Row-column combination of Dyck words

We extend the notion of the Dyck language from words to two-dimensional arrays of symbols, i.e., pictures, using the row-column combination (also known as the crossword) of two Dyck languages over the same alphabet. In a Dyck crossword picture, each column and each row must be a word from the respective Dyck language. The pairing of open and closed parentheses in a Dyck word can be represented by edges connecting corresponding cells in the same row or column. This defines a matching graph, which serves as the two-dimensional analogue of the syntactic tree of a Dyck word. A matching graph is partitioned into simple circuits of unbounded length (always a multiple of four), whose labels form a regular language. These circuits exhibit a wide variety of forms and labelings, which we illustrate and partially classify. With a two-letter alphabet, a Dyck crossword is necessarily empty. The minimal non-trivial case, requiring an alphabet of size four, already generates all possible forms of matching graphs and is the primary focus of our study. We prove that the only picture with a single matching circuit (i.e., a Hamiltonian cycle) has size 2 by 2. Two key properties of Dyck words–cancellation and well-nesting–can be generalized to two dimensions, leading to two alternative definitions of 2D Dyck languages: neutralizable and well-nested. These languages are special cases of Dyck crossword pictures called quaternate, where all circuits have length 4 (i.e., are rectangles). This results in a strict language inclusion hierarchy: well-nested \(\subset \) neutralizable \(\subset \) quaternate \(\subset \) Dyck crosswords. When the alphabet size exceeds four, not all combinations of row and column Dyck languages yield non-empty crosswords. To identify productive combinations, we introduce an alphabetic graph, where nodes represent alphabet symbols and edges represent their couplings. A matching circuit corresponds to the unrolling of an alphabetic graph circuit. Finally, we prove that Dyck crosswords are not tiling-recognizable, as expected for a definition extending Dyck word languages to pictures.