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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. Each entry is a standard text, annotated with what it supports. Only chapters that cite sources appear below; more are added as later parts are written.

Sets, Relations, and Functions: The Language of Structure​

  1. Halmos, P. R. Naive Set Theory. Springer, 1974. — The classic gentle foundation for sets, membership, ordered pairs, relations, and functions; supports the chapter's definitions of a relation as a set of ordered pairs and a function as a single-valued relation. [Evidence tier: standard textbook.]
  2. Enderton, H. B. Elements of Set Theory. Academic Press, 1977. — Rigorous account of ordered pairs, the Cartesian product, relations, and functions, underpinning the set-theoretic definitions used throughout the chapter. [Evidence tier: standard textbook.]
  3. Rosen, K. H. Discrete Mathematics and Its Applications. 8th ed., McGraw-Hill, 2019. — Standard undergraduate treatment of relations and their properties (reflexive, symmetric, transitive), relational composition, and transitive closure. [Evidence tier: standard textbook.]

Order and Lattices: When Things Have Rank​

  1. Davey, B. A., and Priestley, H. A. Introduction to Lattices and Order. 2nd ed., Cambridge University Press, 2002. — The standard reference on partial orders, Hasse diagrams, joins and meets, and lattices; supports the chapter's three order laws, the least upper bound as least common subsumer, and the definition of a (bounded) lattice. [Evidence tier: standard textbook.]
  2. Birkhoff, G. Lattice Theory. 3rd ed., American Mathematical Society, 1967. — The foundational monograph on lattice theory; supports the formal notions of bounded lattices, top and bottom elements, and completeness. [Evidence tier: foundational monograph.]
  3. Ganter, B., and Wille, R. Formal Concept Analysis: Mathematical Foundations. Springer, 1999. — Concept lattices as the bridge from partial orders to knowledge hierarchies; supports the chapter's claim that the join is a least common subsumer and the forward pointer to box-containment orders. [Evidence tier: peer-reviewed monograph.]

Propositional Logic: True, False, and What Follows​

  1. Huth, M., and Ryan, M. Logic in Computer Science: Modelling and Reasoning about Systems. 2nd ed., Cambridge University Press, 2004. — Propositional and predicate logic for computer scientists; supports the treatment of connectives, truth-table semantics, and validity/satisfiability. [Evidence tier: standard textbook.]
  2. Enderton, H. B. A Mathematical Introduction to Logic. 2nd ed., Academic Press, 2001. — Rigorous semantics of propositional logic; supports the model-theoretic notions of assignment, model, satisfiability, and validity, and the complexity remarks (SAT, model counting). [Evidence tier: standard textbook.]
  3. Russell, S., and Norvig, P. Artificial Intelligence: A Modern Approach. 4th ed., Pearson, 2021, ch. 7. — Propositional logic, entailment, Horn clauses, and forward chaining in AI; supports the immediate-consequence operator and the entailment-as-unsatisfiability identity. [Evidence tier: standard textbook.]

First-Order Logic: Objects, Relations, and Quantifiers​

  1. Enderton, H. B. A Mathematical Introduction to Logic. 2nd ed., Academic Press, 2001. — The standard rigorous treatment of first-order syntax, structures, and satisfaction; supports the definition of a structure, the satisfaction relation for the quantifiers, and the undecidability of first-order validity. [Evidence tier: standard textbook.]
  2. Russell, S., and Norvig, P. Artificial Intelligence: A Modern Approach. 4th ed., Pearson, 2021, ch. 8–9. — First-order logic and inference for AI; supports the syntax of terms, predicates, and quantifiers and the role of first-order knowledge in AI. [Evidence tier: standard textbook.]
  3. Abiteboul, S., Hull, R., and Vianu, V. Foundations of Databases. Addison-Wesley, 1995. — Conjunctive queries and their central role in querying; supports the chapter's identification of the existential-conjunctive fragment with database querying and the forward pointer to complex query answering in Volume 4. [Evidence tier: standard textbook.]

