Two-Dimensional Pattern Languages^††thanks: This document is a full version (i. e., it contains all proofs) of the conference paper [3].

Several methods of generation of two-dimensional languages (also called array languages or picture languages) have been proposed in the literature, extending the techniques and results of formal string language theory. A picture is considered as a rectangular array of terminal symbols in the two-dimensional plane. Models based on grammars or automata as well as those based on theoretical properties of the string languages are well-known and have been extensively investigated. We refer the interested readers to books and surveys like the ones by Rosenfeld [12], Wang [15], Rosenfeld and Siromoney [13], Giammarresi and Restivo [7], or Morita [11]. For example, regular string languages (also known as recognizable string languages) can be characterized in terms of local languages and projections. Based on a similar idea, the class REC of recognizable picture languages (see Giammarresi and Restivo [6]) was proposed as a two dimensional counterpart of regular string languages. In this work, we attempt to generalise a class of string languages to the two-dimensional case, which also provides several desirable features and has therefore attracted considerable interest over the last three decades in the formal language theory community as well as in the learning theory community: Angluin’s pattern languages (see [1]).

In this context, a pattern is a string over an alphabet $\{x_{1},x_{2},x_{3},\ldots\}$ of variables, e. g., $\alpha:=x_{1}\,x_{1}\,x_{2}\,x_{2}\,x_{1}$. For some finite alphabet $\Sigma$ of terminal symbols, the pattern language described by $\alpha$ (with respect to $\Sigma$) is the set of all words over $\Sigma$ that can be derived from $\alpha$ by uniformly substituting the variables in $\alpha$ by (non-empty) terminal words. For example, if $\Sigma:=\{\mathtt{a},\mathtt{b},\mathtt{c}\}$, then $u:=\mathtt{b}\mathtt{c}\mathtt{b}\mathtt{b}\mathtt{c}\mathtt{b}\mathtt{c}\mathtt{c}\mathtt{a}\mathtt{c}\mathtt{c}\mathtt{a}\mathtt{b}\mathtt{c}\mathtt{b}$ and $v:=\mathtt{a}\mathtt{b}\mathtt{a}\mathtt{b}\mathtt{a}\mathtt{b}\mathtt{a}\mathtt{a}\mathtt{b}\mathtt{a}\mathtt{a}\mathtt{b}$ are words of the pattern language given by $\alpha$, since replacing $x_{1}$ by $\mathtt{b}\mathtt{c}\mathtt{b}$ and $x_{2}$ by $\mathtt{c}\mathtt{c}\mathtt{a}$ turns $\alpha$ into $u$ and replacing $x_{1}$ by $\mathtt{a}\mathtt{b}$ and $x_{2}$ by $\mathtt{a}\mathtt{b}\mathtt{a}$ turns $\alpha$ into $v$. On the other hand, the word $\mathtt{c}\mathtt{a}\mathtt{b}\mathtt{a}\mathtt{c}\mathtt{a}\mathtt{b}\mathtt{a}\mathtt{b}\mathtt{a}\mathtt{b}\mathtt{a}$ is not a member of the pattern language of $\alpha$.

One of the most notable features of pattern languages is that they have natural and compact human readable descriptors (or generators), namely the patterns. In particular, this advantage becomes evident when patterns are compared to other language descriptors as, e. g., grammars or automata, which are usually quite involved even though the language they describe is rather simple. Nevertheless, patterns can compete with common automata models and grammars in terms of expressive power and their practical relevance is demonstrated by the widespread use of so-called extended regular expressions with backreferences, which implicitly use the concept of patterns and are capable of defining all pattern languages.¹¹1In fact, these extended regular expressions with backreferences are nowadays a standard element of most text editors and programming languages (cf. Friedl [5]).

The main goal of this paper is to generalise the concept of patterns as language descriptors to the two-dimensional case, while preserving the desirable features of (one-dimensional) pattern languages, i. e., the simplicity and compactness of their descriptors. The work done so far on two-dimensional languages demonstrates that there are difficulties that seem to be symptomatic for the task of generalising a class of string languages to the two-dimensional case. Firstly, such a generalisation is usually accompanied with a substantial increase in complexity of the descriptors (e. g., when extending context-free or contextual grammars to the two-dimensional case (see Fernau et al. [2], Freund et al. [4])) and, secondly, there are often many competing and seemingly different ways to generalise a specific class of string languages, which all can be considered natural (e. g., it is still on debate what the appropriate two-dimensional counterpart of the class of regular languages might be (see Giammarresi et al. [8], Matz [10])). Our two-dimensional patterns, to be introduced in this work, are as simple and compact as their one-dimensional counterparts. Although there are several different possibilities of how these two-dimensional patterns can describe two-dimensional languages, one of these sticks out as the intuitively most natural one. Hence, the model of Angluin’s pattern languages seems to be comparatively two-dimensional friendly.

