Computing functions through the orbits of dynamical systems: the case of cellular automata

Abstract

Computing a function traditionally involves constructing a formula, circuit, or program specifically designed for that function. But could a single, fixed computational structure generate an entire universe of functions simply by evolving over time? More generally, to what extent can the expressive power of a dynamical system be measured by the functions emerging along its orbits? We investigate this question in a setting where computation is driven by repeated application of one elementary operation, uniformly distributed in space and unchanged over time. Using cellular automata as a model of homogeneous dynamics, we consider systems whose states represent Boolean functions rather than Boolean values. Starting from a simple configuration containing only input variables and constants, the system evolves autonomously, generating new functions at different spatial positions and times. Unlike conventional function synthesis, no function-specific circuit, program, or operation sequence is constructed: the architecture remains fixed, and functions are identified by their space-time location. We ask which elementary rules can generate every Boolean function of a given number of variables along a single orbit, and whether this functional universality persists as the number of variables increases. We develop algebraic and dynamical criteria separating universal from non-universal behavior, and obtain a complete classification of elementary-rule symmetry classes for three-variable functions. Particularly intriguing is Rule 30, for which simple initial configurations generate all Boolean functions of up to five variables, suggesting possible universality for arbitrarily many variables. Our results support a view of computation in which functional diversity arises not from increasingly elaborate architectures, but from the space-time evolution of a single simple dynamical law.