Row-column combination of Dyck words

Abstract

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.

1 Introduction

The Dyck language is a fundamental concept in theoretical computer science, and in particular for formal language and automata theory. Dyck words represent the last-in-first-out order of events, and the Chomsky-Schützenberger theorem [1] states that any context-free language is the homomorphic image of the intersection of a Dyck language and a regular one.

Motivated by our interest in the theory of two-dimensional (2D), or picture languages (from now on simply “languages”), we investigate the possibility of transporting the Dyck concept to 2D.¹ When moving from 1D to 2D, most formal language concepts and relationships drastically change. In particular, in 2D the Chomsky hierarchy of languages is blurred because the notions of regularity and context-freeness cannot be formulated for pictures without giving up some characteristic properties that hold for words. In fact, it is known [3] that the three equivalent definitions of regular languages by means of finite-state recognizers [4], by regular expressions [5], and by the homomorphism of local languages [6], produce in 2D three distinct language families. The third one gives the family of tiling system recognizable languages (REC) [3], that is perhaps the best known definition for regularity in 2D and has attracted much research effort.

The situation is even less satisfactory for context-free languages where a transposition in 2D remains problematic. None of the existing proposals of “context-free” picture grammars (e.g., [5, 7,8,9,10,11], a survey is [12]) match the expressiveness and richness of formal properties of 1D context-free grammars.

Other researchers have studied the relationship between word and picture languages from a very different viewpoint than ours. The frontier (or row-projection) of a picture is defined as the word in the top row, while the frontier of a picture language is the word language containing the frontiers of all pictures. In [13] it is proved that the frontiers of REC coincide with the family of context-sensitive languages. Other results for certain subclasses of context-sensitive languages are in [14], while the recent study [15] examines the frontiers of languages recognized by various two-dimensional automaton classes.

Since the Dyck language is a simple and typical context-free (non-regular) language, our research objective is to define and compare possible ways to extend from 1D to 2D the underlying concepts–well-parenthesization and well-nesting.

Concerning past related work on anything like a 2D Dyck language, we are aware of just one case. A particular example, the Chinese box language in [10] (reproduced as Example 4 in Sect. 4.3) intuitively consists of embedded or row/column-concatenated rectangles, and was proposed to illustrate the expressiveness of the grammars there introduced. But that language is not a satisfactory proposal, since it is in the family REC, hence “regular” rather than “context-free”.

Here we introduce a new definition that exploits a classical operation for transforming two word languages over the same alphabet into a picture language, named row-column combination (RCC) and also known as crossword.

It is time to describe our approach. We define a Dyck crossword as the picture language obtained by the row-column combination of two Dyck word languages, one for rows and the other for columns, over the same alphabet. In the two languages the coupling between open and closed “parentheses” may be different. All rows and all columns of a Dyck crossword picture are thus well-parenthesized words of the two respective Dyck languages.

The RCC operation, that we prefer to name crossword, has been studied for regular languages (e.g., in [16, 17]). In particular, it is known [16] that the REC family coincides with the projection of the crosswords of two regular languages. But, to our knowledge, the crossword of context-free languages has never been studied before, not even for the Dyck languages. Before entering into the contents and results of this paper, we mention a contemporary related approach in [18] that, instead of using RCC, combines two word languages having different alphabets into a picture language over the Cartesian product of the two alphabets. When applied to Dyck languages, such a Cartesian approach produces a 2D Dyck language more general than Dyck crosswords; but a comparison of the two approaches is out-of-scope here.

Consider the crossword of two Dyck languages with the same even-sized alphabet, partitioned into pairs of coupled letters with not necessarily identical couplings for the rows and the columns. We first prove that if the alphabet contains just two letters, the crossword is empty. The size four alphabet is the minimal one producing a non-empty Dyck crossword, provided the two couplings for rows and for columns are suitably chosen. The corresponding Dyck crossword is denoted $DC_1$ and is the initial and main object of our study, while Dyck crosswords over larger alphabets are studied at the end. There is a strong reason, beside simplicity, for focusing on the minimal alphabet: all the interesting properties of Dyck crosswords are already present in $DC_1$ and their proofs can be extended to the larger alphabet cases.

The matching relations that occur in the Dyck words, laid in the rows and columns of a $DC_1$ picture, can be represented as edges between two cells of the picture array; we formalize the resulting graph as the matching graph of the picture. Clearly every cell is linked to a cell in the same row and another in the same column, and it is immediate to prove that the graph is partitioned into simple, so-called, matching circuits. We prove several properties of such circuits: they must have a length multiple of four, and the same picture may contain circuits of various lengths. In our opinion, the matching graphs can be viewed as the 2D analogue of the syntax trees of Dyck words.

The large variety of visual patterns, represented by the matching graphs, is illustrated in many examples. Some patterns exhibit the symmetries with respect to the vertical and horizontal axes that one intuitively expects to find in a 2D generalization of Dyck word languages; but other pictures present rather surprising non-symmetrical patterns.

We list the primary results, which mostly derive from the matching graph structure of $DC_1$ pictures. The language $DC_1$ is not in REC, yet some of the most interesting $DC_1$ subsets are in REC. The two Dyck languages combined by RCC are saturated by $DC_1$ pictures, in the sense that any of their words respectively occurs in a row or a column of some $DC_1$ picture. For any length multiple of four there exist a picture including a circuit of that length. The set of labels of the nodes traversed by a circuit occurring in $DC_1$ form a regular word language. The only picture whose matching graph is Hamiltonian has size 2 by 2.

An idempotence result says that, if we apply row or column concatenation or their closure to $DC_1$ we remain in $DC_1$; moreover the same property holds if we apply the more general tiling closure operation of Simplot [19].

The definition of Dyck crosswords is quite general, but one may also consider more specialized definitions that more closely fit the basic properties of Dyck words. Any such definition should be also assessed by its conformity to one or the other of the equivalent formal definitions of Dyck word languages. We consider two definitions, omitting details. First, the definition based on a rule canceling two innermost matching parentheses until the word reduces to empty; second, the well-nesting rule that permits to insert a Dyck word inside another. Since for pictures the canceling rule would create a meaningless hole, we rephrase it as a neutralization rule that projects the letters of the Dyck alphabet to a neutral symbol N, until the whole string/picture is neutral.

Accordingly, we define two new 2D Dyck languages: the neutralizable and the well-nested, both over a four letters alphabet. Both of them are based on nesting rectangular circuits, but the two approaches are very different: the neutralizable language imposes a topological (sorting) order on the nesting (by neutralizing from the inside to the outside), while the well-nested language is based on an accretion rule that surrounds a picture with a minimal rectangle containing it. The well-nested language is strictly included in the neutralizable one. Both are found to be special cases of so-called quaternate languages, which are Dyck crosswords restricted to pictures having only circuits of length four, i.e., rectangles. Most properties proved for Dyck crosswords are thus preserved by neutralizable and well-nested languages.

The last development addresses the question of larger Dyck alphabets that we introduce by comparison with Dyck word languages. The alphabet of Dyck words contains $k \ge 1$ pairs of coupled letters and is typically denoted $\Delta _k$ while the language is $D_k$. Notice that the parameter k suffices to characterize all Dyck languages. An important invariance is known between the language $D_k$, $k > 1$, and $D_1$: the set of syntax trees of the two languages are identical, i.e., for any $D_k$ word there is a $D_1$ word with the same tree, disregarding of course the terminal symbols. This means that the size of the terminal alphabet does not influence the richness of the syntactic structures.

For the alphabets of Dyck crosswords the situation is similar, but the tree syntax structure is now replaced by the matching graph. As said earlier, we know that the minimal possible alphabet, used by $DC_1$, has four letters instead of two; and it is easy to prove that $DC_1$ produces all the syntax structures as any larger alphabet can.

But, unlike Dyck word languages, a single integer parameter (the alphabet half-cardinality k for words) does not suffice to characterize all the Dyck crosswords. Clearly, if two (non-empty) Dyck crosswords are defined by two different row/column coupling relations over the alphabet, they are necessarily distinct. For any alphabet the two coupling relations can be summarized into a directed graph with the letters as nodes and with arcs of two sorts, namely horizontal and vertical. Such a graph, named alphabetic to prevent confusion with the matching graph, is partitioned into disjoint simple circuits. Then, two Dyck crosswords having distinct graphs $\Gamma , \Gamma '$ on the same alphabet, are distinct. Therefore, each member of the family of Dyck crosswords is identified by its alphabetic graph and denoted by $DC_\Gamma$. The connection between alphabetic graph and matching graphs is quite simple: intuitively, each circuit of a matching graph is obtained by unrolling a circuit of $\Gamma$ a number of times such that the circuit length is multiple of four.

