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

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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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