Besides the conceptional contribution of this paper, we present a comparison between the expressive power of different classes of two-dimensional pattern languages and an investigation of their closure properties. We conclude the paper by outlining further research questions and possible extensions to the model of two-dimensional pattern languages.

In this section, we briefly recall the standard definitions and notations regarding one- and two-dimensional words and languages.

Let $\mathbb{N}:=\{1,2,3,\ldots\}$ and let $\mathbb{N}_{0}:=\mathbb{N}\cup\{0\}$. For a finite alphabet $\Sigma$, a string or word (over $\Sigma$) is a finite sequence of symbols from $\Sigma$, and $\varepsilon$ stands for the empty string. The notation $\Sigma^{+}$ denotes the set of all nonempty strings over $\Sigma$, and $\Sigma^{*}:=\Sigma^{+}\cup\{\varepsilon\}$. For the concatenation of two strings $w_{1},w_{2}$ we write $w_{1}\cdot w_{2}$ or simply $w_{1}w_{2}$. We say that a string $v\in\Sigma^{*}$ is a factor of a string $w\in\Sigma^{*}$ if there are $u_{1},u_{2}\in\Sigma^{*}$ such that $w=u_{1}\cdot v\cdot u_{2}$. If $u_{1}$ or $u_{2}$ is the empty string, then $v$ is a prefix (or a suffix, respectively) of $w$. The notation $|w|$ stands for the length of a string $w$.

A two-dimensional word (or array) over $\Sigma$ is a tuple

$$W:=((a_{1,1},a_{1,2},\ldots,a_{1,n}),(a_{2,1},a_{2,2},\ldots,a_{2,n}),\ldots,(a_{m,1},a_{m,2},\ldots,a_{m,n}))\,,$$

where $m,n\in\mathbb{N}$ and, for every $i$, $1\leq i\leq m$, and $j$, $1\leq j\leq n$, $a_{i,j}\in\Sigma$. We define the number of columns (or width) and number of rows (or height) of $W$ by $|W|_{c}:=n$ and $|W|_{r}:=m$, respectively. The empty array is denoted by $\lambda$, i. e., $|\lambda|_{c}=|\lambda|_{r}=0$. For the sake of convenience, we also denote $W$ by $[a_{i,j}]_{m,n}$ or by a matrix of one of the following forms:

$$\begin{smallmatrix}a_{1,1}&a_{1,2}&\ldots&a_{1,n}\\ a_{2,1}&a_{2,2}&\ldots&a_{2,n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{m,1}&a_{m,2}&\ldots&a_{m,n}\end{smallmatrix},\>\>\>\>\>\>\begin{bmatrix}a_{1,1}&a_{1,2}&\ldots&a_{1,n}\\ a_{2,1}&a_{2,2}&\ldots&a_{2,n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{m,1}&a_{m,2}&\ldots&a_{m,n}\end{bmatrix}\,.$$

If we want to refer to the $j^{\text{th}}$ symbol in row $i$ of the array $W$, then we use $W[i,j]=a_{i,j}$. By $\Sigma^{++}$, we denote the set of all nonempty arrays over $\Sigma$, and $\Sigma^{**}:=\Sigma^{++}\cup\{\lambda\}$. Every subset $L\subseteq\Sigma^{**}$ is an array language.

Let $W:=[a_{i,j}]_{m,n}$ and $W^{\prime}:=[a^{\prime}_{i,j}]_{m^{\prime},n^{\prime}}$ be two non-empty arrays over $\Sigma$. The column concatenation of $W$ and $W^{\prime}$, denoted by $W\varobar W^{\prime}$, is undefined if $m\neq m^{\prime}$ and is the array

$$\begin{smallmatrix}a_{1,1}&a_{1,2}&\ldots&a_{1,n}&b_{1,1}&b_{1,2}&\ldots&b_{1,n^{\prime}}\\ a_{2,1}&a_{2,2}&\ldots&a_{2,n}&b_{2,1}&b_{2,2}&\ldots&b_{2,n^{\prime}}\\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots\\ a_{m,1}&a_{m,2}&\ldots&a_{m,n}&b_{m^{\prime},1}&b_{m^{\prime},2}&\ldots&b_{m^{\prime},n^{\prime}}\end{smallmatrix}$$

otherwise. The row concatenation of $W$ and $W^{\prime}$, denoted by $W\varominus W^{\prime}$, is undefined if $n\neq n^{\prime}$ and is the array

$$\begin{smallmatrix}a_{1,1}&a_{1,2}&\ldots&a_{1,n}\\ a_{2,1}&a_{2,2}&\ldots&a_{2,n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{m,1}&a_{m,2}&\ldots&a_{m,n}\\ b_{1,1}&b_{1,2}&\ldots&b_{1,n^{\prime}}\\ b_{2,1}&b_{2,2}&\ldots&b_{2,n^{\prime}}\\ \vdots&\vdots&\ddots&\vdots\\ b_{m^{\prime},1}&b_{m^{\prime},2}&\ldots&b_{m^{\prime},n^{\prime}}\end{smallmatrix}$$

otherwise. Intuitively speaking, the vertical line and the horizontal line in the symbols $\varobar$ and $\varominus$, respectively, indicate the edge where the arrays are concatenated. In order to denote that, e. g., $U\varominus V$ is undefined, we also write $U\varominus V=\operatorname{\texttt{undef}}$.

