Computing with Shapes

We introduce a computability model { called shape grammar { based on adjoining polyominoes. A computation in a shape grammar is correct when: (1) it starts by using a speci ed polyomino; (2) it ends when a rectangle is obtained (without holes); (3) at any step the current picture is connected; and (4) a sequencing mapping is given, so that at every step (except the rst one) we use a polyomino depending on the previously used polyomino, as speci ed by this mapping (such a condition is essential for interactive visual languages, as formalized in [2], [3]). If strings of symbols are associated with the polyominoes, then a string language can be associated to a computation. Surprisingly enough, in these circumstances we can characterize the recursively enumerable languages (hence the power of Turing machines). If we preserve only conditions (1), (2), and (3) above, then we cannot generate all linear languages but we can generate all regular languages and strictly more: also some one-letter non-regular languages can be obtained. In particular, we can obtain as correct computations squares only, which is often a di cult task in picture languages (see, e.g., [32]).