The Logic of Hebbian Learning

Abstract

We present the logic of Hebbian learning, a dynamic logic
whose semantics1 are expressed in terms of a layered neural
network learning via Hebb’s associative learning rule. Its lan-
guage consists of modality Tφ (read “typically φ,” formalized
as forward propagation), conditionals φ ⇒ ψ (read “typi-
cally φ are ψ”), as well as dynamic modalities [φ+]ψ (read
“evaluate ψ after performing Hebbian update on φ”). We give
axioms and inference rules that are sound with respect to the
neural semantics; these axioms characterize Hebbian learning
and its interaction with propagation. The upshot is that this
logic describes a neuro-symbolic agent that both learns from
experience and also reasons about what it has learned.