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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]](/integration/references#chapter-id), 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.

Many-Valued Logic: Beyond True and False​

  1. Łukasiewicz, J. "O logice trójwartościowej" ("On Three-Valued Logic"). Ruch Filozoficzny, 1920. — The original three-valued logic; the Ł3 implication and the possible-reading of the middle value. [Evidence tier: peer-reviewed]
  2. Kleene, S. C. Introduction to Metamathematics. North-Holland, 1952. — K3: the strong three-valued connectives and the undefined-reading of the middle value. [Evidence tier: textbook]
  3. Zadeh, L. A. "Fuzzy Sets." Information and Control, 1965. — Degrees of membership on [0,1]; the interval extension the chapter closes with. [Evidence tier: peer-reviewed]
  4. Hájek, P. Metamathematics of Fuzzy Logic. Kluwer, 1998. — The standard mathematical treatment: 1-tautologies (value 1 under every valuation) and the road to t-norm logics. [Evidence tier: textbook]
  5. Priest, G. "The Logic of Paradox." Journal of Philosophical Logic, 1979. — LP: the same strong three-valued tables with both ½ and 1 designated, under which the excluded middle is a law again; the source for the designated-values dial. [Evidence tier: peer-reviewed]

T-norms and T-conorms: Fuzzy AND and OR​

  1. Klement, E. P., Mesiar, R., and Pap, E. Triangular Norms. Kluwer, 2000. — The standard monograph: axioms, the three fundamental t-norms, residuation. [Evidence tier: textbook]
  2. Menger, K. "Statistical Metrics." Proceedings of the National Academy of Sciences, 1942. — The origin of triangular norms in probabilistic metric spaces. [Evidence tier: peer-reviewed]
  3. Hájek, P. Metamathematics of Fuzzy Logic. Kluwer, 1998. — BL logic: the residuated-lattice view; adjunction as the soundness condition for fuzzy modus ponens. [Evidence tier: textbook]
  4. Esteva, F., and Godo, L. "Monoidal t-norm Based Logic: Towards a Logic for Left-Continuous t-norms." Fuzzy Sets and Systems, 2001. — MTL: the general residuation framework over left-continuous t-norms. [Evidence tier: peer-reviewed]
  5. Zadeh, L. A. "Fuzzy Sets." Information and Control, 1965. — min/max as the original fuzzy connectives. [Evidence tier: peer-reviewed]

From Fuzzy to Neural: Softmin, Softmax, and Gradients​

  1. van Krieken, E., Acar, E., and van Harmelen, F. "Analyzing Differentiable Fuzzy Logic Operators." Artificial Intelligence, 2022. — The systematic gradient analysis of fuzzy operators; the vanishing/single-passing/dead-zone pathologies this chapter reproduces in miniature. [Evidence tier: peer-reviewed]
  2. Badreddine, S., d'Avila Garcez, A., Serafini, L., and Spranger, M. "Logic Tensor Networks." Artificial Intelligence, 2022. — The stable product configuration and the π₀/π₁ clamps as the production repair. [Evidence tier: peer-reviewed]
  3. Giunchiglia, E., Stoian, M. C., Khan, S., Cuzzolin, F., and Lukasiewicz, T. "ROAD-R: The Autonomous Driving Dataset with Logical Requirements." Machine Learning, 2023. — t-norm constrained losses (product, Gödel, Łukasiewicz) compared on a real perception task. [Evidence tier: peer-reviewed]
  4. Marra, G., Dumančić, S., Manhaeve, R., and De Raedt, L. "From Statistical Relational to Neurosymbolic Artificial Intelligence: A Survey." Artificial Intelligence, 2024. — The map of differentiable-logic families this Part walks through. [Evidence tier: peer-reviewed]