Furthermore, for every array $U$, $U\varobar\lambda=\lambda\varobar U=U\varominus\lambda=\lambda\varominus U=U$ and $U\varobar\operatorname{\texttt{undef}}=\operatorname{\texttt{undef}}\varobar U=U\varominus\operatorname{\texttt{undef}}=\operatorname{\texttt{undef}}\varominus U=\operatorname{\texttt{undef}}$. Algebraically speaking, if $\Sigma^{**}_{\operatorname{\texttt{undef}}}:=\Sigma^{**}\cup\{\operatorname{\texttt{undef}}\}$, then $(\Sigma^{**}_{\operatorname{\texttt{undef}}},\varobar,\lambda)$ and $(\Sigma^{**}_{\operatorname{\texttt{undef}}},\varominus,\lambda)$ both form monoids with $\operatorname{\texttt{undef}}$ as an absorbing element.

Next, we define some operations for array languages. The row and column concatenation for array languages $L_{1}$ and $L_{2}$ is defined by $L_{1}\varominus L_{2}:=\{U\varominus V\mid U\in L_{1},V\in L_{2},U\varominus V\neq\operatorname{\texttt{undef}}\}$ and $L_{1}\varobar L_{2}:=\{U\varobar V\mid U\in L_{1},V\in L_{2},U\varobar V\neq\operatorname{\texttt{undef}}\}$, respectively. For an array language $L$ and $k\in\mathbb{N}$, $L^{\varominus k}$ denotes the $k$-fold row concatenation of $L$, i. e., $L^{\varominus k}:=L_{1}\varominus L_{2}\varominus\ldots\varominus L_{k}$, $L_{i}=L$, $1\leq i\leq k$. The $k$-fold column concatenation, denoted by $L^{\varobar k}$, is defined analogously. The row and column concatenation closure of an array language $L$ is defined by $L^{\varominus*}:=\bigcup_{k=1}^{\infty}L^{\varominus k}$ and $L^{\varobar*}:=\bigcup_{k=1}^{\infty}L^{\varobar k}$, respectively. Obviously, the row and column concatenation closure of an array language correspond to the Kleene closure of a string language.

Now, we turn our attention to some geometric operations for arrays. The transposition of an array $U$, denoted by ${U}^{\texttt{T}}$, is obtained by reflecting $U$ along the main diagonal. The $\varominus$-reflection and $\varobar$-reflection of $U$, denoted by $U^{\varominus\texttt{R}}$ and $U^{\varobar\texttt{R}}$, respectively, are obtained by reflecting $U$ along the horizontal and vertical axis, respectively. The right turn and left turn of $U$, denoted by $U^{\curvearrowright}$ and $U^{\curvearrowleft}$, respectively, is obtained by turning $U$ through $90$ degrees to the right and to the left, respectively. For example, if $U:=\begin{bmatrix}\mathtt{a}&\mathtt{b}&\mathtt{c}&\mathtt{d}\\ \mathtt{e}&\mathtt{f}&\mathtt{g}&\mathtt{h}\end{bmatrix}$, then

$\displaystyle{U}^{\texttt{T}}=\begin{bmatrix}\mathtt{a}&\mathtt{e}\\ \mathtt{b}&\mathtt{f}\\ \mathtt{c}&\mathtt{g}\\ \mathtt{d}&\mathtt{h}\\ \end{bmatrix}\,,U^{\varominus\texttt{R}}=\begin{bmatrix}\mathtt{e}&\mathtt{f}&\mathtt{g}&\mathtt{h}\\ \mathtt{a}&\mathtt{b}&\mathtt{c}&\mathtt{d}\\ \end{bmatrix}\,,U^{\varobar\texttt{R}}=\begin{bmatrix}\mathtt{d}&\mathtt{c}&\mathtt{b}&\mathtt{a}\\ \mathtt{h}&\mathtt{g}&\mathtt{f}&\mathtt{e}\\ \end{bmatrix}\,,U^{\curvearrowright}=\begin{bmatrix}\mathtt{e}&\mathtt{a}\\ \mathtt{f}&\mathtt{b}\\ \mathtt{g}&\mathtt{c}\\ \mathtt{h}&\mathtt{d}\\ \end{bmatrix}\,,U^{\curvearrowleft}=\begin{bmatrix}\mathtt{d}&\mathtt{h}\\ \mathtt{c}&\mathtt{g}\\ \mathtt{b}&\mathtt{f}\\ \mathtt{a}&\mathtt{e}\\ \end{bmatrix}\,,(U^{\curvearrowright})^{\curvearrowright}=\begin{bmatrix}\mathtt{h}&\mathtt{g}&\mathtt{f}&\mathtt{e}\\ \mathtt{d}&\mathtt{c}&\mathtt{b}&\mathtt{a}\end{bmatrix}\,.$

We address left and right turn also as quarter-turns below. Moreover, the twofold right turn (displayed right-most in the example above) is also known as a half-turn.

