AI 中文总结
该研究从萨伽德的脑-心观点出发，探讨如何构建马库斯的代数思维。提出VaCoAl和PyVaCoAl架构，其绑定操作具有可逆等特性，能填补马库斯代数思维组件空缺，消除卷积退化，还通过相关主张阐述了能力、必要性及自身立场。
AI 中文摘要
对联结主义认知的两种批评聚焦于一种缺失的能力。在《代数思维》中，马库斯分离出任何架构都必须支持的三个组件——对变量的操作、结构化表示以及与类别不同的个体，表明感知机不支持任何一个，并将神经实现留作猜想。在《脑-心》中，萨伽德将绑定作为组装感知、情感、意识和自我的单一机制，但基于循环卷积，这是一种有损代数，在他对自我和情感的递归描述下会退化。我们认为一个基质可以回答这两个问题。VaCoAl是一种基于一个原语（GF(2)上的异或移位，基于原多项式线性反馈移位寄存器）构建的超维架构；PyVaCoAl是其扩展实现，增加了多级救援电路、百万维规模和救援率相变（SRAM-CAM硬件和低功耗声明仍是未来工作）。绑定操作Bind(R,F)=R⊕shift(F)是完全可逆且非交换的：它填补了马库斯的开放寄存器代数（支柱1），支持固定维度的组合捆绑（支柱2），并将个体与类别分离（支柱3），同时消除了萨伽德最深层递归中困扰卷积的深度退化。我们提出三个有范围的主张。能力：在O(N)处的精确可逆绑定产生具有事后可审计性的组合泛化，这是无损或学习基质所不具备的。必要性：两个独立的认知架构程序和一个生物电路（齿状回-CA3）需要相同的可逆组合代数，这通过趋同进化而非仿生学证明它与基质无关。立场：我们不声称超越大语言模型；基质是正交[无关]的，提供了结构上缺乏的可逆、可审计、多跳关系推理统计嵌入。
英文摘要
This paper reports a convergence neither program was looking for. Marcus's The Algebraic Mind named three things any architecture needs -- operations over variables, recursive structure, and individuals distinct from kinds -- showed multilayer perceptrons have none, and left a register-and-treelet implementation as conjecture. Thagard's Brain-Mind ran it from the other end, making binding the mechanism from which the whole of mind is assembled, and circular convolution load-bearing. Marcus leaves his register algebra open; Thagard's is lossy, degrading under the very recursion his own account demands. VaCoAl is a hyperdimensional computing architecture built end-to-end on XOR-and-shift over GF(2) via primitive-polynomial LFSRs; PyVaCoAl is its extended software realization (all results here); the silicon substrate exists as CASRAM. Bind(R,F) = R XOR Shift(F) is exactly reversible and non-commutative: it supplies all three pillars at fixed dimension and removes convolution's depth degradation. Capability: exact reversible binding at O(N) yields compositional generalization with post-hoc auditability, which no lossy or learned substrate offers. Necessity: two independent architecture programs and the dentate gyrus-CA3 circuit require the same algebra -- convergence, not biomimicry. New here: discrimination and failure-tolerance are one. Repair every collision (RR = 1) and the system is bit-identical to a hash dictionary: candidates become indistinguishable and the confidence path-integral collapses; a memory that never fails keeps no record of which routes were hard. Position: we do not surpass large language models but supply the auditable, multi-hop relational reasoning embeddings lack. No consciousness is implemented and no cognitive experiment reported; bit-exactness holds in silicon, approximately under biological noise; speed and power remain unbenchmarked.
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