Distribution Semantics: Logic Meets Probability​

  1. Sato, T. "A Statistical Learning Method for Logic Programs with Distribution Semantics." ICLP, 1995. — The distribution semantics: probabilistic facts + definite rules, P(q) as world-weight. [Evidence tier: peer-reviewed]
  2. Poole, D. "The Independent Choice Logic for Modelling Multiple Agents Under Uncertainty." Artificial Intelligence, 1997. — The independent-choice formulation; the same semantics from the agents side. [Evidence tier: peer-reviewed]
  3. De Raedt, L., Kimmig, A., and Toivonen, H. "ProbLog: A Probabilistic Prolog and its Application in Link Discovery." IJCAI, 2007. — ProbLog: the language this chapter's program is written in, in miniature. [Evidence tier: peer-reviewed]
  4. Sato, T., and Kameya, Y. "PRISM: A Language for Symbolic-Statistical Modeling." IJCAI, 1997. — PRISM: the first system running the distribution semantics, a decade before ProbLog named the p :: f construct. [Evidence tier: peer-reviewed]
  5. Vennekens, J., Verbaeten, S., and Bruynooghe, M. "Logic Programs with Annotated Disjunctions." ICLP, 2004. — Annotated disjunctions: the probabilistic-choice construct where exactly one head atom (or none) fires. [Evidence tier: peer-reviewed]
  6. Fierens, D., Van den Broeck, G., Renkens, J., Shterionov, D., Gutmann, B., Thon, I., Janssens, G., and De Raedt, L. "Inference and Learning in Probabilistic Logic Programs Using Weighted Boolean Formulas." Theory and Practice of Logic Programming, 2015. — The reduction of ProbLog inference to weighted model counting; the bridge to the next chapter. [Evidence tier: peer-reviewed]

Weighted Model Counting: The #P Wall​

  1. Valiant, L. G. "The Complexity of Enumeration and Reliability Problems." SIAM Journal on Computing, 1979. — Valiant's companion catalogue of #P-complete counting problems whose decision versions are easy, monotone 2-SAT counting included; the wall named. [Evidence tier: peer-reviewed]
  2. Chavira, M., and Darwiche, A. "On Probabilistic Inference by Weighted Model Counting." Artificial Intelligence, 2008. — WMC as the common currency of exact probabilistic inference. [Evidence tier: peer-reviewed]
  3. Sang, T., Beame, P., and Kautz, H. "Performing Bayesian Inference by Weighted Model Counting." AAAI, 2005. — The encoding of Bayesian networks into WMC; the paradigm's breadth. [Evidence tier: peer-reviewed]
  4. Toda, S. "PP is as Hard as the Polynomial-Time Hierarchy." SIAM Journal on Computing, 1991. — The polynomial hierarchy sits inside P^#P; how high the counting wall reaches. [Evidence tier: peer-reviewed]
  5. Fierens, D., Van den Broeck, G., Renkens, J., Shterionov, D., Gutmann, B., Thon, I., Janssens, G., and De Raedt, L. "Inference and Learning in Probabilistic Logic Programs Using Weighted Boolean Formulas." Theory and Practice of Logic Programming, 2015. — The ProbLog-to-WMC pipeline this chapter reenacts in miniature. [Evidence tier: peer-reviewed]
  6. Karp, R. M., Luby, M., and Madras, N. "Monte-Carlo Approximation Algorithms for Enumeration Problems." Journal of Algorithms, 1989. — The FPRAS for DNF counting, weighted case included; approximation is tractable on this chapter's explanation formulas. [Evidence tier: peer-reviewed]
  7. Roth, D. "On the Hardness of Approximate Reasoning." Artificial Intelligence, 1996. — Even coarse approximate counting is NP-hard on restricted CNF classes; the honest caveat. [Evidence tier: peer-reviewed]
  8. Chakraborty, S., Meel, K. S., and Vardi, M. Y. "A Scalable Approximate Model Counter." CP, 2013. — Hashing-based approximate counting: random parity constraints plus a SAT oracle, with (ε, δ) guarantees. [Evidence tier: peer-reviewed]