A special operation considered in the context of arrays is the conjugation of an array $U\in\Sigma^{**}$ with $|\Sigma|=2$, denoted by $U^{\texttt{C}}$, which means that the two symbols of $\Sigma$ are exchanged in $U$, e. g., $\left(\begin{bmatrix}\mathtt{a}&\mathtt{b}&\mathtt{a}&\mathtt{a}\\ \mathtt{b}&\mathtt{a}&\mathtt{a}&\mathtt{b}\\ \end{bmatrix}\right)^{\texttt{C}}=\begin{bmatrix}\mathtt{b}&\mathtt{a}&\mathtt{b}&\mathtt{b}\\ \mathtt{a}&\mathtt{b}&\mathtt{b}&\mathtt{a}\\ \end{bmatrix}$. This can be also viewed as a quite restricted form of a two-dimensional morphism defined in the next section.

All these operations for arrays are extended to array languages in the obvious way.

Next, we briefly summarise the concept of (one-dimensional) pattern languages as introduced in [1] by Angluin. Technically, the version of pattern languages used here are called nonerasing terminal-free pattern languages (for an overview of different versions of one-dimensional pattern languages, the reader is referred to [9] by Mateescu and Salomaa).

A (one-dimensional) pattern is a string over an alphabet $X:=\{x_{1},x_{2},x_{3},\ldots\}$ of variables, e. g., $\alpha:=x_{1}\,x_{1}\,x_{2}\,x_{2}\,x_{1}$. In Section 1, we have seen an intuitive definition of the language described by a pattern $\alpha$. This intuition can be formalised in an elegant way by using the concept of (word) morphisms, i. e., mappings $h:\Sigma_{1}^{+}\rightarrow\Sigma_{2}^{+}$, which satisfy $h(u\,v)=h(u)\,h(v)$, for all $u,v\in\Sigma_{1}^{+}$. In this regard, for some finite alphabet $\Sigma$, the (one-dimensional) pattern language of $\alpha$ (with respect to $\Sigma$) is the set $L^{\operatorname{\texttt{1D}}}_{\Sigma}(\alpha):=\{h(\alpha)\mid h:X^{+}\rightarrow\Sigma^{+}\text{ is a morphism}\}$. An alternative, yet equivalent, way to define pattern languages is by means of factorisations. To this end, let $\alpha:=y_{1}\,y_{2}\ldots y_{n}$, $y_{i}\in X$, $1\leq i\leq n$. Then $L^{\operatorname{\texttt{1D}}}_{\Sigma}(\alpha)$ is the set of all words $w\in\Sigma^{+}$ that have a characteristic factorisation for $\alpha$, i. e., a factorisation $w=u_{1}\,u_{2}\cdots u_{n}$, such that, for every $i$, $1\leq i\leq j\leq n$, $y_{i}=y_{j}$ implies $u_{i}=u_{j}$. It can be easily seen, that these two definitions are equivalent. However, for the two-dimensional case, we shall see that a generalisation of these two approaches will lead to different versions of two-dimensional pattern languages. The class of all one-dimensional pattern languages over the alphabet $\Sigma$ is denoted by $\mathcal{L}^{\operatorname{\texttt{1D}}}_{\Sigma}$. We recall the example pattern $\alpha=x_{1}\,x_{1}\,x_{2}\,x_{2}\,x_{1}$ and the words $u:=\mathtt{b}\mathtt{c}\mathtt{b}\mathtt{b}\mathtt{c}\mathtt{b}\mathtt{c}\mathtt{c}\mathtt{a}\mathtt{c}\mathtt{c}\mathtt{a}\mathtt{b}\mathtt{c}\mathtt{b}$ and $v:=\mathtt{a}\mathtt{b}\mathtt{a}\mathtt{b}\mathtt{a}\mathtt{b}\mathtt{a}\mathtt{a}\mathtt{b}\mathtt{a}\mathtt{a}\mathtt{b}$ of Section 1 Since $h(\alpha)=u$ and $g(\alpha)=v$, where $h$ and $g$ are the morphisms induced by $h(x_{1}):=\mathtt{b}\mathtt{c}\mathtt{b}$, $h(x_{2}):=\mathtt{c}\mathtt{c}\mathtt{a}$ and $g(x_{1}):=\mathtt{a}\mathtt{b}$, $g(x_{2}):=\mathtt{a}\mathtt{b}\mathtt{a}$, we can conclude that $u,v\in L^{\operatorname{\texttt{1D}}}_{\Sigma}(\alpha)$, where $\Sigma:=\{\mathtt{a},\mathtt{b},\mathtt{c}\}$.

