Chapter 12: The Principle of Computational Equivalence

Section 8: Undecidability and Intractability

Undecidability in cellular automata

For 1D cellular automata, almost all questions about ultimate limiting behavior are undecidable, even ones that ask about average properties such as density and entropy. (This results in undecidability in classification schemes, as mentioned on page 948.) Questions about behavior after a finite number of steps, even with infinite initial conditions, tend to be decidable for 1D cellular automata, and related to regular languages (see page 957). In 2D cellular automata, however, even questions about a single step are often undecidable. Examples include whether any configurations are invariant under the cellular automaton evolution (see page 942), and, as established by Jarkko Kari in the late 1980s, whether the evolution is reversible, or can generate every possible configuration (see page 959).

From Stephen Wolfram: A New Kind of Science [citation]