| name | algebraic-mind-vacoa |
| description | How to Build Marcus's Algebraic Mind: Algebro-Deterministic Substrate over Galois Fields (arXiv:2605.21379). Maps Gary Marcus's three pillars of cognitive architecture (operations over variables, recursively structured representations, individual/kind distinction) onto the PyVaCoAl/VaCoAl hyperdimensional computing architecture. Uses XOR-and-shift over GF(2) as a single algebraic primitive. Activation: vacoal, hyperdimensional computing, algebraic mind, Gary Marcus, cognitive architecture, reversible variable binding, compositional bundling, Galois fields, PyVaCoAl, counterfactual reasoning. |
Algebraic Mind via VaCoAl: Algebro-Deterministic Substrate over Galois Fields
Methodology from arXiv:2605.21379 (May 2026). Authors: Hiroyuki Chuma, Kanji Otsuk, Yoichi Sato.
Overview
In The Algebraic Mind, Gary Marcus identified three components essential for any adequate cognitive architecture: (1) operations over variables, (2) recursively structured representations, and (3) a distinction between mental representations of individuals and kinds. He argued that standard multilayer perceptrons supported none of these.
This paper demonstrates that the newly developed PyVaCoAl/VaCoAl — a hyperdimensional computing architecture organized end-to-end around a single algebraic primitive, XOR-and-shift over GF(2) — provides the functional substrate meeting Marcus's specifications far more closely than the tensor products, circular convolution, or temporal synchrony available in 2001.
Core Architecture
Single Algebraic Primitive: XOR-and-shift over GF(2)
The entire architecture is built on a single operation implemented by primitive-polynomial linear-feedback shift registers (LFSRs):
- Bind(R, F) = R XOR shift(F)
- All operations are reversible and deterministic
- Operates over Galois Field GF(2)
Three Pillars of Marcus's Cognitive Architecture
| Pillar | VaCoAl Implementation |
|---|
