Estimating the Size of Context-Free Tiling Languages

Abstract

The problem of counting polyominoes motivates this paper. We will develop a general question for study that has counting polyominoes as a special case. We generalize in two ways. Polyominoes are shapes on the tiling made of square tiles. We will consider shapes on other tilings. The set of all polyominoes can be generated by a context-free array grammar, but the size of this set is estimated by counting the words of certain subsets and supersets that are generated by more convenient grammars. Our general question is the problem of counting the words of a context-free array language on a periodic tiling.Counting polyominoes is a difficult problem that has not been completely solved yet. There are various techniques for roughly estimating the number of polyominoes of a given size. We will extend some of these techniques to our general question.