, J., Ji, S., and Wu, F. "ConE: Cone Embeddings for Multi-Hop Reasoning over Knowledge Graphs." NeurIPS, 2021. — Another negation-capable geometry (cones); context for the design space. [Evidence tier: peer-reviewed]
  4. Chen, X., Hu, Z., and Sun, Y. "Fuzzy Logic Based Logical Query Answering on Knowledge Graphs." AAAI, 2022. — FuzzQE: t-norm fuzzy logic as the query algebra; the fuzzy alternative to densities. [Evidence tier: peer-reviewed]
  5. Arakelyan, E., Daza, D., Minervini, P., and Cochez, M. "Complex Query Answering with Neural Link Predictors." ICLR, 2021. — CQD (Continuous Query Decomposition): answering complex queries by fuzzy composition of pretrained 1-hop predictors; the score-level composition style the companion code follows. [Evidence tier: peer-reviewed]

Boxes versus Balls: An Expressiveness Story​

  1. Ren, H., Hu, W., and Leskovec, J. "Query2box: Reasoning over Knowledge Graphs in Vector Space Using Box Embeddings." ICLR, 2020. — Why boxes: closure under intersection for conjunctive queries. [Evidence tier: peer-reviewed]
  2. Vilnis, L., Li, X., Murty, S., and McCallum, A. "Probabilistic Embedding of Knowledge Graphs with Box Lattice Measures." ACL, 2018. — The box lattice: intersection as a lattice meet. [Evidence tier: peer-reviewed]
  3. Xiong, B., Potyka, N., Tran, T.-K., Nayyeri, M., and Staab, S. "Faithful Embeddings for EL++ Knowledge Bases." ISWC, 2022. — BoxEL: the closure argument carried into ontology embedding (next Part’s subject). [Evidence tier: peer-reviewed]
  4. Abboud, R., Ceylan, İ. İ., Lukasiewicz, T., and Salvatori, T. "BoxE: A Box Embedding Model for Knowledge Base Completion." NeurIPS, 2020. — Box calculi for facts and rules; expressiveness analysis. [Evidence tier: peer-reviewed]

EL Embeddings: Geometry Meets Logic​

  1. Kulmanov, M., Liu-Wei, W., Yan, Y., and Hoehndorf, R. "EL Embeddings: Geometric Construction of Models for the Description Logic EL++." IJCAI, 2019. — ELEm: the normal-form losses over balls this chapter reimplements. [Evidence tier: peer-reviewed]
  2. Baader, F., Brandt, S., and Lutz, C. "Pushing the EL Envelope." IJCAI, 2005. — The EL++ logic and normal forms the losses target (Volume 2’s foundation). [Evidence tier: peer-reviewed]
  3. Smaili, F. Z., Gao, X., and Hoehndorf, R. "Onto2Vec: Joint Vector-Based Representation of Biological Entities and Their Ontology-Based Annotations." Bioinformatics, 2018. — The pre-geometric baseline: ontologies as text for embeddings; what ELEm improved on. [Evidence tier: peer-reviewed]
  4. Chen, J., Hu, P., Jiménez-Ruiz, E., Holter, O. M., Antonyrajah, D., and Horrocks, I. "OWL2Vec*: Embedding of OWL Ontologies." Machine Learning, 2021. — The random-walk alternative for whole OWL ontologies. [Evidence tier: peer-reviewed]
  5. Xiong, B., Potyka, N., Tran, T.-K., Nayyeri, M., and Staab, S. "Faithful Embeddings for EL++ Knowledge Bases." ISWC, 2022. — BoxEL: the successor whose subsumption-prediction evaluation on GALEN, the Gene Ontology, and the anatomy ontology reports the ball baseline degrading. [Evidence tier: peer-reviewed]
  6. Jackermeier, M., Chen, J., and Horrocks, I. "Dual Box Embeddings for the Description Logic EL++." WWW, 2024. — Box²EL: the same GALEN / Gene Ontology / anatomy subsumption benchmarks, again with the ELEm ball model as the weaker baseline. [Evidence tier: peer-reviewed]

