On matrices and $K$-relations

TL;DR

This paper establishes a correspondence between a matrix query language and a fragment of positive relational algebra on K-relations, revealing how matrix operations relate to logical frameworks with numerical data.

Contribution

It introduces a new fragment of positive relational algebra that captures matrix queries, linking matrix languages to logical models with restricted arities.

Findings

MATLANG corresponds to a fragment of positive relational algebra on K-relations.

Expressiveness of MATLANG matches that of positive relational algebra with arity restrictions.

Analogy to classical logic with three-variable first-order logic for binary relations.

Abstract

We show that the matrix query language $\mathsf{MATLANG}$ corresponds to a natural fragment of the positive relational algebra on $K$-relations. The fragment is defined by introducing a composition operator and restricting $K$-relation arities to two. We then proceed to show that $\mathsf{MATLANG}$ can express all matrix queries expressible in the positive relational algebra on $K$-relations, when intermediate arities are restricted to three. Thus we offer an analogue, in a model with numerical data, to the situation in classical logic, where the algebra of binary relations is equivalent to first-order logic with three variables.

Tables

Table 1. Table 1: Symbol table for the simulations between 𝖬𝖠𝖳𝖫𝖠𝖭𝖦 𝖬𝖠𝖳𝖫𝖠𝖭𝖦 \mathsf{MATLANG} and ( 𝖠𝖱𝖠 + ζ 2 ) ​ ( 2 ) 𝖠𝖱𝖠 subscript 𝜁 2 2 (\mathsf{ARA}+\zeta_{2})(2) .

Mapping	$\mathrm{𝖬𝖠𝖳𝖫𝖠𝖭𝖦}\to \mathrm{𝖠𝖱𝖠}$	$\mathrm{𝖠𝖱𝖠}\to \mathrm{𝖬𝖠𝖳𝖫𝖠𝖭𝖦}$
attributes $A$/size terms $\alpha$	${\mathrm{row}}_{\alpha }$, ${\mathrm{col}}_{\alpha }$	$\Psi \left(A\right)$
schemas $𝒮$	$\Gamma \left(𝒮\right)$	$\Theta \left(𝒮\right)$
expressions $e$	${\rm Y}\left(e\right)$	$\Phi \left(e\right)$
instances $I$, relations $r$/matrices $M$	${\mathrm{Rel}}_{𝒮,\sigma }\left(I\right)$, ${\mathrm{Rel}}_{s,\sigma }\left(M\right)$	${\mathrm{Mat}}_D\left(I\right)$, ${\mathrm{Mat}}_D\left(r\right)$

Equations

$$({\pi }_Y(r))(t)=\sum _{\begin{array}{c}{t}^{\prime }\in {\mathcal{T}}_D(X),\\ t^{\prime }{|}_Y=t\end{array}}r(t^{\prime }).$$

$$({\sigma }_Y(r))(t)=\{\begin{array}{ll}r(t)& \text{if }t(A)=t(B)\text{ for all }A,B\in Y;\\ 0& \text{otherwise}.\end{array}$$

$$({\zeta }_{A,k}(r_1,\dots ,r_l))(t)=({\hat{\pi }}_A(r_1\bowtie \cdots \bowtie r_l))(t)$$

$$r_i^{\prime }:=\{\begin{array}{ll}{r}_i& \text{if }A\notin X_i;\\ {\rho }_{A\to B}(r_i)& \text{if }A\in X_i,B\notin X_i;\\ {\hat{\pi }}_A({\sigma }_{\{A,B\}}(r_i))& \text{if }A,B\in X_i,\end{array}$$

$${\hat{\pi }}_A({\sigma }_{\{A,B\}}(r_1\bowtie \cdots \bowtie r_n))=r_1^{\prime }\bowtie \cdots \bowtie r_n^{\prime }.$$

$$({\hat{\pi }}_A({\sigma }_{\{A,B\}}(r)))(t)=\sum _{\begin{array}{c}u\in {\mathcal{T}}_D(X),\\ u{|}_{X\setminus \{A\}}=t\end{array}}({\sigma }_{A=B}(r))(u)=\sum _{\begin{array}{c}u\in {\mathcal{T}}_D(X),\\ u{|}_{X\setminus \{A\}}=t,\\ u(A)=u(B)\end{array}}r(u)=r(\tilde{t})$$

