Neurosymbolic Learning and Reasoning

Integration targets and historical foundations

  • KBANN - Towell and Shavlik translate propositional domain theories into a neural network, then refine it by backpropagation. Rules initialize the architecture; subsequent training need not preserve them as hard constraints. Towell and Shavlik 1994.
  • Abductive learning - Alternates learned perception with logical corrections of perceived facts. Dai et al. jointly learn digit recognition and mathematical operations from handwritten equations; logical consistency supplies training feedback for perception. Dai et al. 2019.
  • Neural-symbolic integration - Combines learned representations with symbolic knowledge or computation. Sarker et al. organize integration approaches; injecting a rule into training differs from executing a symbolic program at inference. Sarker et al. 2022.

Logical objectives and differentiable reasoning

  • Semantic loss - Penalizes probability mass assigned to outputs that violate a propositional constraint. Xu et al. connect neural predictions to weighted model counting; a finite soft penalty differs from enforcing validity at decoding. Xu et al. 2018.
  • SATNet - Differentiates through a semidefinite relaxation of MAXSAT to integrate constraint solving with neural perception. Wang et al. test parity and Sudoku; the relaxed solver is not a general exact SAT oracle. Wang et al. 2019.
  • NeuroSAT - A message-passing network learns from SAT/UNSAT labels; satisfying assignments can emerge in its activations. More iterations solve harder tested instances, but the authors report lower reliability than conventional SAT solvers. Selsam et al. 2019.
  • Logical Neural Networks - Associate neurons with components of logical formulas and propagate bounds on truth values. Contradiction-aware learning and an open-world treatment support incomplete knowledge within the proposed logic. Riegel et al. 2020.
  • Logic Tensor Networks - Represent first-order knowledge in a many-valued differentiable logic and optimize satisfaction alongside learning. Truth degrees depend on the chosen logical operators; they are not automatically calibrated probabilities. Badreddine et al. 2022.

Compiled symbolic representations

  • Knowledge compilation - Darwiche and Marquis compare logical representations by succinctness and tractable queries or transformations. Compiling a theory can make repeated reasoning cheap while enlarging the representation; tractability is query- and language-specific. Darwiche and Marquis 2002.
  • Provenance semirings - Annotates input facts with symbols and query results with polynomials. Sums record alternative derivations, products record joint dependencies. For positive relational queries, these annotations preserve how each answer follows from source facts. Green et al. 2007.

Knowledge graphs and ontology learning

  • Text2Onto - Learns concepts and relations from text, attaching confidence scores and source passages. Tracks corpus changes to update affected ontology objects. Conflicting candidates can coexist until a user resolves them. Cimiano and Völker 2005.
  • LogMap - Matches ontology classes through lexical anchors and hierarchy, repairing mappings that make classes unsatisfiable. In biomedical tests, repair raises precision with little recall change. Approximate reasoning can miss conflicts. Jiménez-Ruiz and Cuenca Grau 2011.
  • PARIS - Jointly aligns entities, relations and classes, letting instance matches and schema matches reinforce each other. Weights evidence by how uniquely a relation identifies an entity. YAGO-DBpedia instance alignment reaches 90% precision and 73% recall. Suchanek et al. 2011.
  • Ontop - Maps relational tables to a supplied ontology and rewrites SPARQL queries into SQL. Uses ontology rules and database constraints during rewriting, answering over existing databases without materializing a separate graph. Calvanese et al. 2017.
  • Entity-alignment comparison - With matching procedures standardized, PARIS+ beats the tested neural methods except BERT-INT with descriptions on cross-lingual data. Shared attribute values anchor PARIS matches. PARIS+ also accepts seed entity pairs when attributes are absent. Leone et al. 2022.
  • Text2KGBench - Tests text extraction against a supplied ontology. On two corpora, Vicuna-13B and Alpaca-LoRA score 83-93% relation-vocabulary conformance but 0.22-0.38 fact F1. Conformance checks relation names, not domain/range constraints. Mihindukulasooriya et al. 2023.
  • SPIRES - Fills a supplied schema recursively and resolves names against ontology databases. For 100 Gene Ontology labels, GPT-3.5-turbo gets 98 IDs right versus 3 by prompting alone. With GPT-4-turbo, only 18/100 MONDO disease labels match. Caufield et al. 2024; code.
  • OLLM - Fine-tunes a language model to infer taxonomic paths from documents, masking frequent relations in the loss. On Wikipedia, this improves semantic graph matching over ordinary fine-tuning but lowers exact edge matching. Lo et al. 2024; code.
  • AutoSchemaKG - Induces concept types from extracted entities, events and relations without a supplied schema. In a HippoRAG2 ablation, adding concept nodes raises HotpotQA answer F1 from 77.0 to 78.3 but leaves MuSiQue at 47.3. Bai et al. 2026; code.

