Fractal Grammars which Recover from Perturbations

Neural symbolic integration may be a natural phenomenon of dynamical systems. Attractors—subsets of a state space to which a dynamical system returns when perturbed—are a broadly relevant dynamical systems phenomenon. The mathematical theory has mainly focused on autonomous dynamical systems (i.e., not driven by an environment) of the form f : X → X (where x(t+1) = f(x(t)) [iterated map] or dx dt = f(x) [differential equation]), and discovered a rich inventory of attractors, including stable fixed points, limit cycles, and chaos. Here, I focus on the iterated map case and consider certain nonautonomous dynamical systems characterized by a finite set of functions f1, f2, ..., fk : X → X and a language on alphabet Σ = {1, . . . , k} of one-sided infinite strings which applies the functions in particular orders starting from a specified initial state x(0) in X . I extend the definition of attractor by considering cases where the system returns to an invariant proper subset when perturbed in the environment of the language. The news of this paper is that there is a class of nonautonomous dynamical systems that have attractors for mirror recursion languages, a type of language arguably central to natural language syntax.