Paper organization. Section 2 lists a few basic concepts for picture languages and Dyck word languages. Section 3 introduces by means of the RCC operation the Dyck crosswords $DC_1$ over the minimal alphabet of cardinality four, and investigates their properties, using the picture matching graph and circuits as the main technique. Section 4 deals with the quaternate Dyck crosswords and how they preserve characteristic properties of Dyck words. It defines the neutralizable and well-nested picture languages. Section 5 generalizes Dyck crosswords over larger alphabets, where the two Dyck coupling relations are structurally represented as the alphabetic graph. It revisits the notions of matching graphs and circuits for larger alphabets, and states more general properties. It terminates with results on Dyck crosswords not being recognizable and on saturation, and proves the hierarchy of the Dyck crossword languages considered in the paper. Section 6 presents open problems.

2 Preliminaries

All the alphabets considered are finite and, unless explicitly defined otherwise, they have an even number of symbols. The concepts and notations for picture languages follow mostly [3]. A picture is a rectangular array of letters over an alphabet. The set of all non-empty pictures over $\Sigma$ is denoted by $\Sigma ^{++}$. A subpicture of a picture p is a rectangular array that occurs in p.

Let $p,q \in \Sigma ^{++}$. The horizontal or row concatenation of p and q is denoted as , and it is defined when $|p|_{row}= |q|_{row}$. Similarly, the vertical or column concatenation $p \ominus q$ is defined when $|p|_{col}= |q|_{col}$. We also use the power operations $p^{\ominus k}$ and , $k\ge 1$, their closures $p^{\ominus +},$ and the closure under both concatenations ; concatenations and closures are extended to languages in the obvious way.

The Simplot’s tiling closure operation [19] extends the $\Sigma ^{++}$ notation from an alphabet to a picture language $L\subseteq \Sigma ^{++}$. More precisely, the language $L^{**}$ contains every picture that is partitioned into subpictures such that all of them are in L.

Notice that the concatenations $L^{\ominus k}$, and their closures $L^{\ominus +}$, are included in the tiling closure $L^{**}$, which is the most general way of assembling a picture starting from pictures in a given language.

We illustrate the tiling closure operation with a picture (from [19]) in $L^{**}$ where $L=\{a\obar a, b\ominus b, c\}$, not obtainable by composing row and column concatenations:

$\begin{array}{ccc} a& a& b \\ b& c& b \\ b& a& a \end{array}$.

For a picture language $L\subseteq \Sigma ^{++}$ we define the row language as:

$$\begin{aligned} ROW(L)= \left\{ w \in \Sigma ^+ \mid \exists p\in L,\, p', p'' \in \Sigma ^{++} \text { such that } \begin{array}{l} p=w \ominus p'' \text { or }\\ p= p' \ominus w \ominus p'' \text { or }\\ p= p'\ominus w \end{array} \right\} \end{aligned}$$

(1)

and similarly for the column language COL(L).

We assume familiarity with the essential concepts of the family REC of tiling system recognizable languages [3], defined as the projection of local 2D languages; the relevant properties of REC will be reminded when needed.

Dyck alphabet and language.

The definition and properties of Dyck word languages are basic concepts in formal language theory and it suffices to establish our terminology.

Definition 1

(Dyck alphabet and language) A Dyck alphabet is a set $\Delta$, of even cardinality, associated with a bijection, called Dyck coupling relation, $\widehat{\Delta }\subseteq \Delta _\langle \times \Delta _\rangle$ where $\Delta _\langle$ and $\Delta _\rangle$ are two disjoint sets partitioning $\Delta$. It is customary to refer to the symbols of $\Delta _\langle$ and $\Delta _\rangle$ as the open and closed ones, respectively. Each element $[a,b]\in \widehat{\Delta }$ is called a coupled pair.

The Dyck language over $\Delta$ is the set of words congruent to $\varepsilon$, via the cancellation rule $a b = \varepsilon$ where $[a,b] \in \widehat{\Delta }$, which erases two adjacent letters. Given a word over $\Delta$, two occurrences of the coupled letters a and b are matching if they are erased by the same application of the cancellation rule.

We often define a Dyck alphabet $\Delta$ just by giving its Dyck coupling $\widehat{\Delta }$, since $\Delta$ is uniquely determined by $\widehat{\Delta }$.

3 Dyck crossword over a minimal alphabet

Following [3, 16] we introduce the row-column combination operation, also called crossword, that takes two word languages and produces a picture language. Let $L_1,\,L_2$ be two word languages over the same alphabet $\Sigma$. Their row-column combination (RCC) or crossword language is the picture language over $\Sigma$, denoted by $L_1 \boxplus L_2$, defined as

(2)

The languages $L_1,\,L_2$ are respectively called row and column languages.

The row-column combination of regular languages has received attention in the past since its alphabetic projection coincides with the REC family [3]; some complexity issues for this case have been recently addressed in [17] where the crosswords are called “regex crosswords”.

Example 1

(row-column combination) Consider the picture language $P_{sq}=a^*bc^*\boxplus c^*ba^*$. The elements of $P_{sq}$ are square pictures such that the main diagonal has only letters b and the upper/lower triangular matrix has, respectively, only letters c or a as shown:

$$\begin{aligned} \begin{array}{ccccc} b & c & c & c & c \\ a & b & c & c & c \\ a & a & b & c & c \\ a & a & a & b & c \\ a & a & a & a & b \\ \end{array} \end{aligned}$$

As a second example, the RCC $(a^+\cup b^+)\boxplus (a^+\cup b^+)$ is equal to $a^{++} \cup b^{++}$ since neither rows nor columns may contain both a and b.

3.1 Row-column combination of Dyck languages

The idea of extending the Dyck languages from 1D to 2D by means of the row-column combination of two Dyck languages is very intuitive and straightforward, but, to the best of our knowledge, never studied before.

Let $D^{Row}$ and $D^{Col}$ be two Dyck languages over the same alphabet $\Delta$ with two (possibly different) coupling relations (see Definition 1) denoted as $\widehat{\Delta }^{Row}$ and $\widehat{\Delta }^{Col}$ respectively. A Dyck crossword language (Dyck crossword for brevity) is the row-column combination $D^{Row} \boxplus D^{Col}$. The family of all Dyck crossword languages is denoted as $\mathcal{D}\mathcal{C}$.

Since Dyck words have even length, the crossword of two Dyck languages may only contain pictures of both even height and width. Henceforth, this condition is tacitly assumed to hold for all pictures.

It is perhaps unexpected that the preceding definition fails to produce a non-empty language, unless some constraints on the alphabet cardinality and on the couplings are imposed.

Given a binary relation $\rho$, denote with $\langle \!\langle \rho \rangle \!\rangle$ its symmetric closure.

For instance, if $\rho =\{[a,b], [c,d]\}$ then $\langle \!\langle \rho \rangle \!\rangle = \{[a,b], [c,d], [b,a], [d,c]\}$.

Theorem 1

Let $D^{Row}$ and $D^{Col}$ be two Dyck languages over the same alphabet $\Delta$, with couplings $\widehat{\Delta }^{Row}$ and $\widehat{\Delta }^{Col}$ respectively. If $T=\langle \!\langle \widehat{\Delta }^{Row} \rangle \!\rangle \cap \langle \!\langle \widehat{\Delta }^{Col} \rangle \!\rangle \ne \emptyset$, then all symbols in T never occur in the crossword $D^{Row} \boxplus D^{Col}$.

Proof

Let a, b be two symbols that are coupled for both rows and columns, i.e., if the row coupling is, say, [a, b], then [a, b] or [b, a] is the column coupling. Given a picture with an occurrence of a, say, in the leftmost column, a letter b must occur in the same column, which would require a coupling [b, a] for rows, a contradiction with the definition of Dyck language. $\square$

The following corollary follows from Theorem 1 and from the fact that a Dyck alphabet has even size.

Corollary 1

Let $D',D''$ be two Dyck languages over the same alphabet $\Delta$, with possibly different couplings. If $\Delta$ has fewer than four letters, then $D' \boxplus D''$ is empty.

Starting from the minimal alphabet to be considered, we analyze which couplings define a non-empty Dyck crossword, to be denoted $DC_1$.

Proposition 1

(Dyck Crossword $DC_1)$ Let $\Delta$ be an alphabet of four letters, denoted as $\{a,b,c,d\}$. Then, the coupling $\left\{ [ a, b ], [c,d ]\right\}$ for the rows and $\left\{ [ a, c ], [b,d ]\right\}$ for the columns are (up to isomorphism) the only couplings over $\Delta$ defining two Dyck languages $D^{Row}$ and $D^{Col}$, respectively, such that the Dyck crossword $DC_1 = D^{Row} \boxplus D^{Col}$ is not empty.