$$({\hat{\pi }}_A({\sigma }_{\{A,B\}}(r_1\bowtie \cdots \bowtie r_n)))(t)=(r_1\bowtie \cdots \bowtie r_n)(\tilde{t})=r_1(\tilde{t}{|}_{X_1})\ast \cdots \ast r_n(\tilde{t}{|}_{X_n}).$$

$$r_1(\tilde{t}{|}_{X_1})\ast \cdots \ast r_n(\tilde{t}{|}_{X_n})=r_1^{\prime }(t{|}_{X_1^{\prime }})\ast \cdots \ast r_n^{\prime }(t{|}_{X_n^{\prime }})=(r_1^{\prime }\bowtie \cdots \bowtie r_n^{\prime })(t).\text{qed}$$

$$e={\pi }_{\{B,C\}}({\sigma }_{\{B,C\}}(R\bowtie R\bowtie S\bowtie T\bowtie {\rho }_{\phi }(T))\cup {\sigma }_{\{A,B\}}(R\bowtie S\bowtie T)),$$

$$e$$

$$\equiv {\hat{\pi }}_A({\sigma }_{\{B,C\}}(R\bowtie R\bowtie S\bowtie T\bowtie {\rho }_{\phi }(T)))\cup {\hat{\pi }}_A({\sigma }_{\{A,B\}}(R\bowtie S\bowtie T))$$

$$\equiv {\sigma }_{\{B,C\}}({\hat{\pi }}_A(R\bowtie R\bowtie S\bowtie T\bowtie {\rho }_{\phi }(T)))\cup {\hat{\pi }}_A({\sigma }_{\{A,B\}}(R\bowtie S\bowtie T))$$

$$\equiv {\sigma }_{\{B,C\}}(S\bowtie {\rho }_{\phi }(T)\bowtie {\hat{\pi }}_A(R\bowtie R\bowtie T))\cup {\hat{\pi }}_A({\sigma }_{\{A,B\}}(R\bowtie S\bowtie T))$$

$$\equiv {\sigma }_{\{B,C\}}(S\bowtie {\rho }_{\phi }(T)\bowtie {\zeta }_{A,2}(R\bowtie R,T))\cup {\hat{\pi }}_A({\sigma }_{\{A,B\}}(R\bowtie S\bowtie T))$$

$$\displaystyle\equiv\sigma_{\{B,C\}}(S\Join\rho_{\varphi}(T)\Join\zeta_{A,2}(R\Join R,T))\cup\bigl{(}\hat{\pi}_{A}(\sigma_{\{A,B\}}(R))\Join S\Join\rho_{A\to B}(T)\bigr{)}.$$

$$\mathcal{I}(\text{no\_courses})=\left(\begin{array}{ccc}5& 2& 0\\ 2& 1& 3\end{array}\right)\mathcal{I}(\text{course\_fee})=\left(\begin{array}{c}300\\ 250\\ 330\end{array}\right)$$

$$\Gamma (s):=\{\begin{array}{ll}\{{\mathrm{row}}_{\alpha },{\mathrm{col}}_{\beta }\}& \text{if }s=\alpha \times \beta ;\\ \{{\mathrm{row}}_{\alpha }\}& \text{if }s=\alpha \times 1;\\ \{{\mathrm{col}}_{\beta }\}& \text{if }s=1\times \beta ;\\ \varnothing & \text{if }s=1\times 1,\end{array}$$

$${\rm Y}(e):=\{\begin{array}{ll}{\rho }_{\phi }({\rm Y}(e^{\prime }))& \text{if }\alpha \ne 1\ne \beta ;\\ {\rho }_{{\mathrm{col}}_{\alpha }\to {\mathrm{row}}_{\alpha }}({\rm Y}(e^{\prime }))& \text{if }\alpha \ne 1=\beta ;\\ {\rho }_{{\mathrm{row}}_{\beta }\to {\mathrm{col}}_{\beta }}({\rm Y}(e^{\prime }))& \text{if }\alpha =1\ne \beta ;\\ e^{\prime }& \text{if }\alpha =1=\beta ,\end{array}$$