Circuits: SDD, d-DNNF, and Fast Evaluation​

  1. Darwiche, A., and Marquis, P. "A Knowledge Compilation Map." Journal of Artificial Intelligence Research, 2002. — The languages (NNF, d-DNNF, OBDD), their properties, and the query/succinctness trade-off table. [Evidence tier: peer-reviewed]
  2. Darwiche, A. "SDD: A New Canonical Representation of Propositional Knowledge Bases." IJCAI, 2011. — Sentential decision diagrams: the industrial compilation target. [Evidence tier: peer-reviewed]
  3. Fierens, D., Van den Broeck, G., Renkens, J., Shterionov, D., Gutmann, B., Thon, I., Janssens, G., and De Raedt, L. "Inference and Learning in Probabilistic Logic Programs Using Weighted Boolean Formulas." Theory and Practice of Logic Programming, 2015. — The ProbLog2 pipeline: ground to a weighted Boolean formula, compile to d-DNNF or SDD, evaluate semirings on the circuit. [Evidence tier: peer-reviewed]
  4. Chavira, M., and Darwiche, A. "On Probabilistic Inference by Weighted Model Counting." Artificial Intelligence, 2008. — Compilation-based WMC as the exact-inference workhorse; the ACE system's exhaustive-DPLL compilation. [Evidence tier: peer-reviewed]
  5. Kimmig, A., Van den Broeck, G., and De Raedt, L. "Algebraic Model Counting." Journal of Applied Logic, 2017. — The semiring generalization: one circuit, many tasks (probability, MPE, gradient). [Evidence tier: peer-reviewed]
  6. Choi, Y., Vergari, A., and Van den Broeck, G. "Probabilistic Circuits: A Unifying Framework for Tractable Probabilistic Models." Technical report, UCLA, 2020. — The modern circuit view of tractable inference. [Evidence tier: preprint]
  7. Bollig, B., and Wegener, I. "Improving the Variable Ordering of OBDDs Is NP-Complete." IEEE Transactions on Computers, 1996. — Even improving a given OBDD's variable order is NP-complete; why real compilers settle for ordering heuristics. [Evidence tier: peer-reviewed]

DeepProbLog: Neural Predicates and the Gradient Semiring​

  1. Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., and De Raedt, L. "DeepProbLog: Neural Probabilistic Logic Programming." NeurIPS, 2018. — Neural predicates on ProbLog; the MNIST-addition demonstration. [Evidence tier: peer-reviewed]
  2. Manhaeve, R., Dumančić, S., Kimmig, A., Demeester, T., and De Raedt, L. "Neural Probabilistic Logic Programming in DeepProbLog." Artificial Intelligence, 2021. — The extended source of record: the gradient-semiring pipeline and the exact-inference scaling limits. [Evidence tier: peer-reviewed]
  3. Kimmig, A., Van den Broeck, G., and De Raedt, L. "An Algebraic Prolog for Reasoning about Possible Worlds." AAAI, 2011. — aProbLog: the semiring-labeled inference the gradient semiring instantiates. [Evidence tier: peer-reviewed]
  4. Eisner, J. "Parameter Estimation for Probabilistic Finite-State Transducers." ACL, 2002. — The expectation semiring: the gradient-as-semiring idea in its original habitat. [Evidence tier: peer-reviewed]
  5. Winters, T., Marra, G., Manhaeve, R., and De Raedt, L. "DeepStochLog: Neural Stochastic Logic Programming." AAAI, 2022. — The derivation-semantics sibling: distributions over derivations instead of worlds, and the speed trade; its timing tables show exact DeepProbLog timing out at four-digit MNIST addition. [Evidence tier: peer-reviewed]
  6. van Krieken, E., Thanapalasingam, T., Tomczak, J. M., van Harmelen, F., and ten Teije, A. "A-NeSI: A Scalable Approximate Method for Probabilistic Neurosymbolic Inference." NeurIPS, 2023. — The successor evaluation that locates the exact pipeline's wall: DeepProbLog times out at four-digit MNIST addition, while amortized inference reaches fifteen digits. [Evidence tier: peer-reviewed]

