Vector Symbolic Architectures for Context-Free Grammars

Peter Beim Graben,Matthias Wolff,Werner Meyer,Markus Huber,Ronald Römer

doi:10.1007/s12559-021-09974-y

Abstract

Vector symbolic architectures (VSA) are a viable approach for the hyperdimensional representation of symbolic data, such as documents, syntactic structures, or semantic frames. We present a rigorous mathematical framework for the representation of phrase structure trees and parse trees of context-free grammars (CFG) in Fock space, i.e. infinite-dimensional Hilbert space as being used in quantum field theory. We define a novel normal form for CFG by means of term algebras. Using a recently developed software toolbox, called FockBox, we construct Fock space representations for the trees built up by a CFG left-corner (LC) parser. We prove a universal representation theorem for CFG term algebras in Fock space and illustrate our findings through a low-dimensional principal component projection of the LC parser state. Our approach could leverage the development of VSA for explainable artificial intelligence (XAI) by means of hyperdimensional deep neural computation.

Highlights

We propose a recursive function for mapping context-free grammars (CFG) phrase structure trees onto representation vectors in Fock space and prove a representation theorem for the partial rule-based processing functions
The main result of this study is a Fock space representation theorem for vector symbolic architectures of context-free
We introduced a novel normal form for CFG, called term normal form, and proved that any CFG in Chomsky normal form can be transformed into term normal form

Summary

Introduction

The pioneer of information theory, presented in 1952 a “maze-solving machine” as one of the first proper technical cognitive systems [1].1. It comprises a maze in form of a rectangular board partitioned into discrete cells that are partially separated by removable walls, and a magnetized “mouse” (nicknamed “Theseus”, after the ancient Greek hero) as a cognitive agent. Sensation and memory are implemented by a circuit of relays, switching their states after encounters with a wall In this way, Shannon technically realized a simple, non-hierarchic perception-action cycle (PAC) [2, 3], quite similar to the more sophisticated version depicted in Fig. 1 as a viable generalization of a cybernetic feedback loop

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Conclusion