$${\rm Y}(e):=\{\begin{array}{ll}1({\pi }_{\{{\mathrm{row}}_{\alpha }\}}({\rm Y}(e^{\prime })))& \text{if }\alpha \ne 1;\\ 1({\pi }_{\varnothing }({\rm Y}(e^{\prime })))& \text{if }\alpha =1.\end{array}$$

$${\rm Y}(e):=\{\begin{array}{ll}{\sigma }_{\{{\mathrm{row}}_{\alpha },{\mathrm{col}}_{\alpha }\}}({\rm Y}(e^{\prime })\bowtie 1({\rho }_{{\mathrm{row}}_{\alpha }\to {\mathrm{col}}_{\alpha }}({\rm Y}(e^{\prime }))))& \text{if }\alpha \ne 1;\\ {\rm Y}(e^{\prime })& \text{if }\alpha =1.\end{array}$$

$$\Theta (X):=\{\begin{array}{ll}\Psi (A_1)\times \Psi (A_2)& \text{if }X=\{A_1,A_2\};\\ \Psi (A)\times 1& \text{if }X=\{A\}\text{ for some }A;\\ 1\times 1& \text{if }X=\varnothing .\end{array}$$

$$\Phi (e):=\{\begin{array}{ll}\Phi (e^{\prime })\cdot 1(\Phi (e^{\prime }{)}^T)& \text{if }A=A_2\text{ and }\mathcal{S}(e^{\prime })=\{A_1,A_2\};\\ 1(\Phi (e^{\prime }){)}^T\cdot \Phi (e^{\prime })& \text{otherwise}.\end{array}$$

$$\Phi (e):=\{\begin{array}{ll}\Phi (e^{\prime })& \text{if }|Y|\le 1;\\ \Phi (e^{\prime })\circ \mathrm{diag}(1(\Phi (e^{\prime })))& \text{if }|Y|=2.\end{array}$$

$$\Phi (e):=\{\begin{array}{ll}\Phi (e^{\prime }{)}^T& \text{if }\phi (A_1)>\phi (A_2)\text{ and }\mathcal{S}(e^{\prime })=\{A_1,A_2\};\\ \Phi (e^{\prime })& \text{otherwise}.\end{array}$$

$$\Phi (e):=\{\begin{array}{ll}1(\Phi (e^{\prime }))\cdot 1(\Phi (e^{\prime }){)}^T& \text{if }\mathcal{S}(e^{\prime })=\{A_1,A_2\};\\ 1(\Phi (e^{\prime }))& \text{otherwise}.\end{array}$$

$$\Phi (e):=\{\begin{array}{ll}\Phi (e_1)\circ \Phi (e_2)& \text{if }\mathcal{S}(e_1)=\mathcal{S}(e_2);\\ s(e_1,e_2)\circ \Phi (e_2)& \text{if }\mathcal{S}(e_1)=\varnothing ;\\ \Phi (e_1)\circ s(e_2,e_1)& \text{if }\mathcal{S}(e_2)=\varnothing ;\\ \Phi (e_1)\cdot \Phi (e_2{)}^T& \text{if }\mathcal{S}(e_1)=\{A_1\}\text{ and }\mathcal{S}(e_2)=\{A_2\};\\ (\Phi (e_1)\cdot \Phi (e_2{)}^T{)}^T& \text{if }\mathcal{S}(e_1)=\{A_2\}\text{ and }\mathcal{S}(e_2)=\{A_1\};\\ \mathrm{diag}(\Phi (e_1))\cdot \Phi (e_2)& \text{if }\mathcal{S}(e_1)=\{A_1\}\text{ and }\mathcal{S}(e_2)=\{A_1,A_2\};\\ (\Phi (e_1{)}^T\cdot \mathrm{diag}(\Phi (e_2)){)}^T& \text{if }\mathcal{S}(e_1)=\{A_1,A_2\}\text{ and }\mathcal{S}(e_2)=\{A_1\};\\ \Phi (e_1)\cdot \mathrm{diag}(\Phi (e_2))& \text{if }\mathcal{S}(e_1)=\{A_1,A_2\}\text{ and }\mathcal{S}(e_2)=\{A_2\};\\ (\mathrm{diag}(\Phi (e_1))\cdot \Phi (e_2{)}^T{)}^T& \text{if }\mathcal{S}(e_1)=\{A_2\}\text{ and }\mathcal{S}(e_2)=\{A_1,A_2\},\end{array}$$