As already mentioned, this work deals with the task of generalising pattern languages from the one-dimensional to the two-dimensional case. In order to motivate our approach to solve this task, we first spent some effort on illustrating the general difficulties and obstacles that arise.

Abstractly speaking, a pattern language for a given pattern $\alpha$ is the collection of all elements that satisfy $\alpha$. Thus, a sound definition of how elements satisfy patterns directly entails a sound definition of a class of pattern languages. In the one-dimensional case, the situation that a word satisfies a pattern is intuitively clear and it can be defined in several equivalent ways, i. e., a word $w$ satisfies the pattern $\alpha$ if and only if

• •

  $w$ can be derived from $\alpha$ by uniformly substituting the variables in $\alpha$,

• •

  $w$ is a morphic image of $\alpha$,

• •

  $w$ has a characteristic factorisation for $\alpha$.

We shall now demonstrate that for a two-dimensional pattern, i. e., a two-dimensional word over the set of variables $X$, e. g., $\alpha:=\begin{bmatrix}x_{1}&x_{2}\\ x_{3}&x_{1}\end{bmatrix}$, these concepts do not work anymore or they describe fundamentally different situations. For instance, the basic operation of substituting a single symbol in a word by another word cannot that easily be extended to the two-dimensional case. For example, the replacements $x_{1}\mapsto\begin{bmatrix}\mathtt{a}&\mathtt{a}\end{bmatrix},x_{2}\mapsto\begin{bmatrix}\mathtt{c}\\ \mathtt{c}\end{bmatrix}$ and $x_{3}\mapsto\begin{bmatrix}\mathtt{b}\end{bmatrix}$ may turn $\alpha$ into one of the following objects,

$$\begin{bmatrix}&&\mathtt{c}&\\ \mathtt{a}&\mathtt{a}&\mathtt{c}&\\ &\mathtt{b}&\mathtt{a}&\mathtt{a}\end{bmatrix},\begin{bmatrix}\mathtt{a}&\mathtt{a}&\mathtt{c}\\ &&\mathtt{c}\\ \mathtt{b}&\mathtt{a}&\mathtt{a}\end{bmatrix},\begin{bmatrix}&&\mathtt{c}\\ \mathtt{a}&\mathtt{a}&\mathtt{c}\\ \mathtt{b}&\mathtt{a}&\mathtt{a}\end{bmatrix},\begin{bmatrix}&&&\mathtt{c}\\ \mathtt{a}&\mathtt{a}&&\mathtt{c}\\ &\mathtt{b}&\mathtt{a}&\mathtt{a}\end{bmatrix},$$

which are not two-dimensional words, since they all contain holes or are not of rectangular shape and, most importantly, are not uniquely defined. On the other hand, it is straightforward to generalise the concept of a morphism to the two-dimensional case:

Hence, we may say that a two-dimensional word $W$ satisfies a two-dimensional pattern $\alpha$ if and only if there exists a two-dimensional morphism which maps $\alpha$ to $W$. Unfortunately, this definition seems to be too strong as demonstrated by the following example. From an intuitive point of view, the two-dimensional word $W:=\begin{bmatrix}\mathtt{a}&\mathtt{a}&\mathtt{b}&\mathtt{a}&\mathtt{a}&\mathtt{b}\\ \mathtt{a}&\mathtt{a}&\mathtt{b}&\mathtt{a}&\mathtt{a}&\mathtt{b}\\ \mathtt{c}&\mathtt{c}&\mathtt{c}&\mathtt{c}&\mathtt{c}&\mathtt{c}\end{bmatrix}$ should satisfy the two-dimensional pattern $\alpha:=\begin{bmatrix}x_{1}&x_{1}\\ x_{2}&x_{2}\end{bmatrix}$, but there is no two-dimensional morphism mapping $\alpha$ to $W$. This is due to the fact that, as pointed out by the following proposition (which has also been mentioned by Siromoney et al. in [14]), a two-dimensional morphism is a mapping with a surprisingly strong condition.

Similarly as in the string case, homomorphisms $h:\Sigma_{1}^{++}\rightarrow\Sigma_{2}^{++}$ are uniquely defined by giving the images $h(\Sigma_{1})$. If in particular $h(\Sigma_{1})\subseteq\Sigma_{2}$, we term the resulting morphism a letter-to-letter morphism, while in the even more restricted case when the restriction $h_{\Sigma_{1}}$ of $h$ to $\Sigma_{1}$ yields a surjective mapping $h_{\Sigma_{1}}:\Sigma_{1}\to\Sigma_{2}$, $h$ is referred to as a projection. We can conclude that the existence of a two-dimensional morphism seems to be a reasonable sufficient criterion for the situation that a two-dimensional word satisfies a two-dimensional pattern, but not a necessary one.

