Ai symbolic neuro

Symbolic AI & Neuro-Symbolic Integration

Formal knowledge representation, symbolic reasoning, and their integration with neural methods for knowledge structuring and consolidation pipelines.

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Routing

• USE FOR: Symbolic AI, neuro-symbolic integration, knowledge representation and reasoning (KRR), ontology engineering, description logics, and hybrid symbolic-neural architectures. Use for reasoning about formal knowledge structures, rule-based systems, logic programming, semantic web technologies, and their integration with neural approaches in knowledge graphs and structured reasoning pipelines.
• DO NOT USE FOR: formalising the ontology in Lean (use @lean-knowledge-formalization); commonsense reasoning (use @ai-commonsense-reasoning); agentic AI (use @ai-agentic-evolving).
• TRIGGERS: symbolic AI, neuro-symbolic, knowledge representation, KRR, ontology engineering, description logic, hybrid AI.

Workflow

1. Identify the integration question: symbolic-only, neural-only, or hybrid; pick the integration pattern (loose / tight / unified) from the body.
2. Pick the KRR formalism (description logic, RDF/OWL, conceptual graph, neural-symbolic embedding) that matches the application.
3. Apply the formalism; document where neural and symbolic components communicate + their semantic guarantees.
4. Hand off: to @lean-knowledge-formalization for Lean encoding, to @math-graph-knowledge for graph-theoretic foundations, to @lean-zettelkasten.

Recovery & STOP

• STOP if the question is purely commonsense-reasoning — delegate to @ai-commonsense-reasoning.
• STOP if the question is about causal/deontic logic — delegate to @ai-causal-deontic.
• STOP if the neural component requires empirical evaluation beyond methodological reasoning — escalate to @research-council.

Handoffs

• Predecessors: agent:gateway, skill:lean-research.
• Successors: skill:lean-knowledge-formalization, skill:ai-commonsense-reasoning, skill:math-graph-knowledge.

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Part 1 — Knowledge Representation Foundations

1.1 Representation Languages

Language	Expressiveness	Decidability	Common relevance
Propositional logic	Low	P (SAT: NP-complete)	Gate predicates and finite checks
First-order logic (FOL)	High	Semi-decidable	Theorem formalization
Description logic (ALC)	Medium	ExpTime	Ontology TBoxes
OWL-DL	Medium-high	2NExpTime	Semantic web ontologies
Datalog	Limited FOL	P (data complexity)	Recursive queries on knowledge graphs
Answer Set Programming	Nonmonotonic	ΣP2-complete	Default reasoning
Modal logic	FOL + modalities	Varies	Epistemic/deontic reasoning

1.2 Ontology Engineering

Foundational ontologies:

• DOLCE (Descriptive Ontology for Linguistic and Cognitive Engineering)
• BFO (Basic Formal Ontology) — ISO 21838
• SUMO (Suggested Upper Merged Ontology)

Design patterns:

• Parthood and mereology
• Temporal entities (endurants vs perdurants)
• Roles and role-playing
• Quality and quality spaces

Common application: knowledge-graph quality gates validate ontological consistency:

• TBox coherence (no unsatisfiable concepts)
• ABox consistency (instances satisfy TBox constraints)
• Schema-instance alignment

1.3 Formal Reasoning Systems

System	Approach	Completeness	Use
Tableaux	Model construction attempt	Complete for DL	Ontology reasoning
Resolution	Refutation	Complete for FOL	Theorem proving
Sequent calculus	Structural proof	Complete for FOL	Proof theory
Natural deduction	Introduction/elimination	Complete for FOL	Human-like proofs
Type theory	Curry-Howard	??? (depends on system)	Lean 4 foundation

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Part 2 — Neuro-Symbolic Integration

2.1 Integration Spectrum (Kautz 2020)

Level	Description	Example	Pipeline role
1	Symbolic ← Neural	Neural generates symbolic output	Experience → symbols
2	Symbolic → Neural	Symbolic knowledge guides neural	Structuring → embedding
3	Hybrid pipeline	Neural + symbolic sequential	Full Project pipeline
4	Tightly coupled	Neural and symbolic share representation	Graph of Thoughts
5	Unified	Single system with both capabilities	Target architecture

2.2 Key Architectures

• Neural theorem proving: Neural guide for tactic selection (GPT-f, AlphaProof)
• Knowledge graph embeddings: TransE, RotatE, CompGCN — vector representations of KG
• Graph neural networks: Message passing on knowledge structures
• Neurosymbolic concept learner: Learn visual concepts as logical programs
• Logic tensor networks: Differentiable first-order logic

2.3 Typical Neuro-Symbolic Position

Many verification-oriented pipelines operate at Level 3-4:

• Ingest phase: neural perception, extraction, or embedding.
• Articulation: neural-to-symbolic conversion from natural language to structured claims.
• Structuring: symbolic knowledge graph construction and validation.
• Consolidation: hybrid formal verification plus neural synthesis.
• Application: symbolic-to-neural feedback from validated knowledge to new tasks.