$$\Phi (e):={\left(\Phi (e_1{)}^{T(A_1,A_3)}\cdot \Phi (e_2{)}^{T(A_3,A_2)}\right)}^{T(A_1,A_2)},$$

$$\forall i\exists j\forall k\forall x(M(i,k,x)\to \exists iN(j,i,x))$$

Full text

1 Introduction

Motivated by large-scale data science, there is recent interest in supporting linear algebra operations, such as matrix multiplication, in database systems. This has prompted investigations comparing the expressive power of common matrix operations with the operations on relations provided by the relational algebra and SQL [5, 6, 9, 2].

For boolean matrices, the connection between matrices and relations is very natural and well known. An $m\times n$ boolean matrix $A$ can be viewed as a binary relation $R\subseteq\{1,\dots,m\}\times\{1,\dots,n\}$, where $R$ consists of those pairs $(i,j)$ for which $A_{i,j}=1$. Boolean matrix multiplication then amounts to composition of binary relations. Composition is the central operation in the algebra of binary relations [16, 12, 15]. Besides composition, this algebra has operations such as converse, which corresponds to transposition of a boolean matrix; union and complement, which correspond to disjunction and negation of boolean matrices; and the empty and identity relations, which correspond to the zero and identity matrices.

A common theme in research in the foundations of databases is the expressive power of query languages [1]. When we employ a query language, we would like to understand as well as possible what we can do with it. Of this kind is the classical Codd theorem, stating the equivalence between the standard relational algebra and first-order logic. Likewise, for the algebra of binary relations, a classical result [17] is that it has the same expressive power as the formulas with two free variables in $\mathrm{FO}(3)$, the three-variable fragment of first-order logic. In this sense, we understand quite well the expressive power of a natural set of operations on boolean matrices.

What can now be said in this regard about more general matrices, with entries that are not just boolean values? An $m\times n$ matrix with entries in some semiring $K$ is essentially a mapping from $\{1,\dots,m\}\times\{1,\dots,n\}$ to $K$. This perfectly fits the data model of $K$-relations introduced by Green, Garvounarakis and Tannen [4]. In general, consider an infinite domain $\mathbf{dom}$ and a supply of attributes. In a database instance, we assign to each attribute a range of values, in the form of a finite subset of $\mathbf{dom}$. Attributes can be declared to be compatible; compatible attributes have the same range. A relation schema $S$ is a finite set of attributes. Tuples over $S$ are mappings that assign to each attribute a value of the appropriate range. Now a $K$-relation over $S$ is a mapping that assigns to each tuple over $S$ an element of $K$.

So, an $m\times n$ matrix $X$ can be seen as a $K$-relation over two attributes $A$ and $B$ where the range of $A$ is $\{1,\dots,m\}$ and the range of $B$ is $\{1,\dots,n\}$. We can assume an order on all attributes and choose $A<B$ so that we know which values are row indices and which are column indices. Then an $n\times k$ matrix $Y$ is modeled using attributes $C<D$ where we choose $C$ and $B$ compatible, to reflect that the number of columns of matrix $X$ equals the number of rows of matrix $Y$. We can view vectors as $K$-relations over a single attribute, and scalars as $K$-relations over the empty schema. In general, a $K$-relation of arity $r$ is essentially an $r$-dimensional tensor (multidimensional array). (Because we need not necessarily assume an order on $\mathbf{dom}$, the tensor is unordered.)

Green et al. defined a generalization of the positive relation algebra working on $K$-relations, which we denote here by $\mathsf{ARA}$.111$\mathsf{ARA}$ stands for Annotated-Relation Algebra, as the elements from $K$ that a $K$-relation assigns to its tuples are usually viewed as annotations. When we restrict $\mathsf{ARA}$ to arities of at most three, which we denote by $\mathsf{ARA}(3)$, we obtain an analogue to $\mathrm{FO}(3)$ mentioned above. So, $\mathsf{ARA}$ provides a suitable scenario to reinvestigate, in a data model with numerical values, the equivalence between the algebra of binary relations and $\mathrm{FO}(3)$. In this paper we offer the following contributions.

