How to Build Marcus's Algebraic Mind: Algebro-Deterministic Substrate over Galois Fields
Hiroyuki Chuma, Kanji Otsuk, Yoichi Sato

TL;DR
This paper introduces a novel algebraic computing architecture based on XOR-and-shift over GF(2), supporting key cognitive functions and aligning with Marcus's three pillars of cognitive architecture.
Contribution
It presents PyVaCoAl/VaCoAl, a new hyperdimensional architecture that implements Marcus's components using algebraic primitives over Galois fields, bridging neural and symbolic computation.
Findings
Supports reversible variable binding and non-commutative bundling.
Maps Marcus's three pillars to algebraic operations in the architecture.
Extends to counterfactual reasoning capabilities.
Abstract
In The Algebraic Mind, Gary Marcus identified three components essential for any adequate cognitive architecture: operations over variables, recursively structured representations, and a distinction between mental representations of individuals and kinds. He argued that standard multilayer perceptrons supported none of these, acknowledging that a neural implementation using registers and treelets, constructed via developmental programs rather than gradient descent, remained a programmatic conjecture. Twenty-five years later, the required substrate is now available. Our newly developed PyVaCoAl/VaCoAl is a hyperdimensional computing architecture organized end-to-end around a single algebraic primitive: XOR-and-shift over GF(2), implemented by primitive-polynomial linear-feedback shift registers. The architecture supports reversible variable binding via Bind(R,F) = R XOR shift(F),…
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