In fact, it turns out that characteristic factorisations provide the most promising approach to formalise how a two-dimensional word satisfies a two-dimensional pattern. Recall the example pattern $\alpha=\begin{bmatrix}x_{1}&x_{1}\\ x_{2}&x_{2}\end{bmatrix}$ from above. Since $\alpha=(\begin{bmatrix}x_{1}\end{bmatrix}\varobar\begin{bmatrix}x_{1}\end{bmatrix})\varominus(\begin{bmatrix}x_{2}\end{bmatrix}\varobar\begin{bmatrix}x_{2}\end{bmatrix})$, a characteristic factorisation of a two-dimensional word $U$ for $\alpha$ is a factorisation of the form $U=(V_{1}\varobar V_{1})\varominus(V_{2}\varobar V_{2})$. On the other hand, since $\alpha=(\begin{bmatrix}x_{1}\end{bmatrix}\varominus\begin{bmatrix}x_{2}\end{bmatrix})\varobar(\begin{bmatrix}x_{1}\end{bmatrix}\varominus\begin{bmatrix}x_{2}\end{bmatrix})$, we could as well regard a factorisation $U=(V_{1}\varominus V_{2})\varobar(V_{1}\varominus V_{2})$ as characteristic for $\alpha$. For the sake of convenience, we say that the former factorisation is of column-row type and the latter one is of row-column type. Obviously, the two-dimensional word $W$ from above has a characteristic factorisation of column-row type and a characteristic factorisation of row-column type (with respect to $\alpha$):

	$\displaystyle W=(V_{1}\varobar V_{1})\varominus(V_{2}\varobar V_{2})=(\begin{bmatrix}\mathtt{a}&\mathtt{a}&\mathtt{b}\\ \mathtt{a}&\mathtt{a}&\mathtt{b}\end{bmatrix}\varobar\begin{bmatrix}\mathtt{a}&\mathtt{a}&\mathtt{b}\\ \mathtt{a}&\mathtt{a}&\mathtt{b}\end{bmatrix})\varominus(\begin{bmatrix}\mathtt{c}&\mathtt{c}&\mathtt{c}\end{bmatrix}\varobar\begin{bmatrix}\mathtt{c}&\mathtt{c}&\mathtt{c}\end{bmatrix})=\begin{bmatrix}\mathtt{a}&\mathtt{a}&\mathtt{b}&\mathtt{a}&\mathtt{a}&\mathtt{b}\\ \mathtt{a}&\mathtt{a}&\mathtt{b}&\mathtt{a}&\mathtt{a}&\mathtt{b}\\ \mathtt{c}&\mathtt{c}&\mathtt{c}&\mathtt{c}&\mathtt{c}&\mathtt{c}\end{bmatrix}\,,$
	$\displaystyle W=(V_{1}\varominus V_{2})\varobar(V_{1}\varominus V_{2})=(\begin{bmatrix}\mathtt{a}&\mathtt{a}&\mathtt{b}\\ \mathtt{a}&\mathtt{a}&\mathtt{b}\end{bmatrix}\varominus\begin{bmatrix}\mathtt{c}&\mathtt{c}&\mathtt{c}\end{bmatrix})\varobar(\begin{bmatrix}\mathtt{a}&\mathtt{a}&\mathtt{b}\\ \mathtt{a}&\mathtt{a}&\mathtt{b}\end{bmatrix}\varominus\begin{bmatrix}\mathtt{c}&\mathtt{c}&\mathtt{c}\end{bmatrix})=\begin{bmatrix}\mathtt{a}&\mathtt{a}&\mathtt{b}&\mathtt{a}&\mathtt{a}&\mathtt{b}\\ \mathtt{a}&\mathtt{a}&\mathtt{b}&\mathtt{a}&\mathtt{a}&\mathtt{b}\\ \mathtt{c}&\mathtt{c}&\mathtt{c}&\mathtt{c}&\mathtt{c}&\mathtt{c}\end{bmatrix}\,.$