We define a suitable generalization, to $K$-relations, of the composition operation of classical binary relations. When we add this composition operator to $\mathsf{ARA}$, but restrict arities to at most two, we obtain a natural query language for matrices. We refer to this language here as “$\mathsf{ARA}(2)$ plus composition”. 2. 2.

We show that $\mathsf{ARA}(2)$ plus composition actually coincides with the matrix query language $\mathsf{MATLANG}$, introduced by two of the present authors with Geerts and Weerwag [2] in an attempt to formalize the set of common matrix operations found in numerical software packages. 3. 3.

We show that a matrix query is expressible in $\mathsf{ARA}(3)$ if and only if it is expressible in $\mathsf{MATLANG}$, thus providing an analogue to the classical result about $\mathrm{FO}(3)$ and the algebra of binary relations. More generally, for any arity $r$, we show that an $r$-ary query over $r$-ary $K$-relations is expressible in $\mathsf{ARA}(r+1)$ if and only if it is expressible in $\mathsf{ARA}(r)$ plus composition. For this result, we need the assumption that $K$ is commutative. We stress that the proof is not a trivial adaptation of the proof of the classical result, because we can no longer rely on familiar classical properties like idempotence of union and join.

$\mathsf{ARA}$ has been a very influential vehicle for data provenance.222The paper [4] received the PODS 2017 test-of-time award. The elements from $K$ are typically viewed as annotations, or even as abstract tokens, and the semantics of $\mathsf{ARA}$ operations was originally designed to show how these annotations are propagated in the results of data manipulations. Other applications, apart from provenance, have been identified from the outset, such as security levels, or probabilities [4]. By doing the present work, we have understood that $\mathsf{ARA}$ can moreover serve as a fully-fledged query language for tensors (multidimensional arrays), and matrices in particular. This viewpoint is backed by the recent interest in processing Functional Aggregate Queries (FAQ [10, 11], also known as AJAR [7]). Indeed, FAQ and AJAR correspond to the project-join fragment of $\mathsf{ARA}$, without self-joins.

This paper is further organized as follows. Section 2 recalls the data model of $K$-relations and the associated query language $\mathsf{ARA}$. Section 3 presents the result on $\mathsf{ARA}(r+1)$ and $\mathsf{ARA}(r)$ plus composition. Section 4 relates $\mathsf{ARA}(2)$ plus composition to $\mathsf{MATLANG}$. Section 5 draws conclusions, discusses related work, and proposes directions for further research.

2 Annotated-Relation Algebra

By function we will always mean a total function. For a function $f:X\to Y$ and $Z\subseteq X$, the restriction of $f$ to $Z$, denoted by $f|_{Z}$, is the function $Z\to Y$ where $f|_{Z}(x)=f(x)$ for all $x\in Z$.

Recall that a semiring $K$ is a set equipped with two binary operations, addition ($+$) and multiplication ($*$), such that (1) addition is associative, commutative, and has an identity element [math]; (2) multiplication is associative, has an identity element $1$; and has [math] as an annihilating element; and (3) multiplication distributes over addition. A semiring is called commutative when multiplication is commutative. We fix a semiring $K$.

Remark*.*

We will explicitly indicate where we assume commutativity of $K$.

From the outset, we also fix countable infinite sets $\mathbf{rel}$, $\mathbf{att}$, and $\mathbf{dom}$, the elements of which are called relation names, attributes, and domain elements, respectively. We assume an equivalence relation $\sim$ on $\mathbf{att}$ that partitions $\mathbf{att}$ into an infinite number of equivalence classes that are each infinite. When $A\sim B$, we say that $A$ and $B$ are compatible. Intuitively, $A\sim B$ will mean that $A$ and $B$ have the same set of domain values. A function $f:X\to Y$ with $X$ and $Y$ sets of attributes is called compatible if $f(A)\sim A$ for all $A\in X$.

