Michael Hochman showed that every 1D effectively closed subshift can be simulated by a
3D subshift of finite type and asked whether the same can be done in 2D. It turned out that
the answer is positive and necessary tools were already developed in tilings theory. We
discuss two alternative approaches: first, developed by N. Aubrun and M. Sablik, goes back
to Leonid Levin; the second one, developed by the authors, goes back to Peter Gacs.

… We can make a 2D subshift from it by copying each letter vertically. It is easy to see that
an effectively closed 1D subshift becomes an effectively closed 2D subshift (we use rules
that guarantee the vertical propagation, ie, require that vertical neighbors should have the
same letter, and the rules of the original 1D subshift in horizontal direction). This 2D shift,
denoted by S, is not of finite type, if the original 1D shift was not of finite type. However, S is
sofic, ie, is a projection by a …