As a matter of fact, for every two-dimensional word $U$ there exists a characteristic factorisation for $\alpha=\begin{bmatrix}x_{1}&x_{1}\\ x_{2}&x_{2}\end{bmatrix}$ of column-row type if and only if there exists a characteristic factorisation for $\alpha$ of row-column type. However, this is a particularity of $\alpha$ and, e. g., for $\alpha^{\prime}=\begin{bmatrix}x_{1}&x_{2}&x_{3}\\ x_{2}&x_{3}&x_{1}\end{bmatrix}$ and $W^{\prime}:=\begin{bmatrix}\mathtt{a}&\mathtt{a}&\mathtt{a}&\mathtt{b}&\mathtt{c}\\ \mathtt{b}&\mathtt{c}&\mathtt{a}&\mathtt{a}&\mathtt{a}\\ \end{bmatrix}$, there exists a characteristic factorisation of column-row type $W^{\prime}=(V_{1}\varobar V_{2}\varobar V_{3})\varominus(V_{2}\varobar V_{3}\varobar V_{1})=(\begin{bmatrix}\mathtt{a}&\mathtt{a}&\mathtt{a}\end{bmatrix}\varobar\begin{bmatrix}\mathtt{b}\end{bmatrix}\varobar\begin{bmatrix}\mathtt{c}\end{bmatrix})\varominus(\begin{bmatrix}\mathtt{b}\end{bmatrix}\varobar\begin{bmatrix}\mathtt{c}\end{bmatrix}\varobar\begin{bmatrix}\mathtt{a}&\mathtt{a}&\mathtt{a}\end{bmatrix})$, but no characteristic factorisation of row-column type. Furthermore, the column-row factorisation of $W^{\prime}$ is somewhat at odds with our intuitive understanding of what it means that a two-dimensional word satisfies a two-dimensional pattern. This is due to the fact that factorising $W^{\prime}$ into $(\begin{bmatrix}\mathtt{a}&\mathtt{a}&\mathtt{a}\end{bmatrix}\varobar\begin{bmatrix}\mathtt{b}\end{bmatrix}\varobar\begin{bmatrix}\mathtt{c}\end{bmatrix})\varominus(\begin{bmatrix}\mathtt{b}\end{bmatrix}\varobar\begin{bmatrix}\mathtt{c}\end{bmatrix}\varobar\begin{bmatrix}\mathtt{a}&\mathtt{a}&\mathtt{a}\end{bmatrix})$ means that we associate the two-dimensional factors $\begin{bmatrix}\mathtt{a}&\mathtt{a}&\mathtt{a}\end{bmatrix}$, $\begin{bmatrix}\mathtt{b}\end{bmatrix}$ and $\begin{bmatrix}\mathtt{c}\end{bmatrix}$ with the variables $x_{1}$, $x_{2}$ and $x_{3}$, respectively, but in the pattern $\alpha^{\prime}$ the vertical neighbourship relation between the occurrence of $x_{2}$ in the first row and the occurrence of $x_{3}$ in the second row is not preserved in $W^{\prime}$ with respect to the corresponding two-dimensional factors $\begin{bmatrix}\mathtt{b}\end{bmatrix}$ and $\begin{bmatrix}\mathtt{c}\end{bmatrix}$. More precisely, while a column-row factorisation preserves the horizontal neighbourship relation of the variables, it may violate their vertical neighbourship relation, where for row-column factorisations it is the other way around. Consequently, if we want both the vertical as well as the horizontal neighbourship relation to be preserved, we should require that the two-dimensional word $U$ can be disassembled into two-dimensional factors that induce both a column-row as well as a row-column factorisation. More precisely, we say that $U$ satisfies $\alpha^{\prime}=\begin{bmatrix}x_{1}&x_{2}&x_{3}\\ x_{2}&x_{3}&x_{1}\end{bmatrix}$ if and only if there exist two-dimensional words $V_{1},V_{2}$ and $V_{3}$, such that $U=(V_{1}\varobar V_{2}\varobar V_{3})\varominus(V_{2}\varobar V_{3}\varobar V_{1})=(V_{1}\varominus V_{2})\varobar(V_{2}\varominus V_{3})\varobar(V_{3}\varominus V_{1})$, which we call a proper characteristic factorisation of $U$.

We are now ready to formalise the ideas developed so far and we can finally give a sound definition of two-dimensional pattern languages. Although we consider the class of two-dimensional pattern languages that results from the proper characteristic factorisations as the natural two-dimensional counterpart of the class of one-dimensional pattern languages, we shall also define the other classes of two-dimensional pattern languages which were sketched above.

For the definition of two-dimensional patterns, we use the same set of variables $X$ that has already been used in the definition of one-dimensional pattern languages. An array pattern is a non-empty two-dimensional word over $X$ and a terminal array is a non-empty two-dimensional word over a terminal alphabet $\Sigma$. If it is clear from the context that we are concerned with array patterns and terminal arrays, then we simply say pattern and array, respectively. Any mapping $h:X\rightarrow\Sigma^{++}$ is called a substitution. For any substitution $h$, by $h_{\varobar,\varominus}$, we denote the mapping $X^{++}\rightarrow\Sigma^{++}$ defined in the following way. For any $\alpha:=[y_{i,j}]_{m,n}\in X^{++}$, we define

	$\displaystyle h_{\varobar,\varominus}(\alpha):=\>$	$\displaystyle(h(y_{1,1})\varobar h(y_{1,2})\varobar\ldots\varobar h(y_{1,n}))\varominus$
		$\displaystyle(h(y_{2,1})\varobar h(y_{2,2})\varobar\ldots\varobar h(y_{2,n}))\varominus\ldots\varominus$
		$\displaystyle(h(y_{m,1})\varobar h(y_{m,2})\varobar\ldots\varobar h(y_{m,n}))\,.$

Similarly, $h_{\varominus,\varobar}:X^{++}\rightarrow\Sigma^{++}$ is defined by

	$\displaystyle h_{\varominus,\varobar}(\alpha):=\>$	$\displaystyle(h(y_{1,1})\varominus h(y_{2,1})\varominus\ldots\varominus h(y_{m,1}))\varobar$
		$\displaystyle(h(y_{1,2})\varominus h(y_{2,2})\varominus\ldots\varominus h(y_{m,2}))\varobar\ldots\varobar$
		$\displaystyle(h(y_{1,n})\varominus h(y_{2,n})\varominus\ldots\varominus h(y_{m,n}))\,.$

