From words to pictures: Row-column combinations and Chomsky-Schützenberger theorem

Abstract

The row-column combination RCC maps two (word) languages over the same alphabet onto the set of rectangular arrays, i.e., pictures, such that each row/column is a word of the first/second language. The resulting array is thus a crossword of the component words. Depending on the family of the components, different picture (2D) language families are obtained: e.g., the well-known tiling-system recognizable languages are the alphabetic projection of the crossword of local (regular) languages. We investigate the effect of the RCC operation especially when the components are context-free, also with application of an alphabetic projection. The resulting 2D families are compared with others defined in the past. The classical characterization of context-free languages, known as Chomsky-Schützenberger theorem, is extended to the crosswords in this way: the projection of a context-free crossword is equivalent to the projection of the intersection of a 2D Dyck language and the crossword of strictly locally testable language. The definition of 2D Dyck language relies on a new more flexible so-called Cartesian RCC operation on Dyck languages. The proof involves the version of the Chomsky-Schützenberger theorem that is non-erasing and uses a grammar-independent alphabet.