How to Build Marcus's Algebraic Mind: From Thagard's Brain--Mind Viewpoint
Neural and Evolutionary Computing
Summary
The authors discuss two past critiques by Marcus and Thagard about how current neural networks lack certain cognitive abilities, like working with variables and structured data. They propose a new system called VaCoAl, which uses a special reversible operation that can bind information exactly without losing detail, addressing both critiques. This system supports complex reasoning and keeps individual elements distinct, which older models struggle with. The authors highlight that their approach is different from current language models and aims to add precise, audit-ready reasoning rather than replace them.
Authors
Hiroyuki Chuma, Kanji Otsuka, Yoichi Sato
Abstract
Two critiques of connectionist cognition converge on one missing capacity. In The Algebraic Mind, Marcus isolated three components any architecture must support -- operations over variables, structured representations, and individuals distinct from kinds -- showed perceptrons support none, and left neural implementation as conjecture. In Brain-Mind, Thagard made binding the single mechanism assembling perception, emotion, consciousness, and self, but rested it on circular convolution, a lossy algebra that degrades under the recursion his account of self and emotion demands. We argue one substrate answers both. VaCoAl is a hyperdimensional architecture built on one primitive, XOR-and-shift over GF(2), on primitive-polynomial linear-feedback shift registers; PyVaCoAl is its extended realization, adding a multi-stage rescue circuit, million-dimensional scale, and a Rescue-Rate phase transition (SRAM-CAM hardware and low-power claim remain future work). Binding, Bind(R,F) = R $\oplus$ shift(F), is exactly reversible and non-commutative: it fills Marcus's open register algebra (Pillar 1), supports compositional bundling at fixed dimension (Pillar 2), and separates individuals from kinds (Pillar 3), while removing the depth-degradation that afflicts convolution at Thagard's deepest recursions. We make three scoped claims. Capability: exact reversible binding at O(N) yields compositional generalization with post-hoc auditability no lossy or learned substrate offers. Necessity: two independent cognitive-architecture programs and a biological circuit (dentate gyrus-CA3) requiring the same reversible-compositional algebra is evidence, by convergent evolution not biomimicry, that it is substrate-independent. Position: we do not claim to surpass LLMs; the substrate is orthogonal, supplying the reversible, auditable, multi-hop relational reasoning statistical embeddings structurally lack.