Proof

In the top left corner of any picture there is a letter, say, a, which is an open letter for both rows and columns. Hence, the row language has a coupled pair, say, [a, b] and the column language has a coupled pair, say, [a, c]–we proved in Theorem 1 that if [a, b] is in the row coupling the couplings [a, b] or [b, a] for columns would lead to an empty language. The letter b is thus on the first row; hence, it is an open letter for the column language, therefore the latter must include the coupled pair [b, d]; similarly, the row language must include the coupled pair [c, d]: there is no other letter left and any other choice than d for the closed letter in either case would again lead to the empty language. The corresponding crossword is not empty since it includes the infinite set of pictures of the form: . $\square$

Remark 1

In this and in the following section, the pixels of $DC_1$ will be represented in two ways, either by letters or by corner symbols, as shown in the table:

Symbol associated with each letter
Row	Col	Letter	Corner
Open	Open	a
Closed	Open	b
Open	Closed	c
Closed	Closed	d

Example 2

($DC_1$ picture.)

A $DC_1$ picture is represented in Fig. 1, left, and, using corner symbols, in Fig. 1, right, where the corners tell the presence of four rectangular patterns that are evidenced in Fig. 1, mid. In each rectangle, the four sides correspond to the matching relations [a, b], [c, d] for rows and [a, c], [b, d] for columns.

It is important to observe that such matching relations on the nodes of rectangles are instances of the row/column coupling relations on alphabet $\Delta$. The latter are naturally represented in a graph, called an alphabetic graph. The graph nodes are the letters of $\Delta$ and the edges are of two sorts row and col, to represent the couplings, e.g.:

Alphabetic graphs will be formalized in Sect. 5.1 to study Dyck crosswords over larger alphabets.

3.2 Matching graphs and their circuits

In a $DC_1$ picture, a cell has a matching relation with a cell on the same row and a cell on the same column. It helps to represent such relations as a graph, which in some sense is the analogue of the syntax tree of Dyck words and may be viewed as representing the syntax structure of crosswords. Indeed, this graph may show some quite complex surprising patterns. The simplest patterns are found in $DC_1$ pictures that are partitioned into rectangular circuits connecting four cells, see, e.g., Fig. 1, middle, where an edge connects two symbols, on the same row (or column), which match in the row (column) Dyck word. Notice that the overall picture graph made by the edges contains four disjoint circuits of length four, called rectangles for brevity; three of them are nested inside the outermost one.

Fig. 1

(Left) A $DC_1$ picture whose cells are partitioned into 4 quadruples of matching symbols, identified by the same node size (color). (Middle) An alternative visualization by a graph whose edges connect matching symbols (see Definition 2). (Right) The use of corner symbols instead of letters enhances the visibility of row and column matching

We formally define the graph, situated on the picture grid, made by such circuits.

Definition 2

(Matching graph) The matching graph associated with a picture $p \in {DC_1}$, of size (m, n), is a pair $G=(V,E)$ where the set V of nodes is the set $\{1,\dots n \} \times \{1 \dots m\}$ with the obvious labeling over $\{a,b,c,d\}$, and the set E of edges is partitioned in two sets of row and column edges defined as follows, for all $1 \le i \le n, 1\le j \le m$:

• there is a row (horizontal) edge connecting nodes (i, j) and $(i,j')$, with $j< j'\le m$, if in the i-th row of p the letters occurring at positions j and $j'$ match.

• there is a column (vertical) edge connecting nodes (i, j) and $(i',j)$, with $i< i'\le n$, if in the j-th column of p the letters occurring at positions i and $i'$ match.

For instance, in a $DC_1$ picture, a horizontal edge may connect two matching letters a, b or c, d that occur in the same row; analogously, a vertical edge may connect two matching letters a, c or b, d that occur in the same column.

Theorem 2

(matching circuits) Let p be a $DC_1$ picture.

• 1. The matching graph G of p is partitioned into simple (undirected) circuits, called matching circuits;

• 2. the clockwise visit of a matching circuit of p, starting from any of its nodes with label a, yields a word in $(abdc)^+$, called the circuit label.

Proof

Part (1). By Definition 2, every node of G has degree 2, with one row edge and one column edge, since its corresponding row and column in picture p are Dyck words. Every node must be on a circuit, otherwise there would be a node of degree 1. Each circuit must be simple, and the sets of nodes on two circuits must be disjoint; otherwise, the degree of one of the nodes would be greater than 2.

Part (2). From a node labeled a the clockwise visits continues with the row edge to a node labeled b, which has the column edge to d. The latter node has the row edge to c, etc., finally closing the circuit with the column edge connecting a c to the original a. $\square$

Theorem 2 has a straightforward interpretation in the case of Dyck words: the associated matching graph of a Dyck word of length n is the well-known, “rainbow” representation of the syntax tree, e.g.,

The matching graph on the grid $1,2, \ldots , n$ then corresponds to the binary relation induced by the rainbow arcs; however, the notion of matching circuit vanishes and it is downgraded to a single rainbow arc.

Fig. 2

Two $DC_1$ pictures with long matching circuits. (Left) The picture is partitioned into two circuits of length 12 and 4. (Right) The picture includes a circuit of length 36 and seven rectangles. Its pattern embeds four partial copies (direct or rotated) of the left picture; in, say, the NW copy of the evidenced “triangle” b, d, c, the letters have been changed to a, a, a, also evidenced by larger dots. Such a transformation can be reiterated to grow a series of pictures

The simplest $DC_1$ pictures contain just rectangles. It is maybe unexpected that the circuit length may increase without bound giving rise to a rich range of pictures. Two examples of pictures in $DC_1$ with circuits longer than four are in Fig. 2, left, with a circuit of length 12 labeled by the word $(abdc)^3$, and, right, with a circuit of length 36.

Notice that, when a picture in $DC_1$ is completed with its matching graph, the pixel symbols are redundant since they are uniquely determined on each circuit.

Theorem 3

(Circuits of $DC_1$) The node labels in a $DC_{1}$ picture, where the matching circuits are represented, are redundant.

Proof

In a $DC_{1}$ picture the pixel of each node is uniquely determined by the direction of the incoming and outgoing edge in accordance with the correspondence:

$\square$

3.3 Properties of matching circuits

This section studies the longer circuits in a matching graph. We first observe that a picture partitioned in matching circuits is not necessarily in $DC_1$. Consider the matching graph of an arbitrary picture over the alphabet $\{a,b,c,d\}$ that satisfies all the properties stated in Theorem 2 using the couplings of $DC_1$. Does such a picture belong to $DC_1$? The answer is negative and is witnessed by the picture in Fig. 3.

Fig. 3

Picture not in $DC_1$, with circuits satisfying the conditions of Theorem 2; rows 2 and 3 are not Dyck words

All the columns and the first and fourth rows are legal Dyck words, while the second and third rows are not; for instance, in the second row the edges from a to b cross the edges from c to d, which is illegal for a Dyck word. Still, it is easy to verify that the matching graph is partitioned in two circuits of length 12 and two circuits of length 4 and its label is in $(abcd)^+$.

Fig. 4

(Left) Picture $p_{(1)}$ used as induction basis of Theorem 4. It is covered by a circuit of length $4+8\cdot 1=12$ and by 3 rectangles (not shown). (Middle) Picture $p_{(1)}\ominus p_{(1)}$, the four arcs to be deleted are in green (solid lines), and the four nodes to be relabeled are in blue (also larger dots). (Right) Inductive step: picture $p_{(2)}$ is obtained from $p_{(1)}\ominus p_{(1)}$ by canceling the four green arcs, relabeling the four larger blue nodes as shown (the corresponding rectangle is in solid/blue) and finally adding two solid (blue) arcs that join the double-noose circuits. A circuit of length $4+ 8\cdot 2$ results. All length 4 circuits of $p_{(h-1)}$ and $p_{(1)}$ (not shown for clarity) are unchanged in $p_{(h)}$

Fig. 5

Part (2) of proof of Theorem 4. (Left) The top picture $p'_{(1)}$ with a circuit of length 8 (red/darker edges) is placed over the picture $p''_{(1)}$ that includes as subpicture $p_{(1)}$ of Fig. 4. Notice that the two leftmost columns, covered by two length 4 circuits (not drawn) have been added to picture $p_{(1)}$ to fit picture $p_{(3)}$ width.(Right) Pictures $p''_{(1)}$ and $p'_{(1)}$ are adjusted and connected as in part (1) of the proof, thus obtaining a picture $p'_{(2)}$ with a circuit (in red/dark) of length 16–a total of four nodes (at positions (8, 3), (8, 8), (9, 3), (9, 8) ) have been extracted from the two (now connected) circuits of length 8 and 12; their pixels have changed and form a solid length 4 circuit

We continue with the study of longer circuits.

Theorem 4

(Circuit length) For all $h \ge 1$ there exist a picture $p_{(h)}$ in $DC_1$ that contains a matching circuit of length 4h.

Proof

The case $h=1$ is straightforward. For $h>1$, the proof proceeds in two parts: First, Fig. 4 presents a family of pictures containing circuits of length $4+8h$ for all $h \ge 1$. Second, Fig. 5 shows how to extend these circuits by four additional nodes to cover all remaining cases of length 8h ($h \ge 1$).