A relation schema is a finite subset of $\mathbf{att}$. A database schema is a function $\mathcal{S}$ on a finite set $N$ of relation names, assigning a relation schema $\mathcal{S}(R)$ to each $R\in N$. The arity of a relation name $R$ is the cardinality $|\mathcal{S}(R)|$ of its schema. The arity of $\mathcal{S}$ is the largest arity among relation names $R\in N$.

We now recursively define the expressions of the Annotated-Relation Algebra, abbreviated by $\mathsf{ARA}$. At the same time we assign a relation schema to each $\mathsf{ARA}$ expression by extending $\mathcal{S}$ from relation names to arbitrary $\mathsf{ARA}$ expressions. An $\mathsf{ARA}$ expression $e$ over a database schema $\mathcal{S}$ is equal to

• •

a relation name $R$ of $\mathcal{S}$;

• •

$\mathbf{1}(e^{\prime})$, where $e^{\prime}$ is an $\mathsf{ARA}$ expression, and $\mathcal{S}(e):=\mathcal{S}(e^{\prime})$;

• •

$e_{1}\cup e_{2}$, where $e_{1}$ and $e_{2}$ are $\mathsf{ARA}$ expressions with $\mathcal{S}(e_{1})=\mathcal{S}(e_{2})$, and $\mathcal{S}(e):=\mathcal{S}(e_{1})$;

• •

$\pi_{Y}(e^{\prime})$, where $e^{\prime}$ is an $\mathsf{ARA}$ expression and $Y\subseteq\mathcal{S}(e^{\prime})$, and $\mathcal{S}(e):=Y$;

• •

$\sigma_{Y}(e^{\prime})$, where $e^{\prime}$ is an $\mathsf{ARA}$ expression, $Y\subseteq\mathcal{S}(e^{\prime})$, the elements of $Y$ are mutually compatible, and $\mathcal{S}(e):=\mathcal{S}(e^{\prime})$;

• •

$\rho_{\varphi}(e^{\prime})$, where $e^{\prime}$ is an $\mathsf{ARA}$ expression and $\varphi:\mathcal{S}(e^{\prime})\to Y$ a compatible one-to-one correspondence with $Y\subseteq\mathbf{att}$, and $\mathcal{S}(e):=Y$; or

• •

$e_{1}\Join e_{2}$, where $e_{1}$ and $e_{2}$ are $\mathsf{ARA}$ expressions, and $\mathcal{S}(e):=\mathcal{S}(e_{1})\cup\mathcal{S}(e_{2})$.

A domain assignment is a function $D:\mathbf{att}\to\mathcal{D}$, where $\mathcal{D}$ is a set of nonempty finite subsets of $\mathbf{dom}$, such that $A\sim B$ implies $D(A)=D(B)$. Let $X$ be a relation schema. A tuple over $X$ with respect to $D$ is a function $t:X\to\mathbf{dom}$ such that $t(A)\in D(A)$ for all $A\in X$. We denote by $\mathcal{T}_{D}(X)$ the set of tuples over $X$ with respect to $D$. Note that $\mathcal{T}_{D}(X)$ is finite. A relation $r$ over $X$ with respect to $D$ is a function $r:\mathcal{T}_{D}(X)\to K$. So a relation annotates every tuple over $X$ with respect to $D$ with a value from $K$. If $\mathcal{S}$ is a database schema, then an instance $\mathcal{I}$ of $\mathcal{S}$ with respect to $D$ is a function that assigns to every relation name $R$ of $\mathcal{S}$ a relation $\mathcal{I}(R):\mathcal{T}_{D}(\mathcal{S}(R))\to K$.

Remark*.*

In practice, a domain assignment need only be defined on the attributes that are used in the database schema (and on attributes compatible to these attributes). Thus, it can be finitely specified. While here we have chosen to keep the notion of domain assignment and instance separate, it may perhaps be more natural to think of the domain assignment as being part of the instance.