Semantic Loss and Constraint Layers​

  1. Xu, J., Zhang, Z., Friedman, T., Liang, Y., and Van den Broeck, G. "A Semantic Loss Function for Deep Learning with Symbolic Knowledge." ICML, 2018. — Semantic loss: the definition, the uniqueness axioms, the exactly-one closed form, the semi-supervised results. [Evidence tier: peer-reviewed]
  2. Ahmed, K., Teso, S., Chang, K.-W., Van den Broeck, G., and Vergari, A. "Semantic Probabilistic Layers for Neuro-Symbolic Learning." NeurIPS, 2022. — The layer alternative: constraint-consistent-by-construction prediction via circuit renormalization. [Evidence tier: peer-reviewed]
  3. Giunchiglia, E., and Lukasiewicz, T. "Coherent Hierarchical Multi-Label Classification Networks." NeurIPS, 2020. — The hierarchy constraint layer: coherence by a max construction. [Evidence tier: peer-reviewed]
  4. van Krieken, E., Acar, E., and van Harmelen, F. "Analyzing Differentiable Fuzzy Logic Operators." Artificial Intelligence, 2022. — The fuzzy-penalty alternative this chapter contrasts: t-norm losses and their gradient pathologies. [Evidence tier: peer-reviewed]
  5. Kimmig, A., Van den Broeck, G., and De Raedt, L. "Algebraic Model Counting." Journal of Applied Logic, 2017. — The circuit-evaluation frame that makes loss and gradient one object. [Evidence tier: peer-reviewed]

Scallop: Differentiable Datalog with Provenance​

  1. Huang, J., Li, Z., Chen, B., Samel, K., Naik, M., Song, L., and Si, X. "Scallop: From Probabilistic Deductive Databases to Scalable Differentiable Reasoning." NeurIPS, 2021. — The top-k-proofs provenance and the differentiable reasoning engine. [Evidence tier: peer-reviewed]
  2. Li, Z., Huang, J., and Naik, M. "Scallop: A Language for Neurosymbolic Programming." PLDI, 2023. — The language: provenance framework, 18 semirings, the PyTorch integration. [Evidence tier: peer-reviewed]
  3. Green, T. J., Karvounarakis, G., and Tannen, V. "Provenance Semirings." PODS, 2007. — The algebraic foundation: tagged Datalog over commutative semirings (Volume 2's inheritance). [Evidence tier: peer-reviewed]
  4. Kimmig, A., Van den Broeck, G., and De Raedt, L. "An Algebraic Prolog for Reasoning about Possible Worlds." AAAI, 2011. — aProbLog: semiring-labeled logic programming, the gradient semiring's home. [Evidence tier: peer-reviewed]
  5. Naik, A., Liu, J., Wang, C., Sethi, A., Dutta, S., Naik, M., and Wong, E. "Dolphin: A Programmable Framework for Scalable Neurosymbolic Learning." ICML, 2025. — Vectorized differentiable provenance on GPU; where this line is heading. [Evidence tier: peer-reviewed]

Logic Tensor Networks: Real Logic​

  1. Badreddine, S., d'Avila Garcez, A., Serafini, L., and Spranger, M. "Logic Tensor Networks." Artificial Intelligence, 2022. — Real Logic: groundings, the stable product configuration, p-mean quantifiers, SatAgg learning. [Evidence tier: peer-reviewed]
  2. Serafini, L., and d'Avila Garcez, A. "Logic Tensor Networks: Deep Learning and Logical Reasoning from Data and Knowledge." arXiv:1606.04422, 2016. — The original Real Logic formulation. [Evidence tier: preprint]
  3. van Krieken, E., Acar, E., and van Harmelen, F. "Analyzing Differentiable Fuzzy Logic Operators." Artificial Intelligence, 2022. — The operator analysis that justifies the stable configuration. [Evidence tier: peer-reviewed]
  4. Donadello, I., Serafini, L., and d'Avila Garcez, A. "Logic Tensor Networks for Semantic Image Interpretation." IJCAI, 2017. — LTN applied: grounded reasoning over perception. [Evidence tier: peer-reviewed]
  5. Hájek, P. Metamathematics of Fuzzy Logic. Kluwer, 1998. — The logical backdrop for the connective menu Real Logic draws from, and the standard statement that degrees of truth and degrees of belief are distinct quantities obeying different laws. [Evidence tier: textbook]

