Part IV · Ch. 32 — Neuro-symbolic

Hybrid · Reasoning · Neural network + logic/solver. Couples the perceptual intuition of networks with the verifiable rigor of symbols. Card: ../02-types-of-ai/06-hybrid-neuro-symbolic.kmd.

🎨 Figure F-IV.32.0 — Intuition + proof. Brief: left side "neural" (fuzzy cloud proposing ideas), right side "symbolic" (logical gears verifying); a bridge in the middle closing the cycle.

1. Definition and short history

Unites the connectionist (ch. 02-14) and symbolic (ch. 15-18) paradigms: the network generates candidatesperception, the solver verifiesreasons. Recent advances: AlphaGeometry and AlphaProof (2024) in olympiad mathematics.

2. Foundations

• Mathematical logic / proof theory — formal verification.
• Combinatorics / complexity — search in symbolic spaces.
• Optimization / learning — the neural side (intuition, priors).
• Philosophy / epistemology — verifiable vs. statistical knowledge.

3. Algorithms and architectures

• Neural proposes — the network suggests steps, lemmas, or interprets perception.
• Symbolic verifies/solves — SAT/SMT solver, proof, constraint.
• Coupling — neural-guided search (the network guides the symbolic search).
• Variants — DeepProbLog (probabilistic logic), reasoners over KGs.

4. Inputs

• Hardware: GPU (network) + CPU (solver/search).
• Data: formal problems + solutions; symbolic self-play to generate data.
• Data structures: proof trees, logical graphs, embeddings.
• Systems: network↔solver integration; formal verifiers.

5. Specialized life cycle