Case 1: Circuits of length $4+8h$ We construct $DC_1$ pictures $p_{(h)}$ by induction on $h \ge 1$. The base case $p_{(1)}\in DC_1$ (Fig. 4, left) is a $(4 \times 6)$ picture (where $m_{(1)}=4$) containing a 12-node “double-noose” circuit ($12=4+8$). (1) Case $4+8h$. We define a $DC_1$ picture $p_{(h)}$, by induction on $h\ge 1$. The base case $h=1$ is the $DC_1$ picture $p_{(1)}$ in Fig. 4 (left). Picture $p_{(1)}$ has size $(m_{(1)}, 6)$, where $m_{(1)}=4$, and contains a circuit of length $12=4+8$, referred to as double-noose.

The inductive step extends picture $p_{(h-1)}$, $h>1$, by superimposing a copy of $p_{(1)}$ over $p_{(h-1)}$, as in Fig. 4, middle, and making a few changes (detailed in Fig. 4, right). The construction extracts four nodes from the two circuits of lengths 12 and $4+8(h-1)$, and connects them with two new arcs, called junctions, forming a unique circuit of length $4+8h$. The pixels of the four extracted nodes have changed and form a solid length 4 circuit. The construction preserves $DC_1$ membership of $p_{(h)}$ and yields a circuit of length $4+8h$.

Case 2: Circuits of length 8h ($h \ge 1$) We define a sequence of pictures $p'_{(1)},\ldots p'_{(h)}$ as follows. Picture $p'_{(1)}$ is defined in Fig. 5, top left; it has two circuits of length 8. For every $h\ge 1$, let $q_{(h)}$ be the picture obtained from picture $p_{(h)}$ of the previous case by row concatenation with column concatenations of rectangles, namely . Picture $p'_{(h)}$ is obtained by adding $p'_{(1)}$ on top of $q_{(h-1)}$ and connecting a length 8 circuit of $p'_{(1)}$ with the length $4+8(h-1)= 8h-4$ circuit of $q_{(h-1)}$. As in the previous case, a total of four nodes are removed from the two circuits and their pixels suitably renamed; two new junctions connect the two circuits, forming a unique circuit of length $8 +(8\,h-4)-4= 8\,h$. Figure 5 (right) illustrates this construction for $p'_{(2)}$. $\square$

Another series of pictures that can be enlarged indefinitely is the one in Fig. 2, where the first two terms of the series are shown.

An examination of Fig. 4 shows that there is an infinite subset of $DC_1$ that is in REC, yet it contains quite complex matching circuits. This contrasts with the fact, to be proven in Sect. 5, Theorem 12, that $DC_1$ is not in REC.

Theorem 5

The picture language ${L= \displaystyle \bigcup \nolimits _{h\ge 1}} p_{(h)}$, where each $p_{(h)}$ is the picture defined in the proof of Theorem 4, is in the REC family.

Proof

We first extend the alphabet of L to $\{a,b,c,d,a_1,b_1,c_1,d_1\}$ and mark the circuits longer than 4 over the alphabet $\{a_1,b_1,c_1,d_1\}$, while leaving the remaining circuits over $\{a,b,c,d\}$. The resulting pictures $\widetilde{p}_{(h)}$ have only 6 distinct rows, here identified (from top to bottom) with the letters $1, \dots , 6$:

$$\begin{aligned} \begin{array}{llllllllllll} \text {1:}&a_1aa_1b_1bb_1,&\text {2:}&c_1ab d_1ab,&\text {3:}&a_1cd b_1cd,&\text {4:}&a cc_1d_1db,&\text {5:}&c aa_1b_1bd,&\text {6:}&c_1cc_1d_1dd_1\end{array}. \end{aligned}$$

It is clear from the construction of the pictures $p_{(h)}$ for $h>1$ that $\widetilde{L}$ can be defined as $1\ominus (2\ominus 3\ominus 4\ominus 5)^{*\ominus }2\ominus 3\ominus 6$. Since each of rows $1, \dots , 6$ can be seen as a finite language (thus, in REC) and REC is closed by column concatenation and its closure, also $\widetilde{L}$ is in REC. By closure of REC under projection, also L is in REC (by projecting $a_1$ to a, $b_1$ to b, etc.) $\square$

From an elementary property of Dyck word languages it follows that the distance on the picture grid between two nodes connected by an edge is an odd number, to let room for an even number of letters. This suggests the following Lemma 1.

Given a picture p over an alphabet $\Gamma$, let $p_{i,j}=x$, for $x \in \Gamma$. We say that the occurrence of x in position (i, j) has row parity 1 if i is even, row parity -1 otherwise; similarly, x in (i, j) has column parity 1 if j is even, column parity -1 otherwise.

Lemma 1

(Matching circuit property) Let $\gamma$ be a matching circuit of a picture in $DC_1$, with label in $(abdc)^+$.

• i) All occurrences of a and b in $\gamma$ have the same row parity, but they have opposite row parity to every occurrence of c and d;

• ii) all occurrences of a and c in $\gamma$ have the same column parity, but they have opposite column parity to every occurrence of b and d.

Proof

Given the coordinates (i, j) of a node in $\gamma$, consider the modulo 2 coordinates, namely $(i \mod 2, j \mod 2)$ written as $(i,j)_2$. Modulo 2 coordinates may thus only take values (0, 0), (0, 1), (1, 0), (1, 1). The sequence of modulo 2 coordinates of the nodes traversed in a clockwise visit of a circuit, starting at $(0,0)_2$, has the form: $\left( (0,0)_2 (0,1)_2 (1,1)_2 (1,0)_2\right) ^+$, since the matching circuit alternates between horizontal and vertical edges, and the distance between two matching symbols must be even. By Part (2) of Theorem 2, the circuit label is in $(abdc)^+$. Therefore, if we assume (without loss of generality) that a pixel a in the circuit is at coordinates $(0,0)_2$, then every other a in the circuit is also at $(0,0)_2$, every b is at $(0,1)_2$, every d is at $(1,1)_2$ and every c is at $(1,0)_2$. Therefore, part (i) follows immediately since the row coordinate of a and b is 0 modulo 2 and the one of c, d is 1 modulo 2. The case of part (ii) is analogous. $\square$

An application of Lemma 1 follows.

Let p be a picture in $DC_1$ and G its matching graph. A matching circuit that visits all the nodes of G is called Hamiltonian.

Theorem 6

(Hamiltonian circuits) The only $DC_1$ pictures with a Hamiltonian matching circuit is $\begin{array}{cc}a & b \\ c & d \end{array}$.

Proof

By contradiction, assume that a picture $p \in DC_1$ of size greater than (2, 2), has a Hamiltonian circuit. The first row of any picture is a Dyck word over $\{ a, b \}$ and the leftmost column is a (vertical) word over $\{a,c\}$. By Lemma 1, the second row must be a word over $\{ c, d \}$ and the second column from the left is over $\{b,d\}$. Therefore, the subpicture $( p (1,1) \ominus p (2,1) )$ must be $a \ominus c$ (a cannot occur in the second row, therefore the row must begin with the open letter c for rows) and similarly the subpicture $( p (1,2) \ominus p (2,2) )$ is $b \ominus d$.

Therefore, p contains the subpicture $\begin{array}{cc}a & b \\ c & d \end{array}$ in the top, left corner, i.e., it has a matching circuit of length 4, a contradiction with the existence of a Hamiltonian circuit for a picture of size greater than (2, 2). $\square$

4 Pictures preserving characteristic properties of Dyck words

This section first introduces the pictures like the ones in Figs. 1 and 6, called quaternate, whose matching circuits have length four and are named rectangles. The absence of long and intricate circuits will permit us to introduce a geometrical relation of containment between rectangles. If such a relation is acyclic we call the corresponding language quaternate acyclic.

We then consider a few alternative definitions of Dyck picture languages, not based on the crossword operation but on extending “natural” definitions of Dyck words to pictures. We first show that the standard definition of Dyck words by means of the cancellation rule can be extended to 2D by means of a cancellation rule suitably transformed into a neutralization operation, thus defining the neutralizable picture language $DN_1$. Then, we prove that the neutralizable and the quaternate acyclic languages are the same, and we list some of their properties.

Subsequently, we introduce an operation called accretion that is analogue to surrounding a Dyck word between a pair of parentheses, thus increasing the nesting level by one. The pictures generated by accretions and concatenations, actually by the more general tiling closure, are called well-nested Dyck crosswords. We show that the neutralizable crosswords strictly contain the well-nested ones.

Acyclic quaternate languages.

Our intuition of Dyck words is based on a mirror symmetry between open and closed parentheses, which is lost in 2D when the matching graph becomes complicated with long looping and twisted circuits. To recover this intuition, we focus on pictures, called quaternate, which contain only circuits of length four, i.e., rectangles. In fact, the four corners of a rectangle have symmetrical positions in a picture and may be viewed as analogous to Dyck parentheses.

Definition 3

(Quaternate $DC_1$) A Dyck crossword picture, such that all its matching circuits are of length 4, is called quaternate. The quaternate Dyck language $DQ_1$ is the language of quaternate pictures in $DC_1$.