GPU-Native NeSy: KLay and Lobster​

  1. Maene, J., Derkinderen, V., and Zuidberg Dos Martires, P. "KLay: Accelerating Arithmetic Circuits for Neurosymbolic AI." ICLR, 2025. — Layerization, tensorization, and GPU evaluation of d-DNNF/SDD circuits. [Evidence tier: peer-reviewed]
  2. Cohen, W. W., Yang, F., and Mazaitis, K. "TensorLog: A Probabilistic Database Implemented Using Deep-Learning Infrastructure." Journal of Artificial Intelligence Research, 2020. — Relations as matrices; logical inference as differentiable linear algebra. [Evidence tier: peer-reviewed]
  3. Biberstein, P., Li, Z., Devietti, J., and Naik, M. "Lobster: A GPU-Accelerated Framework for Neurosymbolic Programming." ASPLOS, 2026. — GPU-native Datalog with provenance: the language-level answer. [Evidence tier: peer-reviewed]
  4. Kimmig, A., Van den Broeck, G., and De Raedt, L. "Algebraic Model Counting." Journal of Applied Logic, 2017. — The semiring-parametric evaluation KLay implements on tensors. [Evidence tier: peer-reviewed]
  5. Darwiche, A. "SDD: A New Canonical Representation of Propositional Knowledge Bases." IJCAI, 2011. — The compiled representation being accelerated. [Evidence tier: peer-reviewed]

Neural-LP and DRUM: Learning Chain Rules​

  1. Yang, F., Yang, Z., and Cohen, W. W. "Differentiable Learning of Logical Rules for Knowledge Base Reasoning." NeurIPS, 2017. — Neural-LP: attention over TensorLog operators; the recurrence and extraction algorithm. [Evidence tier: peer-reviewed]
  2. Sadeghian, A., Armandpour, M., Ding, P., and Wang, D. Z. "DRUM: End-to-End Differentiable Rule Mining on Knowledge Graphs." NeurIPS, 2019. — The rank-1 entanglement theorem and the rank-L CP repair. [Evidence tier: peer-reviewed]
  3. Cohen, W. W., Yang, F., and Mazaitis, K. "TensorLog: A Probabilistic Database Implemented Using Deep-Learning Infrastructure." Journal of Artificial Intelligence Research, 2020. — The operator substrate. [Evidence tier: peer-reviewed]
  4. Evans, R., and Grefenstette, E. "Learning Explanatory Rules from Noisy Data." Journal of Artificial Intelligence Research, 2018. — ∂ILP: the template-based differentiable ILP sibling; context for the design space. [Evidence tier: peer-reviewed]

Neural Theorem Proving: NTP and CTP​

  1. Rocktäschel, T., and Riedel, S. "End-to-End Differentiable Proving." NeurIPS, 2017. — NTP: kernel unification inside backward chaining, rule templates, the min/max proof algebra. [Evidence tier: peer-reviewed]
  2. Minervini, P., Bošnjak, M., Rocktäschel, T., Riedel, S., and Grefenstette, E. "Differentiable Reasoning on Large Knowledge Bases and Natural Language." AAAI, 2020. — GNTP: top-k filtering that made differentiable proving scale. [Evidence tier: peer-reviewed]
  3. Minervini, P., Riedel, S., Stenetorp, P., Grefenstette, E., and Rocktäschel, T. "Learning Reasoning Strategies in End-to-End Differentiable Proving." ICML, 2020. — CTP: the learned rule-selection module and its variants. [Evidence tier: peer-reviewed]
  4. Robinson, J. A. "A Machine-Oriented Logic Based on the Resolution Principle." Journal of the ACM, 1965. — The hard unification being relaxed; Volume 1's foundation. [Evidence tier: peer-reviewed]

RNNLogic and the Symbolic Baseline AnyBURL​

  1. Qu, M., Chen, J., Xhonneux, L.-P., Bengio, Y., and Tang, J. "RNNLogic: Learning Logic Rules for Reasoning on Knowledge Graphs." ICLR, 2021. — Rules as latent variables; the generator-reasoner EM. [Evidence tier: peer-reviewed]
  2. Meilicke, C., Chekol, M. W., Ruffinelli, D., and Stuckenschmidt, H. "Anytime Bottom-Up Rule Learning for Knowledge Graph Completion." IJCAI, 2019. — AnyBURL: path sampling, the three templates, sampled confidence, max-aggregation ranking. [Evidence tier: peer-reviewed]
  3. Rossi, A., Barbosa, D., Firmani, D., Matinata, A., and Merialdo, P. "Knowledge Graph Embedding for Link Prediction: A Comparative Analysis." ACM TKDD, 2021. — The benchmark landscape in which rule-based systems remain competitive. [Evidence tier: peer-reviewed]
  4. Galárraga, L., Teflioudi, C., Hose, K., and Suchanek, F. "AMIE: Association Rule Mining under Incomplete Evidence in Ontological Knowledge Bases." WWW, 2013. — The mining tradition's confidence measures under the open world. [Evidence tier: peer-reviewed]