Inference and Proof: Chaining Facts into Conclusions​

  1. Russell, S., and Norvig, P. Artificial Intelligence: A Modern Approach. 4th ed., Pearson, 2021, ch. 9. — Inference, forward and backward chaining, and soundness/completeness; supports the treatment of modus ponens, the data-driven versus goal-driven directions of proof, and the soundness and completeness of forward chaining for Horn clauses. [Evidence tier: standard textbook.]
  2. Ceri, S., Gottlob, G., and Tanca, L. Logic Programming and Databases. Springer, 1990. — Datalog, the immediate-consequence operator, and least-model semantics; supports the chapter's definition of T_P, the least fixpoint, and its identity with the least Herbrand model. [Evidence tier: standard textbook.]
  3. Abiteboul, S., Hull, R., and Vianu, V. Foundations of Databases. Addison-Wesley, 1995. — The theory of Datalog evaluation, least fixpoint, and its correctness; supports the completeness argument that fixpoint iteration derives exactly the entailed ground atoms. [Evidence tier: standard textbook.]

Resolution and SLD: How a Machine Proves​

  1. Lloyd, J. W. Foundations of Logic Programming. 2nd ed., Springer, 1987. — The standard rigorous account of substitutions, unification, and the SLD-resolution proof procedure; supports the chapter's treatment of unification as finding a variable-to-term substitution, the standardize-apart discipline, and backward chaining as goal-directed search. [Evidence tier: standard textbook.]
  2. Robinson, J. A. "A Machine-Oriented Logic Based on the Resolution Principle." Journal of the ACM, vol. 12, no. 1, 1965, pp. 23–41. — The paper that introduced the resolution inference rule and unification as its engine; supports the chapter's identification of SLD as a specialization of general resolution to definite (Horn) clauses. [Evidence tier: foundational paper.]
  3. Sterling, L., and Shapiro, E. The Art of Prolog: Advanced Programming Techniques. 2nd ed., MIT Press, 1994. — The classic hands-on treatment of Prolog and SLD resolution; supports the chapter's description of goal-directed proof, the proof tree, and query answering as the machinery behind Prolog. [Evidence tier: standard textbook.]

Fixpoints: Reasoning as Reaching a Limit​

  1. Tarski, A. "A Lattice-Theoretical Fixpoint Theorem and Its Applications." Pacific Journal of Mathematics, vol. 5, no. 2, 1955, pp. 285–309. — The paper that proved every order-preserving (monotone) map on a complete lattice has a complete lattice of fixpoints, in particular a least one; supports the chapter's Knaster–Tarski theorem and the existence of lfp(F) as a non-constructive guarantee. [Evidence tier: foundational paper.]
  2. Davey, B. A., and Priestley, H. A. Introduction to Lattices and Order. 2nd ed., Cambridge University Press, 2002. — Complete lattices, monotone and continuous maps, and fixed-point theorems; supports the chapter's constructive Kleene climb reaching the least fixpoint as the least upper bound of the ascending chain, and the finiteness/continuity condition under which that climb terminates. [Evidence tier: standard textbook.]
  3. Ceri, S., Gottlob, G., and Tanca, L. Logic Programming and Databases. Springer, 1990. — Datalog, the immediate-consequence operator, and least-model semantics; supports the chapter's definition of T_P, the least fixpoint, and its identity with the least Herbrand model. [Evidence tier: standard textbook.]
  4. Abiteboul, S., Hull, R., and Vianu, V. Foundations of Databases. Addison-Wesley, 1995. — The theory of Datalog evaluation, least fixpoint, and its correctness; supports the identity of Datalog evaluation with least-fixpoint iteration and transitive closure as the textbook example of genuine recursion. [Evidence tier: standard textbook.]

What Is Learning: Fitting Functions to Data​

  1. Hastie, T., Tibshirani, R., and Friedman, J. The Elements of Statistical Learning. 2nd ed., Springer, 2009. — Supervised learning, generalization, and model fitting; supports the chapter's treatment of generalization versus memorization, the train/test split, and the principled limits on inferring unseen behavior from a finite sample. [Evidence tier: standard textbook.]
  2. Bishop, C. M. Pattern Recognition and Machine Learning. Springer, 2006. — The standard graduate text on linear and logistic regression, loss functions, and optimization; supports the chapter's model/loss framing, the least-squares line, and the logistic classifier with its sigmoid and cross-entropy loss. [Evidence tier: standard textbook.]
  3. Goodfellow, I., Bengio, Y., and Courville, A. Deep Learning. MIT Press, 2016. — Gradient-based learning and the machine-learning basics; supports the gradient-descent loop, the learning rate, and the one-paragraph taxonomy of supervised, unsupervised, and reinforcement learning. [Evidence tier: standard textbook.]