A key feature of any context-free word language is that every word has a syntax tree. Following the analogy between matching circuits and syntax trees mentioned earlier in Sect. 3, we can restrict the relative positions of matching circuits to resemble a tree-like structure. These considerations suggest the following definitions.

Consider two rectangles $R_1,R_2$ in a picture. We say that $R_1$ is partially contained in $R_2$, writing $R_1\,< \,R_2$, if some nodes of $R_1$ are inside or on a side of $R_2$. The partial containment relation of a picture is the set of all such relations. Note that the number of nodes contained in $R_2$ may be 1, 2 or 4, but never 3 (which is geometrically impossible).

Some containment relations are illustrated in Fig. 6, including: $R_1<R_2$, $R_2<R_1$, $R_3<R_1$ (in row 1), $R_3<R_2$ (in column 3), $R_4 < R_1$. The letters are marked with subscripts to identify the rectangles.

Restricting the partial containment to form a tree (even by ignoring transitive containment), results in a very severe constraint in 2D. The resulting family of languages, defined in Example 5 below, has too simple a structure to be interesting. Instead, we impose just a typical characteristic of trees on the partial containment relation: the existence of a topological order in the visit of nodes. This can be achieved simply by requiring acyclicity on the partial containment relation, as in the following definition.

Definition 4

(Acyclic quaternate picture) A quaternate picture in $DQ_1$ is called acyclic if its partial containment relation “<” is acyclic.

We observe that the quaternate pictures in Fig. 6 are not acyclic, because they respectively contain the cycles $R_1< R_2 < R_1$ and $R_1<R_2< R_3< R_4 < R_1$. On the other hand, in the pictures presented in Figs. 7 and 9 below, the partial containment relations (displayed in the figures) are acyclic. Hence, the rectangles can be topologically ordered.

Fig. 6

Both pictures are quaternate but not acyclic, hence not neutralizable by Theorem 7. The nodes relevant for illustrating partial containment are indexed. To avoid clogging, the rectangles in the specular right half of the left picture are not drawn

In the next section we prove that the acyclic pictures are exactly those which obey a neutralization rule simulating the cancellation rule of Dyck words.

4.1 Neutralizable languages

Preliminarily, we transform the cancellation rule for words, $a b \rightarrow \varepsilon$, which erases innermost matching letters, into a length-preserving rule. This modification is necessary because in 2D, erasing an internal subpicture would create a “hole”, resulting in an object that no longer qualifies as a proper picture. The Dyck cancellation rule is thus rephrased as the neutralization rule $a b \rightarrow NN$, where N is a new “neutral” (i.e., uncoupled) letter. Through a series of neutralization steps, this transformation maps a Dyck word to a word in $N^+$.

We introduce a neutralization rule that maps the letters in a quadruple (representing the corners of a subpicture) to the neutral letter N. The neutralizable Dyck language $DN_1$ is obtained by iteratively applying neutralization, starting from 2-by-2 subpictures, until the entire picture is neutralized. More precisely, given a picture p, all subpictures of the form are neutralized by replacing them with the subpicture $\begin{array}{cc} N & N \\ N & N \end{array}$. If p contains a subpicture with four matching corners whose interior and sides are completely neutralized, then the four corners are also neutralized. Thus the subpicture:

is replaced by a subpicture of the same size having only N as letters. The procedure is repeated until no neutralization is possibile: it successfully terminates if the resulting picture is in $N^{++}$.

The notation $N^{m,n}$, where $m,n>0$, stands for a homogeneous picture in $N^{++}$ of size m, n. For streamlining the following Definition 5, we extend this notation to the case where either m or n are 0, to introduce identity elements for row and column concatenations: given a picture p of size (m, n), by definition and $p \ominus N^{0,n}= N^{0,n}\ominus p = p$.

Definition 5

(Neutralizable Dyck language) Let $\Delta =\{a,b,c,d\}$ and let N be a new symbol not in $\Delta$. The neutralization relation $\xrightarrow {\nu } \; \subseteq \left( \{N\}\cup \Delta \right) ^{++}\times \left( \{N\}\cup \Delta \right) ^{++}$ is the smallest relation such that $p \xrightarrow {\nu } p'$, for $p,p'$ in $\left( \{N\}\cup \Delta \right) ^{++}$, if there are $m,n\ge 2$ such that $p'$ is obtained from p by replacing a subpicture of p of the form:

(3)

with the picture (of the same size) $N^{m,n}$.

The neutralizable Dyck language, denoted by $DN_1\subseteq \Delta ^{++}$, is the set of pictures p such that there exists $p' \in N^{++}$ with $p \xrightarrow {\nu }\mathrel {\phantom {\rightarrow }^+} p'$.

To sum up, a $DN_1$ picture is transformed into a picture in $N^{++}$ by a series of neutralizations, applied in any feasable order.

Example 3

(Neutralizations) The following picture p on the alphabet $\Delta _1$ is in $DN_1$ since it reduces to the neutral one by means of a sequence (here in left to right order) of six neutralization steps:

Fig. 7

Left: The neutralizable picture of Example 3, with every rectangle labeled with an index; Right: its partial containment relation “<”, here represented by a graph, is acyclic

Fig. 7 shows that the partial containment relation is acyclic, thus the picture is quaternate acyclic.

It is no coincidence that the pictures of Example 3 and of Fig. 9 are both neutralizable and acyclic; the next theorem proves that the two definitions define the same set of pictures.

Theorem 7

(Acyclic equals neutralizable) A picture is neutralizable if, and only if, it is quaternate acyclic.

Proof

Let relation < be acyclic. Then sort the rectangles in topological order and apply neutralization starting from a rectangle without predecessors. When a rectangle is processed, all of its predecessors have already been neutralized, allowing neutralization to proceed until all rectangles are neutralized. The converse is immediate: if relation < contains a cycle, no rectangle in the cycle can be neutralized first, making complete neutralization impossible. $\square$

This result supports the analogy between the neutralization rule for Dyck words and the rule of the same name for pictures: both rely on topological sorting to define an order to be applied to perform neutralization. For Dyck words, the order is a partial order, whereas for pictures it is a directed acyclic graph.

4.2 Well-nested Dyck languages

We now focus on defining Dyck words through well-nested accretion and concatenation, extending this concept to pictures by considering rectangles. The resulting picture language is called well-nested Dyck, denoted by DW.

A Dyck word can be extended either by enclosing it with matching parentheses or by concatenating it with another Dyck word. Similarly, we define well-nested 2D Dyck languages through accretion of well-nested pictures or via concatenations - more precisely, through the more general tiling closure operation.

Definition 6

(Well-nested Dyck language) Let $\Delta =\{a,b,c,d\}$. Define two partial maps $h_r,h_c:\Delta \rightarrow \Delta$ as: $h_r(a)=c$, $h_r(b)=d$ and $h_c(a)=b$, $h_c(c)=d$.

For every picture $p \in \Delta ^{++}$, for all rows $w_r$ in the (word) Dyck language over the parentheses [a, b], and for all columns $w_c$ in the Dyck language over the parentheses [a, c], such that $|w_r|=|p|_{col}$, $|w_c|=|p|_{row}$, the nesting accretion of p is the set of the following two pictures:

The well-nested Dyck language, denoted by $DW_1$, is the smallest set including the square picture $\begin{array}{cc} a & b \\ c & d \end{array}$ and closed under nesting accretion and tiling closure.

For instance, in the case of row accretion, the mapping of $w_r = \{aa\,babb\}$ is $h_r(w_r) = ccdcdd$, which represents a row that closes the column parentheses a, b opened in $w_r$. The case of column accretion is analogous. Fig. 8 illustrates the generation process of a picture by a sequence of steps of row and column accretion.

Fig. 8

A sequence of steps in $DW_1$ to construct the same (neutralizable) picture of Example 3. Starting from the square crossword, we apply row accretion, then row concatenation with a “double square” picture, and finally column accretion

Although $DW_1$ is defined through a different mechanism than neutralizable languages, the inclusion $DW_1 \subseteq DN_1$ is immediate since any picture in $DW_1$ can be neutralized according to Definition 5. The following theorem establishes that this inclusion is strict.

Theorem 8

The language $DW_1$ is strictly included in $DN_1$.

Proof

The thesis follows since the neutralizable picture in Fig. 9, left, is not well-nested. $\square$

In particular, we observe that rectangle $R_1$ is not obtained by an accretion of the rectangles $R_3$ and $R_4$ that are partially contained in $R_1$.

Fig. 9

Left: a picture in $DN_1$ but not in $DW_1$. Right: its partial containment relation is acyclic

4.3 Alternative, unsatisfactory definitions of well-nested Dyck languages

We are going to show that other quite natural definitions of well-nesting are possible, but that in our opinion are less appropriate than the preceding one.