Query Embedding: The Computation DAG​

  1. Ren, H., and Leskovec, J. "Beta Embeddings for Multi-Hop Logical Reasoning in Knowledge Graphs." NeurIPS, 2020. — The 14-type benchmark, the easy/hard split, and the evaluation protocol this chapter reenacts. [Evidence tier: peer-reviewed]
  2. Ren, H., Hu, W., and Leskovec, J. "Query2box: Reasoning over Knowledge Graphs in Vector Space Using Box Embeddings." ICLR, 2020. — The 9 EPFO structures and the train-on-5, zero-shot-on-4 convention. [Evidence tier: peer-reviewed]
  3. Chandra, A. K., and Merlin, P. M. "Optimal Implementation of Conjunctive Queries in Relational Data Bases." STOC, 1977. — The classical hardness result: conjunctive-query evaluation is NP-hard in general, which is why the field's fragment stays tree-shaped and anchored. [Evidence tier: peer-reviewed]
  4. Hamilton, W. L., Bajaj, P., Zitnik, M., Jurafsky, D., and Leskovec, J. "Embedding Logical Queries on Knowledge Graphs." NeurIPS, 2018. — The query DAG and topological-order evaluation. [Evidence tier: peer-reviewed]
  5. 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: the fragment's boundaries and the field's map. [Evidence tier: preprint]
  6. Yin, H., Wang, Z., Fei, W., and Song, Y. "EFO_k-CQA: Towards Knowledge Graph Complex Query Answering beyond Set Operation." KDD, 2025. — Beyond the fragment: multiple free variables and cyclic query graphs. [Evidence tier: peer-reviewed]

From GQE to BetaE: Points, Boxes, Distributions​

  1. Hamilton, W. L., Bajaj, P., Zitnik, M., Jurafsky, D., and Leskovec, J. "Embedding Logical Queries on Knowledge Graphs." NeurIPS, 2018. — GQE: points, DeepSets intersection, the conjunctive fragment. [Evidence tier: peer-reviewed]
  2. Ren, H., Hu, W., and Leskovec, J. "Query2box: Reasoning over Knowledge Graphs in Vector Space Using Box Embeddings." ICLR, 2020. — Boxes, the DNF union theorem, the inside/outside distance. [Evidence tier: peer-reviewed]
  3. Ren, H., and Leskovec, J. "Beta Embeddings for Multi-Hop Logical Reasoning in Knowledge Graphs." NeurIPS, 2020. — BetaE: closed negation, the full 14-type coverage. [Evidence tier: peer-reviewed]
  4. Zaheer, M., Kottur, S., Ravanbakhsh, S., Póczos, B., Salakhutdinov, R., and Smola, A. J. "Deep Sets." NeurIPS, 2017. — The permutation-invariant architecture family that GQE's intersection instantiates. [Evidence tier: peer-reviewed]
  5. Zhang, Z., Wang, J., Chen, J., Ji, S., and Wu, F. "ConE: Cone Embeddings for Multi-Hop Reasoning over Knowledge Graphs." NeurIPS, 2021. — The cone alternative with negation; the design space beyond Betas. [Evidence tier: peer-reviewed]
  6. Chen, X., Hu, Z., and Sun, Y. "Fuzzy Logic Based Logical Query Answering on Knowledge Graphs." AAAI, 2022. — FuzzQE: t-norm query algebra; the bridge to the next chapter. [Evidence tier: peer-reviewed]