BoxEL and Box²EL: Faithful Ontology Embeddings​

  1. Kulmanov, M., Liu-Wei, W., Yan, Y., and Hoehndorf, R. "EL Embeddings: Geometric Construction of Models for the Description Logic EL++." IJCAI, 2019. — The ball baseline being compared against. [Evidence tier: peer-reviewed]
  2. Xiong, B., Potyka, N., Tran, T.-K., Nayyeri, M., and Staab, S. "Faithful Embeddings for EL++ Knowledge Bases." ISWC, 2022. — BoxEL: boxes with volume-based semantics for EL++. [Evidence tier: peer-reviewed]
  3. Peng, X., Tang, Z., Kulmanov, M., Niu, K., and Hoehndorf, R. "Description Logic EL++ Embeddings with Intersectional Closure." arXiv:2202.14018, 2022. — ELBE: boxes chosen expressly for intersectional closure. [Evidence tier: preprint]
  4. Jackermeier, M., Chen, J., and Horrocks, I. "Dual Box Embeddings for the Description Logic EL++." WWW, 2024. — Box²EL: concept and role boxes with bump vectors. [Evidence tier: peer-reviewed]

TransBox and mOWL: Closure and Tooling​

  1. Yang, H., Chen, J., and Sattler, U. "TransBox: EL++-closed Ontology Embedding." WWW, 2025. — EL++-closure, role boxes, and the soundness theorem. [Evidence tier: peer-reviewed]
  2. Zhapa-Camacho, F., Kulmanov, M., and Hoehndorf, R. "mOWL: Python Library for Machine Learning with Biomedical Ontologies." Bioinformatics, 2023. — The toolkit: normalization, datasets, models, evaluation. [Evidence tier: peer-reviewed]
  3. Jackermeier, M., Chen, J., and Horrocks, I. "Dual Box Embeddings for the Description Logic EL++." WWW, 2024. — Box²EL, whose codebase TransBox extends; the bump mechanism. [Evidence tier: peer-reviewed]
  4. Chen, J., Hu, P., Jiménez-Ruiz, E., Holter, O. M., Antonyrajah, D., and Horrocks, I. "OWL2Vec*: Embedding of OWL Ontologies." Machine Learning, 2021. — The graph/text projection alternative in the same ecosystem. [Evidence tier: peer-reviewed]

Hyperbolic Embeddings: Hierarchy in Curved Space​

  1. Sala, F., De Sa, C., Gu, A., and Ré, C. "Representation Tradeoffs for Hyperbolic Embeddings." ICML, 2018. — The distortion/dimension trade-off made precise. [Evidence tier: peer-reviewed]
  2. Nickel, M., and Kiela, D. "Poincaré Embeddings for Learning Hierarchical Representations." NeurIPS, 2017. — The Poincaré-ball embedding and Riemannian SGD recipe. [Evidence tier: peer-reviewed]
  3. Chami, I., Wolf, A., Juan, D.-C., Sala, F., Ravi, S., and Ré, C. "Low-Dimensional Hyperbolic Knowledge Graph Embeddings." ACL, 2020. — Hyperbolic KG embeddings at low dimension. [Evidence tier: peer-reviewed]
  4. Ganea, O.-E., Bécigneul, G., and Hofmann, T. "Hyperbolic Entailment Cones for Learning Hierarchical Embeddings." ICML, 2018. — Entailment cones in hyperbolic space. [Evidence tier: peer-reviewed]

Message Passing: The GNN Blueprint​

  1. Gilmer, J., Schoenholz, S. S., Riley, P. F., Vinyals, O., and Dahl, G. E. "Neural Message Passing for Quantum Chemistry." ICML, 2017. — The message-passing abstraction (MPNN). [Evidence tier: peer-reviewed]
  2. Battaglia, P. W., et al. "Relational Inductive Biases, Deep Learning, and Graph Networks." arXiv:1806.01261, 2018. — The blueprint generalized; graphs as an inductive bias. [Evidence tier: preprint]
  3. Kipf, T. N., and Welling, M. "Semi-Supervised Classification with Graph Convolutional Networks." ICLR, 2017. — GCN: the normalized-adjacency layer. [Evidence tier: peer-reviewed]
  4. Hamilton, W. L., Ying, R., and Leskovec, J. "Inductive Representation Learning on Large Graphs." NeurIPS, 2017. — GraphSAGE: sampled aggregation, inductive setting. [Evidence tier: peer-reviewed]