Example 4

(Chinese Boxes) The natural question of what constitutes a Dyck-like language in two dimensions found a preliminary answer in [10], which introduced a language of well-embedded rectangular boxes (called Chinese boxes) as an example demonstrating the generative power of tile rewriting grammars.

The alphabet is defined as , where the corner symbols represent the four vertices of a box, and horizontal or vertical sequences of bullets represent box sides. Rather than using the original grammatical formalism, we present here an equivalent recursive definition. Given a picture p of size (n, m), with $n,m\ge 0$, its Chinese accretion is the picture:

i.e.,

when p is empty.

The Chinese box language is defined as the smallest set containing the empty picture and closed under row and column concatenations, as well as under Chinese accretion.

Fig. 10 shows a Chinese box picture alongside a same-sized picture from $DW_1$ for comparison (right). A row word resembles a Dyck word, with the pairs and acting as matching pairs, but additionally contains the neutral symbol $\bullet$ which has no matching counterpart. The case for column words is analogous, with matching pairs and .

Initially conjectured to lie outside REC, the Chinese boxes language was later shown to admit a tiling system (see [20]), disproving the initial conjecture. Consequently, if we accept that REC languages represent the natural 2D analogue of regular 1D languages, then Chinese boxes fail to properly generalize Dyck languages to two dimensions, since Dyck languages are not regular. In contrast, we will demonstrate in Sect. 5 (Theorem 12) that $DW_1$, $DN_1$ and $DC_1$ are not REC languages.

Fig. 10

(Left) An example of Chinese box. (Right) A similar version in $DW_1$

Example 5

(Perfect nesting) An alternative characterization of the well-nesting property for Dyck words states that substrings delimited by matching pairs must be either disjoint or properly nested. This suggests a natural 2D generalization where we consider, instead of substrings, subpictures bounded by rectangle corners.

A quaternate $DC_1$ picture is perfectly nested if the subpictures associated with rectangles are either pairwise disjoint or included one in the other. A quaternate $DC_1$ picture is perfectly nested if the subpictures corresponding to its rectangles are either pairwise disjoint or properly contained. While conceptually simple, the language of perfectly nested pictures is not tiling-recognizable and exhibits an overly restrictive structure: its partial containment relation (ignoring transitive containment) forms a forest of trees. An alternative, illustrative definition can be given in terms of width-2 pictures.

Let $\Delta =\{a,b,c,d\}$. For any picture $p\in \Delta ^{++}$, let $p^\dagger$ denote its transpose (extended to languages in the standard way). Define the horizontal tape language as , with its transpose $T^\dagger$ being the vertical tape language. It can be verified that the language of perfectly nested pictures equals the tiling closure $(T\cup T^\dagger )^{**}$ of these tape languages, revealing its highly constrained nature. While representing a natural 2D extension of 1D nesting, perfect nesting is too weak to be significant.

5 Dyck crosswords over larger alphabets

Having previously restricted our alphabet to four letters, we now investigate the effects of larger alphabets in the two Dyck coupling relations, by introducing a convenient representation called alphabetic graph. The strong connection between alphabetic graph (one per alphabet) and matching graph (one per picture) permits to compute the label of the matching circuits that may occur in non-$DC_1$ pictures. A key result shows that every circuit appearing in any Dyck crossword picture also appears in some $DC_1$ picture under alphabetic projection. Indeed, since $DC_1$ pictures are completely characterized by their unlabeled matching graphs (with pixel values being redundant), our study essentially constitutes a theory of these matching graphs. This immediate, but fundamental, observation justifies a posteriori our initial focus on $DC_1$, as all syntactic structures of general Dyck crosswords are already present in the four-letter case. Indeed, since $DC_1$ pictures are completely characterized by their unlabeled matching graphs (with pixel values being redundant), our study essentially constitutes a theory of these unlabeled matching graphs.

We pay special attention to the particular case where alphabetic circuits are rectangles.

The section concludes with some general results about Dyck crosswords and their subfamilies (neutralizable and well-nested languages), particularly their non-recognizability by tiling systems.

5.1 Alphabetic graphs

The alphabet of Dyck words contains $k \ge 1$ pairs of coupled letters and is often denoted $\Delta _k$, while the language is indexed as $D_k$. i.e., the parameter k suffices to characterize all Dyck languages. A major change emerges in 2D: unlike Dyck word languages, a single integer parameter (the alphabet half-cardinality k for Dyck words) does not suffice to characterize all the Dyck crosswords. Instead, the set of Dyck crosswords is characterized by a more combinatorial parameter that expresses the way of combining the row/column coupling relations of the alphabet. For any alphabet the particular combination chosen defines a directed graph with the letters as nodes and with bipartite row/column arcs. A distinct Dyck crossword corresponds to each such graph, which generalizes the alphabetic graph anticipated in Example 2.

We formalize the notion of alphabetic graph.

Definition 7

(Alphabetic graph) Let $\Delta$ be an alphabet and let $\widehat{\Delta }^{Row}$,$\widehat{\Delta }^{Col}$ be two Dyck coupling relations. The alphabetic graph of $\widehat{\Delta }^{Row}$, $\widehat{\Delta }^{Col}$ is a directed graph $\Gamma =(V,E)$ such that:

• 1. the set of nodes is $V=\Delta$. The name of a node is called node label;

• 2. the set of edges E is partitioned in the two row/column sorts $\widehat{\Delta }^{Row}$, $\widehat{\Delta }^{Col}$.

The following proposition uses the undirected version of the alphabetic graph.

Proposition 2

(alphabetic circuits) Let $\Delta$ be an alphabet and let $\widehat{\Delta }^{Row}$,$\widehat{\Delta }^{Col}$ be two Dyck coupling relations.

• 1. The alphabetic graph $\Gamma$ of $\widehat{\Delta }^{Row}$,$\widehat{\Delta }^{Col}$ is partitioned into simple, undirected circuits, called alphabetic circuits;

• 2. every alphabetic circuit has even length; if a circuit has length 2 its node labels never occur in a picture.

Proof

The proof of Part (1) is the same as the proof of Theorem 2, Part (1): every symbol in $\Delta$ is exactly in one row coupling and in one column coupling; therefore, every node of the alphabetic graph belongs to two edges (in $\widehat{\Delta }^{Row}$,$\widehat{\Delta }^{Col}$, respectively), leading to simple, undirected circuits. Part (2) follows essentially from Corollary 1. $\square$

Remark 2

(family $\mathcal{D}\mathcal{C}$ indexed by alphabetic graphs) Each alphabet with two Dyck couplings determines an alphabetic graph $\Gamma$, hence also a set of alphabetic circuits, and finally the language $DC_{\Gamma }$. $\mathcal{D}\mathcal{C}$ is the family of all languages $DC_{\Gamma }$ for every possible alphabetic graph $\Gamma$.

Example 6

(Alphabetic graphs over the same alphabet) Let $\Delta = \{a_1, a_2, \dots , a_{12}\}$. Two different alphabetic graphs are illustrated below.

• 1. The couplings

  $$\begin{aligned} \widehat{\Delta }^{Row}=\{[a_1,a_{2}], [a_4,a_3], [a_5,a_6], [a_8,a_7], [ a_9, a_{10} ], [a_{12},a_{11} ]\} \\ \widehat{\Delta }^{Col}=\{[a_2,a_{3}], [a_5,a_4], [a_6,a_7], [a_1,a_8], [ a_{10},a_{11} ], [a_{9},a_{12} ]\}. \end{aligned}$$

  are represented by the alphabetic graph $\Gamma _1$ in Fig. 11, top, with the alphabetic circuits $\gamma _1$ and $\gamma _2$.

• 2. With different Dyck couplings (omitted for brevity), another alphabetic graph $\Gamma _2$, is shown in Fig. 11, bottom, that contains the same circuit $\gamma _2$ and two circuits isomorphic to $\gamma _2$.

Notice that length 8 matching circuits do occur in both $DC_{\Gamma _1}$ and $DC_{\Gamma _2}$ pictures, but with different labels. In $DC_{\Gamma _1}$ such circuits are labeled $a_1 a_2 a_3 a_4 a_5 a_6 a _7 a_8$ or $(a_9 a_{10} a_{11} a_{12})^2$; in $DC_{\Gamma _2}$ they are labeled $(a_1 a_{2} a_{3} a_{4})^2$ or $(a_5 a_{6} a_{7} a_{8})^2$ or $(a_9 a_{10} a_{11} a_{12})^2$.

We observe that all the circuits of graph $\Gamma _2$ have length 4, therefore they partition the alphabet into 3 quadruples of letters, $\{a_1, a_{2}, a_{3}, a_{4}\}$, etc. Dyck crosswords, such as $DC_{\Gamma _2}$, with the alphabet partitioned into $k \ge 1$ quadruples have been named $DC_k$ in previous work [2].