Fuzzy and Training-Free CLQA: CQD, GNN-QE, QTO​

  1. Arakelyan, E., Daza, D., Minervini, P., and Cochez, M. "Complex Query Answering with Neural Link Predictors." ICLR, 2021. — CQD: t-norm composition of a pretrained 1-hop predictor; beam and continuous variants. [Evidence tier: peer-reviewed]
  2. Bai, Y., Lv, X., Li, J., and Hou, L. "Answering Complex Logical Queries on Knowledge Graphs via Query Computation Tree Optimization." ICML, 2023. — QTO: the neural adjacency matrix, exact tree DP, the optimality theorem, faithful proofs. [Evidence tier: peer-reviewed]
  3. Zhu, Z., Galkin, M., Zhang, Z., and Tang, J. "Neural-Symbolic Models for Logical Queries on Knowledge Graphs." ICML, 2022. — GNN-QE: explicit fuzzy sets over entities with product fuzzy logic; interpretable intermediates. [Evidence tier: peer-reviewed]
  4. Chen, X., Hu, Z., and Sun, Y. "Fuzzy Logic Based Logical Query Answering on Knowledge Graphs." AAAI, 2022. — FuzzQE: the parameter-free t-norm query algebra trained on 1p alone. [Evidence tier: peer-reviewed]
  5. Trouillon, T., Welbl, J., Riedel, S., Gaussier, É., and Bouchard, G. "Complex Embeddings for Simple Link Prediction." ICML, 2016. — The 1-hop workhorse everything here composes. [Evidence tier: peer-reviewed]

Foundation Models for CLQA: LMPNN and UltraQuery​

  1. Wang, Z., Song, Y., Wong, G. Y., and See, S. "Logical Message Passing Networks with One-hop Inference on Atomic Formulas." ICLR, 2023. — LMPNN: frozen KGE + closed-form logical messages on the query graph. [Evidence tier: peer-reviewed]
  2. Galkin, M., Zhou, J., Ribeiro, B., Tang, J., and Zhu, Z. "A Foundation Model for Zero-shot Logical Query Reasoning." NeurIPS, 2024. — UltraQuery: the inductive projector inside the fuzzy executor; zero-shot CLQA on unseen graphs. [Evidence tier: peer-reviewed]
  3. Galkin, M., Yuan, X., Mostafa, H., Tang, J., and Zhu, Z. "Towards Foundation Models for Knowledge Graph Reasoning." ICLR, 2024. — ULTRA: the relation-of-relations graph and vocabulary-independent link prediction. [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: the source-seeded propagation backbone. [Evidence tier: peer-reviewed]

Soft Reasoners: RuleTaker and ProofWriter​

  1. Clark, P., Tafjord, O., and Richardson, K. "Transformers as Soft Reasoners over Language." IJCAI, 2020. — RuleTaker: the generator, the depth-stratified protocol, the depth-generalization findings. [Evidence tier: peer-reviewed]
  2. Tafjord, O., Dalvi Mishra, B., and Clark, P. "ProofWriter: Generating Implications, Proofs, and Abductive Statements over Natural Language." Findings of ACL-IJCNLP, 2021. — The CWA/OWA re-release, and iterative vs all-at-once proof generation. [Evidence tier: peer-reviewed]
  3. Saha, S., Ghosh, S., Srivastava, S., and Bansal, M. "PRover: Proof Generation for Interpretable Reasoning over Rules." EMNLP, 2020. — Joint answer + proof-graph prediction over rule language. [Evidence tier: peer-reviewed]
  4. Sanyal, S., Liao, Z., and Ren, X. "RobustLR: A Diagnostic Benchmark for Evaluating Logical Robustness of Deductive Reasoners." EMNLP, 2022. — The perturbation-invariance instrument this chapter miniaturizes. [Evidence tier: peer-reviewed]

