  Background / introduction. Vector symbolic architectures (VSA) are a viable
approach for the hyperdimensional representation of symbolic data, such as
documents, syntactic structures, or semantic frames. Methods. 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.
Results. 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 states. Conclusions. Our approach could
leverage the development of VSA for explainable artificial intelligence (XAI)
by means of hyperdimensional deep neural computation. It could be of
significance for the improvement of cognitive user interfaces and other
applications of VSA in machine learning.