Relational GNNs: R-GCN and Beyond​

  1. Schlichtkrull, M., Kipf, T. N., Bloem, P., van den Berg, R., Titov, I., and Welling, M. "Modeling Relational Data with Graph Convolutional Networks." ESWC, 2018. — R-GCN: relational message passing, basis decomposition. [Evidence tier: peer-reviewed]
  2. Zhu, J., Yan, Y., Zhao, L., Heimann, M., Akoglu, L., and Koutra, D. "Beyond Homophily in Graph Neural Networks: Current Limitations and Effective Designs." NeurIPS, 2020. — Heterophily: where neighbor averaging fails, and the designs that survive it. [Evidence tier: peer-reviewed]
  3. Vashishth, S., Sanyal, S., Nitin, V., and Talukdar, P. "Composition-Based Multi-Relational Graph Convolutional Networks." ICLR, 2020. — CompGCN: composing relation embeddings into messages. [Evidence tier: peer-reviewed]
  4. Zhu, Z., Zhang, Z., Xhonneux, L.-P., and Tang, J. "Neural Bellman-Ford Networks: A General Graph Neural Network Framework for Link Prediction." NeurIPS, 2021. — NBFNet: path-formulation of relational message passing. [Evidence tier: peer-reviewed]
  5. Galkin, M., Yuan, X., Mostafa, H., Tang, J., and Zhu, Z. "Towards Foundation Models for Knowledge Graph Reasoning." ICLR, 2024. — ULTRA: relation-agnostic transfer; where relational GNNs are heading. [Evidence tier: peer-reviewed]

The Expressiveness Ceiling: 1-WL, C², and Graded Modal Logic​

  1. Xu, K., Hu, W., Leskovec, J., and Jegelka, S. "How Powerful are Graph Neural Networks?" ICLR, 2019. — GIN; GNN power is bounded by 1-WL; sum-aggregation injectivity. [Evidence tier: peer-reviewed]
  2. Morris, C., Ritzert, M., Fey, M., Hamilton, W. L., Lenssen, J. E., Rattan, G., and Grohe, M. "Weisfeiler and Leman Go Neural: Higher-Order Graph Neural Networks." AAAI, 2019. — The 1-WL equivalence and k-WL hierarchy. [Evidence tier: peer-reviewed]
  3. Cai, J.-Y., Fürer, M., and Immerman, N. "An Optimal Lower Bound on the Number of Variables for Graph Identification." Combinatorica, 1992. — k-WL equals C^{k+1}: the counting-logic characterization. [Evidence tier: peer-reviewed]
  4. Barceló, P., Kostylev, E. V., Monet, M., Pérez, J., Reutter, J., and Silva, J.-P. "The Logical Expressiveness of Graph Neural Networks." ICLR, 2020. — GNN node classifiers and graded modal logic (every unary C² classifier is expressible by an ACR-GNN). [Evidence tier: peer-reviewed]
  5. Grohe, M. "The Logic of Graph Neural Networks." LICS, 2021. — The survey uniting WL, logic, and GNNs. [Evidence tier: peer-reviewed]
  6. Immerman, N., and Lander, E. "Describing Graphs: A First-Order Approach to Graph Canonization." Complexity Theory Retrospective, Springer, 1990. — Color refinement equals C²: the original two-variable counting-logic characterization. [Evidence tier: peer-reviewed]

Attention: Reasoning by Relevance​

  1. Graves, A., Wayne, G., and Danihelka, I. "Neural Turing Machines." arXiv:1410.5401, 2014. — Content-addressed differentiable memory; attention as soft lookup made explicit. [Evidence tier: preprint]
  2. Bahdanau, D., Cho, K., and Bengio, Y. "Neural Machine Translation by Jointly Learning to Align and Translate." ICLR, 2015. — The original content-based attention. [Evidence tier: peer-reviewed]
  3. Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A. N., Kaiser, Ł., and Polosukhin, I. "Attention Is All You Need." NeurIPS, 2017. — Scaled dot-product and multi-head attention; the √d_k argument. [Evidence tier: peer-reviewed]
  4. Veličković, P., Cucurull, G., Casanova, A., Romero, A., Liò, P., and Bengio, Y. "Graph Attention Networks." ICLR, 2018. — Attention as learned edge weights on a graph (the message-passing tie). [Evidence tier: peer-reviewed]