Gradient Descent: Learning by Following the Slope​

  1. MacKay, D. J. C. Information Theory, Inference, and Learning Algorithms. Cambridge University Press, 2003, ch. 39 ("The Single Neuron as a Classifier"). — The calibration reference for this chapter: the weight-space picture, the sigmoid activation, the cross-entropy (information-content) objective, the chain-rule derivation of the gradient, the on-line and batch gradient-descent algorithms, the learning-rate schedule, and regularization by weight decay. [Evidence tier: standard textbook.]
  2. Robbins, H., and Monro, S. "A Stochastic Approximation Method." Annals of Mathematical Statistics, vol. 22, no. 3, 1951, pp. 400–407. — The founding paper of stochastic approximation; supports the treatment of stochastic gradient descent as a noisy estimate of the true gradient and the diminishing step-size schedule needed for convergence. [Evidence tier: foundational paper.]
  3. Goodfellow, I., Bengio, Y., and Courville, A. Deep Learning. MIT Press, 2016, chs. 4 and 8. — Numerical optimization for machine learning; supports the convex-versus-non-convex distinction, the Hessian and condition-number account of ill-conditioning, mini-batch estimation, momentum, and the prevalence of saddle points in high-dimensional non-convex losses. [Evidence tier: standard textbook.]

Neural Networks: Differentiable Function Approximators​

  1. Goodfellow, I., Bengio, Y., and Courville, A. Deep Learning. MIT Press, 2016. — The standard graduate text on deep learning; supports the treatment of the neuron, the sigmoid and other activations, multilayer perceptrons, and gradient-based training. [Evidence tier: standard textbook.]
  2. Rumelhart, D. E., Hinton, G. E., and Williams, R. J. "Learning Representations by Back-Propagating Errors." Nature, 1986. — The paper that established backpropagation as a practical training method; supports the chapter's chain-rule gradient computation and the training of a multi-layer perceptron by error propagation. [Evidence tier: foundational paper.]
  3. Cybenko, G. "Approximation by Superpositions of a Sigmoidal Function." Mathematics of Control, Signals and Systems, 1989. — The universal approximation theorem for sigmoidal networks; supports the chapter's one-sentence claim that a single hidden layer can approximate any continuous function on a bounded region. [Evidence tier: peer-reviewed article.]

Embeddings: Meaning as Geometry​

  1. Hinton, G. E., McClelland, J. L., and Rumelhart, D. E. "Distributed Representations." In Parallel Distributed Processing: Explorations in the Microstructure of Cognition, Vol. 1, MIT Press, 1986. — The founding account of distributed representation; supports the chapter's claim that a symbol's meaning is a pattern spread across every coordinate rather than a flag in one slot, so that nearby patterns mean similar things. [Evidence tier: foundational chapter.]
  2. Bordes, A., Usunier, N., Garcia-Duran, A., Weston, J., and Yakhnenko, O. "Translating Embeddings for Modeling Multi-relational Data." Advances in Neural Information Processing Systems (NeurIPS), 2013. — The TransE paper; supports the chapter's relation-as-translation model, the h + r ≈ t scoring, margin-ranking training against corrupted triples, and the per-epoch renormalization of entity vectors to the unit norm. [Evidence tier: foundational paper.]
  3. Wang, Q., Mao, Z., Wang, B., and Guo, L. "Knowledge Graph Embedding: A Survey of Approaches and Applications." IEEE Transactions on Knowledge and Data Engineering, 29(12), 2017. — A broad survey of knowledge-graph embedding methods; supports the chapter's claim that such embeddings became a workhorse for link prediction and knowledge-base completion, and the open-world generalization they provide. [Evidence tier: peer-reviewed survey.]

