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What Do Hebbian Learners Learn? Reduction Axioms for Iterated Hebbian Learning
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Gives reduction axioms that translate dynamic [φ] formulas about iterated Hebbian learning into base-logic formulas on binary feed-forward neural nets, yielding a completeness theorem for the learning logic.
Problem
Completeness of logics for neural network learning operators; prior work gave only soundness for Hebbian learning on binary feed-forward nets, leaving completeness open.
Approach
Maps a dynamic operator [φ] to iterated Hebbian update (Hebb*, repeated until fixed point) and proves reduction axioms reducing [φ]-formulas to forward-propagation (Prop) and reachability (Reach) formulas, so completeness follows from base-logic completeness; the axioms read as a plausibility upgrade.
Key results
- Main reduction theorem: Prop after Hebb* equals Prop(B ∪ (Prop(A) ∩ Reach(Prop(A) ∩ Prop(B)))), giving reduction axioms for [φ]T.
- Completeness theorem: iterated Hebbian learning logic is completely axiomatized by the reduction axioms plus base-logic axioms, conditional on base-logic completeness.
- Reduction axioms and main theorems verified in the Lean 4 theorem prover.
Limitations
- Base logic completeness for K and T is not established; the paper leaves it as future work.
- Restricted to binary, feed-forward, fully connected nets with unstable Hebb rule; fuzzy activations left open.
- Fixed-point Hebb* rather than single-step Hebb; no guarantees per update step.
Experimental setup
Builds on: Hebb's learning rule Leitgeb forward propagation semantics dynamic epistemic logic plausibility upgrade (Baltag and Smets) Kisby, Blanco, and Moss 2022
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