Probabilistic and answer-set programs

  • DeepProbLog - Adds neural predicates to probabilistic logic programs and trains neural and symbolic components jointly from examples. Program structure supplies explicit reasoning; perceptual uncertainty enters through neural probabilities. Manhaeve et al. 2018.
  • NeurASP - Treats neural outputs as distributions over atomic facts in answer-set programs. Rules can constrain perception and training; the paper reports substantial symbolic-reasoning overhead relative to neural training alone. Yang 2020.
  • DeepStochLog - Neural grammar rules define probabilities over derivations in stochastic logic programs. DeepProbLog instead uses possible-world semantics; the choice changes what a probability represents and how inference is computed. Winters et al. 2022.
    • DeepProofLog - Learns neural guidance for derivation steps in stochastic logic programs. A mapping from resolution to Markov decision processes connects proof search with dynamic programming and reinforcement learning. Jiao et al. 2026.
  • Scallop - Combines Datalog-style rules, recursion, aggregation and negation with differentiable provenance-based reasoning. It offers a programming interface for integrating neural inputs with structured computations. Li et al. 2023.

Learning rules and programs

  • Neural theorem proving - Rocktäschel and Riedel replace exact symbolic unification with similarity between learned embeddings inside backward chaining. This permits gradient learning of rules and symbols while changing the semantics of a proof score. Rocktäschel and Riedel 2017.
  • NeuralLP - Learns compositions of differentiable logical operations for knowledge-base reasoning. A neural controller jointly selects rule structure and weights; results concern relational queries rather than unrestricted theorem proving. Yang et al. 2017.
  • Differentiable ILP - Evans and Grefenstette learn explanatory rules through a differentiable inductive-logic framework. Experiments address noisy symbolic data and coupling to neural perception; the available rule language bounds expressivity. Evans and Grefenstette 2018.
  • DreamCoder - Alternates program search, abstraction discovery and training a neural search guide on replayed or generated tasks. Learned libraries reuse symbolic components across tasks; the supplied programming language shapes the search. Ellis et al. 2023.
  • Neuro-Symbolic Concept Learner - Learns object representations and question-to-program parsing from image/question/answer supervision. Symbolic programs execute over learned scene concepts; correct answers alone do not establish correct concept grounding. Mao et al. 2019.
  • LeanDojo / ReProver - Retrieves premises for language-model-guided proof search in Lean. Its novel-premise split tests proofs using lemmas absent from training proofs; checked formal proofs provide a different output guarantee from soft neural proof scores. Yang et al. 2023.
  • AlphaGeometry - A neural language model proposes constructions for a symbolic geometry engine. Trinh et al. solve 25 of 30 selected olympiad problems using synthetic training data; the result concerns Euclidean plane geometry. Trinh et al. 2024.
    • AlphaGeometry2 - Extends the geometry language, symbolic engine and search-tree knowledge sharing. The authors report solving 84% of IMO geometry problems from 2000-2024; this expands both language coverage and search beyond AlphaGeometry. Chervonyi et al. 2025.

Evidence and failure modes

  • Concept identifiability - Marconato et al. characterize reasoning shortcuts as unintended optima of the learning objective. Reconstruction and partial concept supervision have limits: matching task labels and rules can leave multiple incompatible concept meanings. Marconato et al. 2023.
  • Reasoning shortcuts - rsbench tests whether correct downstream answers hide incorrect learned concepts. Bortolotti et al. find concept quality remains difficult for neural and neurosymbolic models; task accuracy alone misses these failures. Bortolotti et al. 2024.

Software

  • DeepProbLog implementation - Official probabilistic-programming system with neural predicates and example tasks. The probabilistic program, neural architecture and inference configuration jointly define the experiment. code.
  • LTNtorch - PyTorch implementation of Logic Tensor Networks with logical operators, quantifiers and satisfaction objectives. Record the fuzzy operators and aggregation choices when comparing models. code.
  • NVIDIA Auto Ontology - Uses LLMs to map database columns to business terms, infer relationships and build a semantic catalog. Retrieves catalog entries and schema context to generate SQL for natural-language questions. code.