Fig. 11

Two alphabetic graphs $\Gamma _1$ and $\Gamma _2$ on the same alphabet $\{a_1, a_2, \dots , a_{12}\}$, but with different couplings. To ease reading, straight lines are used for row edges and curves for column edges

The Definition 2 of matching graph as well as Theorem 2, Part (1) on matching circuits, immediately extend from $DC_1$ to any language in the family $\mathcal{D}\mathcal{C}$, and we need not repeat them here. On the other hand, we have to reconsider the definition of label for alphabetic circuits and matching circuits since the simple definition given for $DC_1$ does not generalize to every language in $\mathcal{D}\mathcal{C}$.

We first need the following property on the length of matching circuits, which is obvious in $DC_1$ but not for any $DC_{\Gamma }$.

Theorem 9

For every picture in a Dyck Crossword, the length of any matching circuit is a multiple of 4,

Proof

Given the coordinates (i, j) of a node in p, consider the modulo 2 coordinates, namely $(i \mod 2, j \mod 2)$ written as $(i,j)_2$. Modulo 2 coordinates may thus only take values (0, 0), (0, 1), (1, 0), (1, 1). Pick now a pixel in a matching circuit and assume, without loss of generality, that its node is at $(0,0)_2$. It is connected by a horizontal edge to another node in the same row. To ensure the parity condition that there must be an even number of positions in p between two matching symbols, the coordinates of this second pixel must be $(0,1)_2$. This pixel is connected by a vertical edge to a third pixel that must be at coordinates $(1,1)_2$. The third pixel is connected by a horizontal edge to a fourth pixel, at $(1,0)_2$. The fourth pixel must be connected by a vertical edge to a pixel at $(0,0)_2$. No other sequence is possible, i.e., the sequence of modulo 2 coordinates of the grid nodes traversed in a clockwise visit of the (distinct) nodes of a circuit, starting with a node at coordinates $(0,0)_2$, has the form: $\left( (0,0)_2 (0,1)_2 (1,1)_2 (1,0)_2\right) ^+$. To close the circuit, there must be a vertical edge connecting a pixel at $(1,0)_2$ to the first pixel at $(0,0)_2$, therefore the number of arcs of a circuit must be a multiple of four. $\square$

We recall a few standard definitions. Let $w=w_1 \ldots w_n \in \Sigma ^+$, with each $w_i \in \Sigma$, be a word; a circular permutation of w is a word $w_{i+1} w_{i+2} \dots w_n w_1 \dots w_i$, for some $0\le i\le n-1$. The set of all circular permutations of w is denoted by circ(w) and it is called a circular word. Notice that two circular words $circ(w'),circ(w'')$ may be equal even if $w'\ne w''$. The Kleene plus $circ(w)^+$ of a circular word circ(w) is defined as $circ(w^+)$.

Given a circuit $\gamma$ of length n, whether alphabetic or matching, the label of the circuit is denoted as $lab(\gamma )$ and it is defined as the circular word (over $\Delta$) corresponding to the labels encountered during one clockwise visit of the circuit starting from any of its nodes.

We can now examine the circular words made by the labels of the matching circuits that are present in Dyck crosswords pictures, first considering a single matching circuit.

Proposition 3

(Label of a matching circuit) Let $\Gamma$ be an alphabetic graph, let $DC_{\Gamma }=D' \boxplus D''$, and $p \in DC_{\Gamma }$; given a matching circuit $\mu$ of p and its label $lab(\mu )=$  circ(w), where $w \in \Delta ^+$, there exists an alphabetic circuit $\gamma$ such that $circ(w)\in lab(\gamma )^+$.

Loosely speaking, the matching circuit label is obtained by unrolling one or more times the label of an alphabetic circuit.

We now compute the language of the matching circuits in $DC_\Gamma$ having node labels in one of the alphabetic circuits of $\Gamma$.

Theorem 10

(Matching circuit language) Let $DC_{\Gamma }$ be a non-empty Dyck crossword over an alphabetic graph $\Gamma$. Let $\gamma \subseteq \Gamma$ be an alphabetic circuit of length $n\ge 4$. Consider the set $C_\gamma$ of all the matching circuits having node labels in $lab(\gamma )$. Then the language of the labels of $C_\gamma$ is $lab(\gamma )^+$ if n is a multiple of four, $\left( lab(\gamma )lab(\gamma )\right) ^+$ otherwise.

Proof

Let $lab(\gamma )$ be $a_1 \dots a_n$. By Theorem 9, any matching circuit has length multiple of four. Consider a matching circuit of length m such that $\frac{m}{n}$ is a nonzero integer.

From a node labeled $a_j$ in the matching circuit there is an edge connecting it with a node labeled $a_{j+1 \mod n}$, from which there is an edge connecting it with $a_{j+2 \mod n}$, etc., until an edge connecting back to $a_j$ is found.

If the length n of the alphabetic circuit is a multiple of four, then $a_1 \dots a_n$ must be repeated $\frac{m}{n}\ge 1$ times as the circuit label.

If n is not a multiple of four, then the alphabetic circuit must be repeated $\frac{m}{n}\ge 2$ times in the matching circuit. $\square$

We illustrate with two cases. The labels of matching circuits in $DC_1$ form the language $circ(abdc)^+$. If an alphabetic circuit has label $circ(a_1 a_2 a_3 a_4 a_5 a_6)$, the corresponding matching circuits have label language $\left( circ(\left( a_1 a_2 a_3 a_4 a_5 a_6\right) ^2)\right) ^+$.

Theorem 10 reveals that matching circuit labels are generated by unrolling alphabetic circuit labels one or more times, with the number of repetitions determined by whether $lab(\gamma )$ is a multiple of four. This fundamental relationship between alphabetic and matching circuits has important consequences for language inclusion. When considering alphabetic graphs $\Gamma ' \subseteq \Gamma ''$, we observe the language $DC_{\Gamma '}$ is strictly included into $DC_{\Gamma ''}$ because the pictures of the latter may exhibit matching circuits not possible in the former.

A crucial invariance property parallels the one-dimensional case: just as all Dyck word languages $D_k$ (for $k > 1$ parenthesis pairs) share isomorphic syntax tree structures with $D_1$ (when disregarding leaf labels), Dyck crosswords exhibit the same structural independence from alphabet size. The matching graphs–our two-dimensional analogue of syntax trees–remain equally rich regardless of alphabet cardinality.

In fact, we establish that the minimal four-letter alphabet suffices to generate all possible matching graph structures that appear in any larger-alphabet Dyck crossword. Hence, the syntax structure of Dyck crosswords pictures, intended as the presence and disposition of matching circuits, does not depend on the cardinality of the alphabet. This alphabet-minimality result justifies our focused study of $DC_1$ in Sect. 3.2.

Theorem 11

(Circuits of $\mathcal{D}\mathcal{C}$) For every picture in a Dyck crossword, there exists a picture in $DC_1$ of the same size and having the same matching graph when ignoring the node labels.

Proof

Given a picture p in a Dyck crossword over an alphabet $\Delta$, consider one of its matching circuits. The circuit has length m multiple of 4 and its label is a sequence of node labels, from a set of n symbols $\{a_1, \dots ,a_n\}\subseteq \Delta$, with n such that $m \mod n =0$. Assume without loss of generality that $a_1$ is the label of the leftmost pixel of the top row of the circuit. The alphabetic circuit of $\{a_1,\dots ,a_n\}$ has the form:

Let $\pi :\Delta \rightarrow \{a,b,c,d\}$ be the projection such that for all i, with $0\le i\le \frac{n}{4} -1$:

$$\begin{aligned} \pi (a_{4i+1})=a, \pi (a_{4i+2})=b, \pi (a_{4i+3})={ d}, \pi (a_{4i+4})={ c}. \end{aligned}$$

By applying the projection to every matching circuit $\mathcal {c}$ with symbols in $\{a_1, \dots ,a_n\}$, we obtain matching circuits for $DC_1$. By repeating the procedure for the remaining circuits of p, picture p is replaced by a picture $p'$ in $DC_1$, having matching circuits superposable to the circuits of p. $\square$

5.2 Generalization of $DC_1,DN_1,DW_1$ to languages over k quadruples

In analogy with the fact that a Dyck alphabet consists of a finite number of parentheses pairs, we introduce the subfamily $\{DC_k\mid k\ge 1\}$ of $\mathcal{D}\mathcal{C}$, already hinted at in Example 6, whose alphabetic circuits are quadruples.

Proposition 4

(Dyck crossword $DC_k$) For all $k\ge 1$, let $\Delta$ be an alphabet with 4k symbols, partitioned into k quadruplets, denoted as $\Delta =\{a_i, b_i, c_i, d_i \mid 1 \le i \le k\}$. The two coupling relations:

$$\begin{aligned} \left\{ \begin{array}{ll} \widehat{\Delta }^{Row}: & \left\{ [ a_i, b_i] \mid i \le 1 \le k \right\} \cup \left\{ [c_i,d_i]\mid 1 \le i \le k \right\} \\ \widehat{\Delta }^{Col}: & \left\{ [ a_i, c_i] \mid i \le 1 \le k \right\} \cup \left\{ [b_i,d_i]\mid 1 \le i \le k \right\} \end{array} \right. \end{aligned}$$

(4)

define, respectively, two Dyck languages $D^{Row}, D^{Col}$ over $\Delta$, such that the Dyck crossword $DC_k= D^{Row} \boxplus D^{Col}$ is not empty, and all the letters of $\Delta$ are used

The alphabetic graph of $DC_k$ has the form illustrated in Fig. 11, bottom, for the case $k=3$.