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Part 3 — Commonsense Reasoning

3.1 Commonsense Knowledge Types

Type	Description	Formalization Challenge
Physical	Objects fall, liquids flow	Qualitative physics, spatial reasoning
Social	People have intentions, emotions	Theory of mind, folk psychology
Temporal	Events have duration, causation	Allen's interval algebra
Taxonomic	Dogs are animals	Inheritance hierarchies
Default	Birds fly (unless penguin)	Nonmonotonic logic
Causal	Pushing causes movement	Causal models (Pearl)

3.2 Formal Approaches

• Circumscription (McCarthy): Minimize abnormality
• Default logic (Reiter): Default rules with exceptions
• Answer set programming: Stable models with negation-as-failure
• Probabilistic logic: Weight rules by confidence
• Large language models: Implicit commonsense from training data

3.3 Pipeline Integration

Commonsense reasoning commonly enters structured-AI pipelines in:

• Articulation: Interpreting tacit knowledge requires commonsense context
• Structuring: Default rules for knowledge graph completion
• Quality gates: Commonsense consistency checks on extracted knowledge
• Trust dynamics: Social commonsense for multi-agent coordination

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Part 4 — Knowledge Graph Reasoning

4.1 Reasoning Tasks

Task	Input	Output	Method
Link prediction	$(h, r, ?)$	Missing tail entity	Embedding + scoring
Type inference	Entity features	Entity type	Classification
Rule mining	KG triples	Horn rules	AMIE, AnyBURL
Query answering	Complex query	Answer entities	Query embedding
KG completion	Partial KG	Complete KG	Combination methods
Consistency checking	KG + constraints	Violations	DL reasoning

4.2 Formal Quality Metrics

Metric	Formula	Interpretation
Precision	TP / (TP + FP)	Fraction of predicted links that are correct
Recall	TP / (TP + FN)	Fraction of true links predicted
MRR	$\frac{1}{	Q
Hits@K	Fraction of correct in top-K	Retrieval quality
Semantic validity	Ontology-consistency rate	Structural soundness

4.3 Knowledge Graph Architecture

Experience → Raw triples (LLM extraction)
  → Articulation → Candidate assertions with confidence
    → Structuring → Validated KG with ontological constraints
      → Consolidation → Proven properties (Lean formalization)
        → Innovation → Novel inferences from verified base

Quality gates at each transition enforce:

• Confidence thresholds (configurable per domain)
• Ontological consistency (TBox reasoning)
• Provenance tracking (W3C PROV-O)
• Temporal validity (assertions expire)

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Part 5 — Host-Repository Lean Extension Points

Do not assume repository-local Lean modules, tactics, or namespaces exist unless the host repository explicitly provides them. When a repository has local knowledge-graph or quality-gate modules, map them to this generic checklist:

Local extension point	Symbolic AI aspect	Typical structures
Provenance / audit trail module	DAG well-formedness, trust composition	stages, edges, well-formedness predicates
Causal / knowledge-graph module	Causal DAG and graph formalization	causal links, typed edges, confidence scores
Quality-gate module	Predicate logic on quality measures	gates, thresholds, monotonicity theorems
Classification / regime module	Decision procedures for regimes	regime types, decidability, monotonicity
Causal-reasoning skill handoff	Causal DAGs and counterfactual reasoning	delegate to @lean-causal-reasoning

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Part 6 — Research Connections

6.1 Epistemic Mapping Targets

KK (Known)	KU (Gap)	UU (To Discover)
DAG formalization	Full DL reasoning in Lean	OWL-to-Lean translation
Propositional gates	Nonmonotonic formalization	Commonsense integration
Trust composition	Neuro-symbolic verification	LLM-generated proof verification
Provenance chains	Temporal knowledge evolution	Ontology evolution formalization

6.2 Key References

• Brachman & Levesque (2004) — Knowledge Representation and Reasoning
• Hitzler et al. (2020) — Neuro-Symbolic AI: state of the art
• Marcus & Davis (2019) — Rebooting AI (commonsense perspective)
• Hogan et al. (2021) — Knowledge Graphs (comprehensive survey)
• Davis (2015) — Commonsense reasoning: an event calculus perspective