Example 1**.**

Let us record for a university both the number of courses each student takes in each department and the course fee for each department. Let $K$ be the set of integers and let $\mathcal{S}$ be a database schema on $\{\text{no{\_}courses},\allowbreak\text{course{\_}fee}\}$ with $\mathcal{S}(\text{no{\_}courses})=\{\text{student},\text{dptm}\}$ and $\mathcal{S}(\text{course{\_}fee})=\{\text{dptm}\}$. Let $D$ be a domain assignment with $D(\text{student})=\{\text{Alice},\text{Bob}\}$ and $D(\text{dptm})=\{\text{CS},\text{Math},\text{Bio}\}$. A database instance $\mathcal{I}$ of $\mathcal{S}$ with respect to $D$ is shown in Figure 1.

We now define the relation $\mathbf{1}_{X}$, as well as the generalizations of the classical operations from the positive relational algebra to work on $K$-relations.

One

For every relation schema $X$, we define $\mathbf{1}_{X}:\mathcal{T}_{D}(X)\to K$ where $\mathbf{1}_{X}(t)=1$ for every $t\in\mathcal{T}_{D}(X)$.

Union

Let $r_{1},r_{2}:\mathcal{T}_{D}(X)\to K$. Define $r_{1}\cup r_{2}:\mathcal{T}_{D}(X)\to K$ as $(r_{1}\cup r_{2})(t)=r_{1}(t)+r_{2}(t)$.

Projection

Let $r:\mathcal{T}_{D}(X)\to K$ and $Y\subseteq X$. Define $\pi_{Y}(r):\mathcal{T}_{D}(Y)\to K$ as

[TABLE]

Selection

Let $r:\mathcal{T}_{D}(X)\to K$ and $Y\subseteq X$ where the elements of $Y$ are mutually compatible. Define $\sigma_{Y}(r):\mathcal{T}_{D}(X)\to K$ such that

[TABLE]

Renaming

Let $r:\mathcal{T}_{D}(X)\to K$ and $\varphi:X\to Y$ a compatible one-to-one correspondence. We define $\rho_{\varphi}(r):\mathcal{T}_{D}(Y)\to K$ as $\rho_{\varphi}(r)(t)=r(t\circ\varphi)$.

Join

Let $r_{1}:\mathcal{T}_{D}(X_{1})\to K$ and $r_{2}:\mathcal{T}_{D}(X_{2})\to K$. Define $r_{1}\Join r_{2}:\mathcal{T}_{D}(X_{1}\cup X_{2})\to K$ as $(r_{1}\Join r_{2})(t)=r_{1}(t|_{X_{1}})*r_{2}(t|_{X_{2}})$.

The above operations provide semantics for $\mathsf{ARA}$ in a natural manner. Formally, let $\mathcal{S}$ be a database schema, let $e$ be an $\mathsf{ARA}$ expression over $\mathcal{S}$, and let $\mathcal{I}$ be an instance of $\mathcal{S}$. The output relation $e(\mathcal{I})$ of $e$ under $\mathcal{I}$ is defined as follows. If $e=R$ with $R$ a relation name of $\mathcal{S}$, then $e(\mathcal{I}):=\mathcal{I}(R)$. If $e=\mathbf{1}(e^{\prime})$, then $e(\mathcal{I}):=\mathbf{1}_{\mathcal{S}(e^{\prime})}$. If $e=e_{1}\cup e_{2}$, then $e(\mathcal{I}):=e_{1}(\mathcal{I})\cup e_{2}(\mathcal{I})$. If $e=\pi_{X}(e^{\prime})$, then $e(\mathcal{I}):=\pi_{X}(e^{\prime}(\mathcal{I}))$. If $e=\sigma_{Y}(e^{\prime})$, then $e(\mathcal{I}):=\sigma_{Y}(e^{\prime}(\mathcal{I}))$. If $e=\rho_{\varphi}(e^{\prime})$, then $e(\mathcal{I}):=\rho_{\varphi}(e^{\prime}(\mathcal{I}))$. Finally, if $e=e_{1}\Join e_{2}$, then $e(\mathcal{I}):=e_{1}(\mathcal{I})\Join e_{2}(\mathcal{I})$.

Remark*.*

The language $\mathsf{ARA}$ is a slight variation of the $K$-annotated relational algebra as originally defined by Green et al. [4] to better suit our purposes.

