# 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]( "Knowledge-based artificial neural networks; Artificial Intelligence"). - **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]( "Bridging Machine Learning and Logical Reasoning by Abductive Learning; NeurIPS"). - **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]( "Neuro-Symbolic Artificial Intelligence: Current Trends; AI Communications"). ## 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]( "A Semantic Loss Function for Deep Learning with Symbolic Knowledge; ICML"). - **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]( "SATNet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver; ICML"). - **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]( "Learning a SAT Solver from Single-Bit Supervision; ICLR"). - **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]( "Logical Neural Networks; arXiv"). - **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]( "Logic Tensor Networks; Artificial Intelligence"). ## 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]( "A Knowledge Compilation Map; JAIR"). - **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]( "Provenance Semirings; PODS"). ## 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]( "Text2Onto - A Framework for Ontology Learning and Data-driven Change Discovery; NLDB"). - **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]( "LogMap: Logic-based and Scalable Ontology Matching; ISWC"). - **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]( "PARIS: Probabilistic Alignment of Relations, Instances, and Schema; PVLDB"). - **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]( "Ontop: Answering SPARQL queries over relational databases; Semantic Web"). - **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]( "A Critical Re-evaluation of Neural Methods for Entity Alignment; PVLDB"). - **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]( "Text2KGBench: A Benchmark for Ontology-Driven Knowledge Graph Generation from Text; ISWC"). - **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]( "Structured Prompt Interrogation and Recursive Extraction of Semantics (SPIRES): a method for populating knowledge bases using zero-shot learning; Bioinformatics"); [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]( "End-to-End Ontology Learning with Large Language Models; NeurIPS"); [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]( "AutoSchemaKG: Autonomous Knowledge Graph Construction through Dynamic Schema Induction from Web-Scale Corpora; ACL"); [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]( "DeepProbLog: Neural Probabilistic Logic Programming; NeurIPS"). - **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]( "Extending Answer Set Programs with Neural Networks; Electron. Proc. Theor. Comput. Sci."). - **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]( "DeepStochLog: Neural Stochastic Logic Programming; AAAI"). - **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]( "DeepProofLog: Efficient Proving in Deep Stochastic Logic Programs; AAAI"). - **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]( "Scallop: A Language for Neurosymbolic Programming; PACMPL"). ## 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]( "End-to-End Differentiable Proving; NeurIPS"). - **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 Learning of Logical Rules for Knowledge Base Reasoning; NeurIPS"). - **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]( "Learning Explanatory Rules from Noisy Data; Journal of Artificial Intelligence Research"). - **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]( "DreamCoder: Growing generalizable, interpretable knowledge with wake-sleep Bayesian program learning; Philosophical Transactions A"). ## Grounded reasoning and theorem search - **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]( "The Neuro-Symbolic Concept Learner: Interpreting Scenes, Words, and Sentences From Natural Supervision; ICLR"). - **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]( "LeanDojo: Theorem Proving with Retrieval-Augmented Language Models; NeurIPS"). - **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]( "Solving olympiad geometry without human demonstrations; Nature"). - **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]( "Gold-medalist Performance in Solving Olympiad Geometry with AlphaGeometry2; JMLR"). ## 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]( "Not All Neuro-Symbolic Concepts Are Created Equal: Analysis and Mitigation of Reasoning Shortcuts; NeurIPS"). - **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]( "A Neuro-Symbolic Benchmark Suite for Concept Quality and Reasoning Shortcuts; NeurIPS"). ## 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](). ## Related topics - [Symbolic regression](https://mlpapers.org/symbolic-regression/) - [Interpretability](https://mlpapers.org/interpretability/) - [Bayesian networks](https://mlpapers.org/bayesian-nets/) - [Graph neural networks](https://mlpapers.org/graph-neural-networks/) - [Natural language processing](https://mlpapers.org/nlp/) - [Meta-learning](https://mlpapers.org/meta-learning/) - [Generalization](https://mlpapers.org/generalization/)