Translate-Then-Prove: Logic-LM and LINC​

  1. Pan, L., Albalak, A., Wang, X., and Wang, W. Y. "Logic-LM: Empowering Large Language Models with Symbolic Solvers for Faithful Logical Reasoning." Findings of EMNLP, 2023. — The three-stage pipeline, per-dataset solvers, and self-refinement with its executable-rate lifts. [Evidence tier: peer-reviewed]
  2. Olausson, T., Gu, A., Lipkin, B., Zhang, C., Solar-Lezama, A., Tenenbaum, J., and Levy, R. "LINC: A Neurosymbolic Approach for Logical Reasoning by Combining Language Models with First-Order Logic Provers." EMNLP, 2023. — Sampled translations, error-filtered majority voting, and the three-way prover protocol. [Evidence tier: peer-reviewed]
  3. Ye, X., Chen, Q., Dillig, I., and Durrett, G. "SatLM: Satisfiability-Aided Language Models Using Declarative Prompting." NeurIPS, 2023. — The declarative variant: specifications solved by an SMT solver. [Evidence tier: peer-reviewed]
  4. Kazemi, M., Kim, N., Bhatia, D., Xu, X., and Ramachandran, D. "LAMBADA: Backward Chaining for Automated Reasoning in Natural Language." ACL, 2023. — The inverted design: symbolic control flow with LLM inference primitives. [Evidence tier: peer-reviewed]
  5. Robinson, J. A. "A Machine-Oriented Logic Based on the Resolution Principle." Journal of the ACM, 1965. — The engine's guarantee, inherited from Volume 1. [Evidence tier: peer-reviewed]

NL Reasoning Benchmarks: FOLIO, LogicBench, PrOntoQA​

  1. Han, S., et al. "FOLIO: Natural Language Reasoning with First-Order Logic." EMNLP, 2024. — Expert-written three-way entailment with prover-verified FOL gold. [Evidence tier: peer-reviewed]
  2. Parmar, M., Patel, N., Varshney, N., Nakamura, M., Luo, M., Mashetty, S., Mitra, A., and Baral, C. "LogicBench: Towards Systematic Evaluation of Logical Reasoning Ability of Large Language Models." ACL, 2024. — Single-inference-rule isolation; per-rule competence reporting. [Evidence tier: peer-reviewed]
  3. Saparov, A., and He, H. "Language Models Are Greedy Reasoners: A Systematic Formal Analysis of Chain-of-Thought." ICLR, 2023. — PrOntoQA: fictional ontologies, machine-checkable chains-of-thought, the greedy-reasoner finding. [Evidence tier: peer-reviewed]
  4. Saparov, A., Pang, R. Y., Padmakumar, V., Joshi, N., Kazemi, M., Kim, N., and He, H. "Testing the General Deductive Reasoning Capacity of Large Language Models Using OOD Examples." NeurIPS, 2023. — The out-of-distribution extension: depth, width, and rule-type generalization. [Evidence tier: peer-reviewed]
  5. Tafjord, O., Dalvi Mishra, B., and Clark, P. "ProofWriter: Generating Implications, Proofs, and Abductive Statements over Natural Language." Findings of ACL-IJCNLP, 2021. — The per-depth evaluation precedent and proof-accuracy metrics. [Evidence tier: peer-reviewed]

The Honest Verdict: When Integration Pays Off​

  1. De Raedt, L., Dumančić, S., Manhaeve, R., and Marra, G. "From Statistical Relational to Neuro-Symbolic Artificial Intelligence." IJCAI, 2020. — The integration landscape's dimensions; the survey frame for the verdict. [Evidence tier: peer-reviewed]
  2. Marra, G., Dumančić, S., Manhaeve, R., and De Raedt, L. "From Statistical Relational to Neurosymbolic Artificial Intelligence: A Survey." Artificial Intelligence, 2024. — The mature map: semantics, inference, and learning choices as a design space. [Evidence tier: peer-reviewed]
  3. van Krieken, E., Acar, E., and van Harmelen, F. "Analyzing Differentiable Fuzzy Logic Operators." Artificial Intelligence, 2022. — The gradient-pathology evidence behind the fuzzy side of the ledger. [Evidence tier: peer-reviewed]
  4. Garcez, A. d'Avila, and Lamb, L. C. "Neurosymbolic AI: The 3rd Wave." Artificial Intelligence Review, 2023. — The field-level case for integration; where the division of labor is heading. [Evidence tier: peer-reviewed]
  5. Marconato, E., Teso, S., Vergari, A., and Passerini, A. "Not All Neuro-Symbolic Concepts Are Created Equal: Analysis and Mitigation of Reasoning Shortcuts." NeurIPS, 2023. — Reasoning shortcuts formalized: the unresolved column's first entry and Volume 5's opening theme. [Evidence tier: peer-reviewed]