Quantum Representation Theory for Nonlinear Dynamical Automata

Abstract

Nonlinear dynamical automata (NDAs) are implementations of Turing machines by nonlinear dynamical systems. In order to use them as parsers, the whole string to be processed has to be encoded in the initial conditions of the dynamics. This is, however, rather unnatural for modeling human language processing. I shall outline an extension of NDAs that is able to cope with that problem. The idea is to encode only a “working memory” by a set of initial conditions in the system’s phase space, while incoming new material then acts like “quantum operators” upon the phase space thus mapping a set of initial conditions onto another set. Because strings can be concatenated non-commutatively, they form the word semigroup, whose algebraic properties must be preserved by this mapping. This leads to an algebraic representation theory of the word semigroup by quantum operators acting upon the phase space of the NDA.