Hey Pentti, We Did (More of) It!: A Vector-Symbolic Lisp With Residue Arithmetic

TL;DR

This paper addresses the challenge of representing and executing structured programs within neural models by proposing a Turing-complete vector-space Lisp implemented with Fourier-domain HRR (FHRR) representations and Residue Hyperdimensional Computing (RHC) for integer arithmetic. The approach combines a VSA Lisp with auto-associative Resonator networks to decode composite representations, enabling carry-free integer operations and a Lisp interpreter that operates entirely in vector space. Key contributions include a novel RHC-based integer encoding, new arithmetic primitives (int?, add, mul, sub, div), and a modified lambda-based evaluation strategy that preserves readability and efficiency. The work demonstrates that high-dimensional vector representations can express executable, structured programs with interpretable neural states, potentially advancing general intelligent agents through algebraic, structure-aware representations.

Abstract

Using Frequency-domain Holographic Reduced Representations (FHRRs), we extend a Vector-Symbolic Architecture (VSA) encoding of Lisp 1.5 with primitives for arithmetic operations using Residue Hyperdimensional Computing (RHC). Encoding a Turing-complete syntax over a high-dimensional vector space increases the expressivity of neural network states, enabling network states to contain arbitrarily structured representations that are inherently interpretable. We discuss the potential applications of the VSA encoding in machine learning tasks, as well as the importance of encoding structured representations and designing neural networks whose behavior is sensitive to the structure of their representations in virtue of attaining more general intelligent agents than exist at present.

Figures (1)

• Figure 1: RHC integer encoding vs. list encoding, $D = 1000$