Classic?

Classic? is a cellular automata based on string rewriting, and has an infinite number of states. Below is its semantics.

Semantics

Cell contents/States

Each individual cell holds a string. This string can be of any length. Each program must have an untitled symbol defined, which is given to all cells that have not been defined.

Rules

First lets define these symbols:

C = the cell we are currently checking
L = cell to the left of C
R = cell to the right of C
A = cell above C
B = cell below C
LM = leftmost symbol of C
RM = rightmost symbol of C

Then take the rules, written in pseudo-code:

1 If LM = L then C=C+R else delete RM
2 If RM = R then C=L+C else delete LM
3 If LM = A then C=C+A else delete RM
4 If RM = B then C=B+A else delete LM

Etymology

Classic? gets its name because its operations by themselves are pretty classic within rewriting, however the language itself is not. As such it is a question if it is Classic?