Intuitively speaking, both mappings $h_{\varobar,\varominus}$ and $h_{\varobar,\varominus}$, when applied to an array pattern $\alpha$, first substitute every variable occurrence of $\alpha$ by a terminal array according to the substitution $h$ and then these $m\times n$ individual terminal arrays are assembled to one terminal array by either first column-concatenating all the $n$ terminal arrays in every individual row and then row-concatenating the resulting $m$ terminal arrays, or by first row-concatenating all the $m$ terminal arrays in every individual column and then column-concatenating the resulting $n$ terminal arrays.

Let $\alpha\in X^{++}$, $W\in\Sigma^{++}$ and let $h:X\rightarrow\Sigma^{++}$. The array $W$ is a (1) column-row image of $\alpha$ (with respect to $h$), (2) a row-column image of $\alpha$ (with respect to $h$) or (3) a proper image of $\alpha$ (with respect to $h$) if and only if (1) $h_{\varobar,\varominus}(\alpha)=W$, (2) $h_{\varominus,\varobar}(\alpha)=W$ or (3) $h_{\varobar,\varominus}(\alpha)=h_{\varominus,\varobar}(\alpha)=W$, respectively. The mapping $h$ is called a column-row substitution for $\alpha$ and $W$, a row-column substitution for $\alpha$ and $W$ or a proper substitution for $\alpha$ and $W$, respectively. We say that $W$ is a column-row, a row-column or a proper image of $\alpha$ if there exists a column-row, a row-column or a proper substitution, respectively, for $\alpha$ and $W$.

A nice and intuitive way to interpret the different kinds of images of array patterns is to imagine a grid to be placed over the terminal array. The vertical lines of the grid represent a column concatenation and the horizontal lines of the grid represent a row concatenation of the corresponding factorisation. This means that every rectangular area of the grid corresponds to an occurrence of a variable $x$ in the array pattern or, more precisely, to the array $h(x)$ substituted for $x$. The fact that an array satisfies a pattern is then represented by the situation that each two rectangular areas of the grid that correspond to occurrences of the same variable must have identical content. In Figure 1, an example for each a morphic image, a proper image, a column-row image and a row-column image of a $5\times 4$ pattern is represented in this illustrative way.

Alternatively, we can interpret the property that a terminal array $W$ is a certain type of image of an array pattern as a tiling of $W$. More precisely, $W$ satisfies a given array pattern $\alpha$ with $n$ different variables if and only if $n$ tiles can be allocated to the $n$ variables of $\alpha$ such that combining the tiles as indicated by the structure of $\alpha$ yields $W$. The grids depicted in Figure 1 then illustrate the structure of such a tiling. The definitions of the corresponding classes of pattern languages are now straightforward:

Since, for a fixed array pattern $\alpha$, every morphic image is a proper image and every proper image is a row-column image as well as a column-row image, the following subset relations between the different types of pattern languages hold (in the following diagram, an arrow denotes a subset relation):

In this section, we state some general lemmas about two-dimensional morphisms and array pattern languages, which shall be important for proving the further results presented in this paper. First, we refine Proposition 1, by giving a convenient characterisation for the morphism property for mappings on arrays. To this end, we define a substitution $h$ to be $(m,n)$-uniform if, for every $x\in X$, $|h(x)|_{r}=m$ and $|h(x)|_{c}=n$ and a substitution is uniform if it is $(m,n)$-uniform, for some $m,n\in\mathbb{N}$.

The next lemma states that the composition of two two-dimensional morphisms is again a two-dimensional morphism.

It is intuitively clear that the structure of a pattern fully determines the corresponding pattern language and the actual names of the variables are irrelevant, e. g., the patterns $\begin{bmatrix}x_{1}&x_{2}&x_{1}\\ x_{2}&x_{3}&x_{3}\end{bmatrix}$ and $\begin{bmatrix}x_{7}&x_{3}&x_{7}\\ x_{3}&x_{5}&x_{5}\end{bmatrix}$ should be considered identical. In the following we formalise this intuition. Two array patterns $\alpha:=[y_{i,j}]_{m,n}$ and $\beta:=[z_{i,j}]_{m^{\prime},n^{\prime}}$ are equivalent up to a renaming, denoted by $\alpha\sim\beta$, if and only if $m=m^{\prime}$, $n=n^{\prime}$ and, for every $i,j,i^{\prime},j^{\prime}$, $1\leq i,i^{\prime}\leq m$, $1\leq j,j^{\prime}\leq n$, $y_{i,j}=y_{i^{\prime},j^{\prime}}$ if and only if $z_{i,j}=z_{i^{\prime},j^{\prime}}$.