Two Cultures: Symbols versus Vectors​

  1. Newell, A., and Simon, H. A. "Computer Science as Empirical Inquiry: Symbols and Search." Communications of the ACM, vol. 19, no. 3, 1976, pp. 113–126. — The ACM Turing Award lecture that states the physical symbol system hypothesis; supports the chapter's claim that a physical symbol system "has the necessary and sufficient means for general intelligent action" and the reading of Part II's forward- and backward-chaining engines as symbol systems in miniature. [Evidence tier: foundational paper.]
  2. Rumelhart, D. E., McClelland, J. L., and the PDP Research Group. Parallel Distributed Processing: Explorations in the Microstructure of Cognition. 2 vols., MIT Press, 1986. — The two-volume founding statement of connectionism; supports the chapter's account of the connectionist program in which knowledge lives distributed across the connection weights of many simple units rather than in hand-written rules. [Evidence tier: foundational monograph.]
  3. Garcez, A. d'Avila, and Lamb, L. C. "Neurosymbolic AI: The 3rd Wave." Artificial Intelligence Review, vol. 56, 2023, pp. 12387–12406. — A survey framing neuro-symbolic AI as the field's "third wave"; supports the chapter's thesis that the natural resolution of the symbolic and connectionist traditions is a system that reasons like a prover and generalizes like a learned geometry at the same time. [Evidence tier: peer-reviewed survey.]

The Kautz Taxonomy: Six Ways to Combine​

  1. Kautz, H. "The Third AI Summer." AAAI Robert S. Engelmore Memorial Lecture; AI Magazine, vol. 43, no. 1, 2022, pp. 105–125. — The keynote that introduced the six-category taxonomy of neuro-symbolic systems; supports the chapter's naming and ordering of the six couplings from loose pipelines to a reasoning engine nested inside a network. [Evidence tier: peer-reviewed article.]
  2. Garcez, A. d'Avila, and Lamb, L. C. "Neurosymbolic AI: The 3rd Wave." Artificial Intelligence Review, vol. 56, 2023, pp. 12387–12406. — A survey organizing the field; supports the chapter's claim that real systems blur the taxonomy's categories and that the boundaries between cells are porous. [Evidence tier: peer-reviewed survey.]
  3. Sarker, M. K., Zhou, L., Eberhart, A., and Hitzler, P. "Neuro-Symbolic Artificial Intelligence: Current Trends." AI Communications, vol. 34, no. 3, 2021, pp. 197–209. — A map of neuro-symbolic approaches; supports the chapter's placement of methods across the coupling spectrum and the observation that the best coupling for a task remains an open, empirical question. [Evidence tier: peer-reviewed article.]

The Running Example: One Knowledge Base, Five Volumes​

  1. Hitzler, P., Krötzsch, M., and Rudolph, S. Foundations of Semantic Web Technologies. CRC Press, 2010. — The standard text on TBox/ABox, class hierarchies, and ontologies; supports the chapter's reading of the rules as an implicit terminology box (professor ⊑ researcher ⊑ person) sitting above the asserted ABox. [Evidence tier: standard textbook.]
  2. Ren, H., Hu, W., and Leskovec, J. "Query2box: Reasoning over Knowledge Graphs in Vector Space Using Box Embeddings." ICLR, 2020. — Answers multi-hop existential positive first-order (EPFO) queries directly in vector space; supports the chapter's multi-hop query framing and the Volume 4 preview of differentiable complex query answering. [Evidence tier: peer-reviewed conference paper.]
  3. Hogan, A., et al. "Knowledge Graphs." ACM Computing Surveys, 2021. — A comprehensive survey of knowledge graphs, their schemas, and reasoning over them; supports the chapter's claim that the nodes-and-edges knowledge-graph shape is the shared substrate every lens reads. [Evidence tier: peer-reviewed survey.]

The Honest Verdict: What Foundations Buy You​

  1. Kautz, H. "The Third AI Summer." AI Magazine, vol. 43, no. 1, 2022, pp. 105–125. — The keynote that framed the neuro-symbolic program and its promise; supports the chapter's naming of neuro-symbolic AI as the project of joining neural learning with symbolic reasoning. [Evidence tier: peer-reviewed article.]
  2. Garcez, A. d'Avila, and Lamb, L. C. "Neurosymbolic AI: The 3rd Wave." Artificial Intelligence Review, vol. 56, 2023, pp. 12387–12406. — A survey of the state and stakes of combining the two traditions; supports the chapter's claim that the two halves are complementary and that closing the gap between them is the field's central work. [Evidence tier: peer-reviewed survey.]
  3. Marcus, G. "The Next Decade in AI: Four Steps Towards Robust Artificial Intelligence." arXiv:2002.06177, 2020. — The argument that robust AI requires both learning and symbols; supports the chapter's framing of the exact-versus-robust, explainable-versus-opaque, and complete-versus-scalable tensions. [Evidence tier: preprint.]