For all $k\ge 1$, the quaternate Dyck language $DQ_k$ is the language of quaternate pictures in $DC_k$. The definitions of $DN_1, DW_1$ can be immediately generalized to $DN_k,DW_k$, with the same properties already studied for $k=1$. Then, the family $\mathcal{D}\mathcal{Q}$ of all quaternate Dyck languages is $\bigcup _{k\ge 1} DQ_k$, the family $\mathcal{D}\mathcal{N}$ is $\bigcup _{k\ge 1} DN_k$ and the family $\mathcal{D}\mathcal{W}$ is $\bigcup _{k\ge 1} DW_k$.

Dyck crosswords are not tiling recognizable

When moving from one to two dimensions the family REC of tiling system recognizable languages is often taken as corresponding to the regular, i.e., recognizable word language family. Since Dyck languages are context-free, but not regular, any language pretending to extend them in 2D should be expected not to belong to REC. This turns out to be the case for all three 2D Dyck language definitions in this paper.

We need the following lemma.

Lemma 2

(Language not in REC) Let and for every $n\ge 1$, let $X=\bigcup _n\{ a^nb^n\}, Y=\bigcup _n\{c^nd^n\}$. The language $L= X \ominus R \ominus Y$ is not in the REC family.

Proof

From closure properties of REC, the language R is in REC. By contradiction, assume that L is in REC. By applying the Horizontal Iteration Lemma of [3] (Lemma 9.1) to L, there exists a (suitably large) picture p in L which can be written as the row concatenation of the three (nonempty) pictures x, q, y, i.e., , such that is also in L, thus contradicting the fact that the top row of the pictures in L must have the form $a^n b^n$. $\square$

Theorem 12

(Comparison with REC) All the languages in $\mathcal{D}\mathcal{W}$, in $\mathcal{D}\mathcal{N}$, in $\mathcal{D}\mathcal{Q}$ and in $\bigcup _{k\ge 1} DC_k$ are not in the REC family.

Proof

Without loss of generality, we consider only the languages $DC_1, DQ_1, DN_1, DW_{1}$. Let R be the REC language defined in Lemma 2. Define the language By closure properties of REC, T is in REC as well. The language L of Lemma 2 can be defined as $L= DC_1\cap T = DQ_1\cap T =DN_1 \cap T = DW_{1}\cap T$ (e.g., the first row of every picture in L has the form $a^n b^n$ since this is the only Dyck word in $a^+b^+$). By contradiction, if any of $DC_1, DQ_1, DN_1, DW_{1}$ is in REC, then by closure properties of REC also L would be in REC, contradicting Lemma 2. $\square$

Remark 3

The result of Theorem 12 for $DC_k$ can be immediately extended to any Dyck crossword $L'$ whose matching graph has at least one alphabetic circuit of length 4. In fact, in this case $DC_1$ is (up to isomorphism) included in $L'$: the non-recognizable language of Lemma 2 can thus be defined as $L= L'\cap T$ and the same argument used for $DC_1$, in the proof of the Theorem, applies. Notice that it is currently not known whether a Dyck crossword is empty when its alphabetic graph has no circuit of length 4.

Closures, hierarchy and saturation

Another property of every language L in $\mathcal{D}\mathcal{C}$ is that its tiling closure is L itself. This is obvious since each row of a picture p in L is the concatenation of Dyck words, and similarly for columns. An analogous result holds for each language $DN_1$ and $DQ_1$ (for $DW_1$ this holds by definition).

Theorem 13

(Invariance under tiling closure) For every $k\ge 1$, $(DQ_k)^{**} = DQ_k$, $(DN_k)^{**} = DN_k$, $(DW_k)^{**} = DW_k$.

From Theorems 7,8, from the examples of Fig. 6 and from the definition of alphabetic graph, we have the strict inclusions:

Corollary 2

(Hierarchy) For all $k\ge 1$:

$$\begin{aligned} \begin{array}{l} DW_k\subset DN_k \subset DQ_k\subset DC_k, \text { and } \\ DW_k\subset DW_{k+1}, DN_k\subset DN_{k+1}, DC_k\subset DC_{k+1}. \end{array} \end{aligned}$$

Remark 4

Corollary 2 can partially be extended to all languages in $\mathcal{D}\mathcal{C}$, since for all alphabetic graphs $\Gamma ,\Gamma '$, if the circuits of $\Gamma$ are included in the circuits of $\Gamma '$, then $DC_\Gamma \subseteq DC_{\Gamma '}$. The inclusion is strict if $\Gamma '$ includes an alphabetic circuit of length 4, otherwise the problem is open.

In general, given two regular languages $S',S''$, it is undecidable whether their crossword is empty (Theorem 9.1 of [3]). This implies that there are crosswords that do not saturate their component languages, i.e., such that $ROW(S' \boxplus S'')\subsetneq S'$ and $COL(S' \boxplus S'')\subsetneq S''$. Such a negative property does not hold for Dyck crosswords. Actually, we prove that any of the Dyck-like picture languages defined in this paper does saturate its component language. In fact, although the languages $DW_k, DN_k$ and $DC_k$ are strictly included one in the other, their row (column) languages are identical.

Theorem 14

(Saturation of row/column languages)  For all $k\ge 1$, let $D^{Row}, D^{Col}$ be the row/column Dyck languages of Proposition 4 such that $D^{Row}\boxplus D^{Col}=DC_k$. Then:

$$\begin{aligned}\begin{array}{l} \text {ROW}(DC_k)=\text {ROW}(DN_k) = \text {ROW}(DW_k)= D^{Row}, \\ \text {COL}(DC_k)=\text {COL}(DN_k)= \text {COL}(DW_k) = D^{Col}. \end{array} \end{aligned}$$

Proof

By Corollary 2, it is enough to prove that $D^{Row}\subseteq \text {ROW}(DW_k)$. The reverse inclusion is obvious and the case for columns is symmetrical. Without loss of generality, we consider only the case $k=1$, i.e., we consider the language $DW_1$ over the alphabet $\Delta =\{{a,b,c,d}\}$. We prove by induction on $n\ge 2$, that for every word $w \in D^{Row}_1$ of length n there exists a picture $p \in DW_1$ of the form $w_1 \ominus w_2 \ominus w \ominus w_3$ for $w_1,w_2,w_3 \in D^{Row}_1$.

There are two base cases, the words ab and cd. The word ab is (also) the third row in the $DW_1$ picture $ab\ominus cd\ominus ab \ominus cd$, while cd is (also) the third row in the $DW_1$ picture $ab\ominus ab\ominus cd \ominus cd$.

The induction step has three cases: a word $w\in D^{Row}$ of length $n>2$ has the form $w'w''$, or the form $a w' b$ or the form $c w' d$, for some $w',w'' \in D^{Row}$ of length less than n. Let $p',p''$ be the pictures verifying the induction hypothesis for $w'$ and $w''$ respectively. The case of concatenation $w'w''$ is obvious (just consider the picture ). The case $a w' b$ can be solved by considering the picture , which is in $DW_1$. Similarly, for the case $c w' d$ just consider the $DW_1$ picture . $\square$

6 Conclusion

The mathematical study of Dyck crosswords presents a rich research area with many open questions, particularly concerning pictures containing longer matching circuits. These crosswords can exhibit remarkably diverse patterns whose classification we have only begun to explore, even in the simpler case of rectangular circuits. The variety of possible patterns depends on multiple circuit parameters including: circuit length, number of crossings (both within a circuit and between different circuits), and the relative spatial arrangement of circuits on the grid. We highlight several specific open problems:

First, the empirical observation that every alphabetic graph must contain a length-4 circuit (implying each Dyck crossword picture contains at least one matching circuit of length four) awaits formal proof or disproof. Resolving this would directly address the open problems posed in Remarks 3 and 4.

Second, the number of circuits present in a picture is unbounded for the picture series used in the proof of Theorem 4. This raises the question whether bounding the number of circuits per picture also bounds the total number of such pictures. Theorem 6 provides a partial answer by showing there exists exactly one $DC_1$ picture with a single circuit.

A third problem concerns characterizing which subsets of Dyck crosswords belong (or don’t belong) to REC. For example, despite their visual complexity, the infinite family of pictures in Fig. 4 surprisingly lies within REC (Theorem 5).

Finally, extending classical enumeration results for Dyck paths (e.g., [21]) to two dimensions presents an intriguing challenge for counting $DC_1$ pictures. Developing such enumeration techniques could reveal new combinatorial properties of these structures.