Chapter 12: The Principle of Computational Equivalence

Section 8: Undecidability and Intractability

Undecidability in tiling problems

The question of whether a particular set of constraints like those on page 220 can be satisfied over the whole 2D plane is in general undecidable. For much as on page 943, one can imagine setting up a 1D cellular automaton with the property that, say, the absence of a particular color of cell throughout the 2D pattern formed by its evolution signifies satisfaction of the constraints. But even starting from a fixed line of cells, the question of whether a given color will ever occur in the evolution of a 1D cellular automaton is in general undecidable, as discussed in the main text. And although it is somewhat more difficult to show, this question remains undecidable even if one allows any possible configuration of cells on the starting line. (There are several different detailed formulations; the first explicit proof of undecidability in a tiling problem was given by Hao Wang in 1960; the version with no fixed cells by Robert Berger in 1966 by setting up an elaborate emulation of a register machine.) (See also page 943.)

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