Vector Symbolic Architectures: Binding and Superposition​

  1. Plate, T. A. "Holographic Reduced Representations." IEEE Transactions on Neural Networks, 1995. — HRR: circular-convolution binding, capacity analysis. [Evidence tier: peer-reviewed]
  2. Smolensky, P. "Tensor Product Variable Binding and the Representation of Symbolic Structures in Connectionist Systems." Artificial Intelligence, 1990. — The exact tensor-product ancestor of all binding. [Evidence tier: peer-reviewed]
  3. Kanerva, P. "Hyperdimensional Computing: An Introduction to Computing in Distributed Representation with High-Dimensional Random Vectors." Cognitive Computation, 2009. — The hyperdimensional view: robustness from dimensionality. [Evidence tier: peer-reviewed]
  4. Kleyko, D., Rachkovskij, D. A., Osipov, E., and Rahimi, A. "A Survey on Hyperdimensional Computing aka Vector Symbolic Architectures, Part I: Models and Data Transformations." ACM Computing Surveys, 2022. — The family, unified and compared. [Evidence tier: peer-reviewed]
  5. Elhage, N., Hume, T., Olsson, C., Schiefer, N., Henighan, T., Kravec, S., Hatfield-Dodds, Z., Lasenby, R., Drain, D., Chen, C., Grosse, R., McCandlish, S., Kaplan, J., Amodei, D., Wattenberg, M., and Olah, C. "Toy Models of Superposition." arXiv:2209.10652, 2022. — Feature superposition in trained networks: more features than dimensions, packed as nearly orthogonal directions. [Evidence tier: preprint]

Soft Unification: Matching Symbols in Vector Space​

  1. Robinson, J. A. "A Machine-Oriented Logic Based on the Resolution Principle." Journal of the ACM, 1965. — The hard unification being softened (Volume 1’s foundation). [Evidence tier: peer-reviewed]
  2. Rocktäschel, T., and Riedel, S. "End-to-End Differentiable Proving." NeurIPS, 2017. — NTP: backward chaining with RBF-kernel soft unification. [Evidence tier: peer-reviewed]
  3. Minervini, P., Bošnjak, M., Rocktäschel, T., Riedel, S., and Grefenstette, E. "Differentiable Reasoning on Large Knowledge Bases and Natural Language." AAAI, 2020. — GNTP: making differentiable proving scale. [Evidence tier: peer-reviewed]
  4. Minervini, P., Riedel, S., Stenetorp, P., Grefenstette, E., and Rocktäschel, T. "Learning Reasoning Strategies in End-to-End Differentiable Proving." ICML, 2020. — Conditional Theorem Provers: learned rule selection. [Evidence tier: peer-reviewed]

The Honest Verdict: What Vectors Can and Cannot Hold​

  1. Nickel, M., Murphy, K., Tresp, V., and Gabrilovich, E. "A Review of Relational Machine Learning for Knowledge Graphs." Proceedings of the IEEE, 2016. — What embedding methods deliver and where they stop. [Evidence tier: peer-reviewed]
  2. Xu, K., Hu, W., Leskovec, J., and Jegelka, S. "How Powerful are Graph Neural Networks?" ICLR, 2019. — The expressiveness ceiling as a hard limit on the neural pillar. [Evidence tier: peer-reviewed]
  3. Ruffinelli, D., Broscheit, S., and Gemulla, R. "You CAN Teach an Old Dog New Tricks! On Training Knowledge Graph Embeddings." ICLR, 2020. — The sobering methodology result: training protocol rivals model choice. [Evidence tier: peer-reviewed]
  4. Garcez, A. d’Avila, and Lamb, L. C. "Neurosymbolic AI: The 3rd Wave." Artificial Intelligence Review, 2023. — Why the division of labor points to integration (Volume 4’s thesis). [Evidence tier: peer-reviewed]