Two-dimensional Dyck words

TL;DR

This paper explores extending Dyck languages to 2D pictures through various definitions, analyzing their properties, relationships, and potential as a foundation for 2D context-free languages.

Contribution

It introduces new 2D Dyck language definitions, compares their properties, and establishes their relationships, advancing the understanding of 2D formal languages.

Findings

$DN_k$ properly includes $DW_k$

$DN_k$ coincides with the quaternate $DC_k$

The languages form a strict inclusion hierarchy

Abstract

We propose different ways of lifting the notion of Dyck language from words to 2-dimensional (2D) pictures, by means of new definitions of increasing comprehensiveness. Two of the proposals are based on alternative definitions of a Dyck language, which are equivalent over words but not on pictures. First, the property that any two pairs of matching parentheses are well-nested or disjoint is rephrased for rectangular boxes and leads to the well-nested Dyck, $DW_k$. This is a generalization of the known Chinese box language, but, unlike the Chinese boxes, $DW_k$ is not recognizable by a tiling system. Second, the Dyck cancellation rule is rephrased as a neutralization rule, mapping a quadruple of symbols representing the corners of a subpicture onto neutral symbols.The neutralizable Dyck language $DN_k$ is obtained by iterating neutralizations, starting from 2-by-2 subpictures, until the picture is wholly neutralized. Third, we define the Dyck crossword $DC_k$ as the row-column combination of Dyck word languages, which prescribes that each column and row is a Dyck word. The relation between matching parentheses is represented in $DC_k$ by an edge of a graph situated on the picture grid. Such edges form a circuit, of path length multiple of four, of alternating row and column matches. Length-four circuits have rectangular shape, while longer ones exhibit a large variety of forms. A proper subset of $DC_k$, called quaternate, is also introduced by excluding all circuits of length greater than 4. We prove that $DN_k$ properly includes $DW_k$, and that it coincides with the quaternate $DC_k$ such that the neutralizability relation between subpictures induces a partial order. The 2D languages well-nested, neutralizable, quaternate and Dyck crossword are ordered by strict inclusions. This work can be also seen as a first step towards the definition of context-free picture languages.