First of all, the original definition does not have a domain assignment $D:\mathbf{att}\to\mathcal{D}$ but instead a single domain common to all attributes (and it therefore also does not have a compatibility relation $\sim$). As such, the original definition corresponds to the case where database schemas and $\mathsf{ARA}$ expressions use only mutually compatible attributes. We need our more general setting when we compare $\mathsf{ARA}$ to $\mathsf{MATLANG}$ in Section 4.

Also, here, we focus on equality selections, while the original paper does not fix the allowed selection predicates. Finally, the original definition assumes zero-relations $\mathbf{0}_{X}$, while we instead use one-relations $\mathbf{1}_{X}$.

The following observations, to the effect that some (but not all) classical relational-algebra equivalences carry over to the $K$-annotated setting, were originally made by Green et al.

Proposition 2** (Proposition 3.4 of [4]).**

The following properties and equivalences hold, where, for each given equivalence, we assume that the left-hand side is well defined.

• •

Union is associative and commutative.

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Join is associative and distributive over union, i.e., $(r_{1}\cup r_{2})\Join r_{3}=(r_{1}\Join r_{3})\cup(r_{2}\Join r_{3})$.

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Any two selections commute.

• •

Projection and selection commute when projection retains the attributes on which selection takes place.

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Projection distributes over union, i.e., $\pi_{Y}(r_{1}\cup r_{2})=\pi_{Y}(r_{1})\cup\pi_{Y}(r_{2})$.

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Selection distributes over union, i.e., $\sigma_{Y}(r_{1}\cup r_{2})=\sigma_{Y}(r_{1})\cup\sigma_{Y}(r_{2})$.

• •

We have $\sigma_{Y}(r_{1})\Join r_{2}=\sigma_{Y}(r_{1}\Join r_{2})$ and $r_{1}\Join\sigma_{Y}(r_{2})=\sigma_{Y}(r_{1}\Join r_{2})$.

• •

If $K$ is commutative, then join is commutative.

Note that idempotence of union and of join, i.e., $r\Join r=r\cup r=r$, which holds for the classical relational algebra, does not in general hold for $\mathsf{ARA}$.

We supplement Proposition 2 with the following easy-to-verify properties.

Lemma 3**.**

Let $r_{1}:\mathcal{T}_{D}(X_{1})\to K$ and $r_{2}:\mathcal{T}_{D}(X_{2})\to K$.

• •

If $X_{1}\cap X_{2}\subseteq X\subseteq X_{1}\cup X_{2}$, then $\pi_{X}(r_{1}\Join r_{2})=\pi_{X\cap X_{1}}(r_{1})\Join\pi_{X\cap X_{2}}(r_{2})$.

• •

If $Y_{1},Y_{2}\subseteq X_{1}$ where $Y_{1}\cap Y_{2}\neq\emptyset$ and the attributes of $Y_{1}$ and of $Y_{2}$ are mutually compatible, then $\sigma_{Y_{2}}(\sigma_{Y_{1}}(r_{1}))=\sigma_{Y_{1}\cup Y_{2}}(r_{1})$.

• •

If $\varphi:X_{1}\cup X_{2}\to X$ is a compatible one-to-one correspondence, then $\rho_{\varphi}(r_{1}\Join r_{2})=\rho_{\varphi|_{X_{1}}}(r_{1})\Join\rho_{\varphi|_{X_{2}}}(r_{2})$. If moreover $X_{1}=X_{2}$, then $\rho_{\varphi}(r_{1}\cup r_{2})=\rho_{\varphi}(r_{1})\cup\rho_{\varphi}(r_{2})$.

• •

If $Y\subseteq X_{1}$ and $\varphi:X_{1}\to X$ is a compatible one-to-one correspondence, then $\rho_{\varphi}(\sigma_{Y}(r_{1}))=\sigma_{\varphi(Y)}(\rho_{\varphi}(r_{1}))$, where $\varphi(Y)=\{\varphi(y)\mid y\in Y\}$.

We also use the operation of projecting away an attribute, i.e., $\hat{\pi}_{A}(e):=\pi_{\mathcal{S}(e)\setminus\{A\}}(e)$ if $A\in\mathcal{S}(e)$. Note that conversely, $\pi_{X}(e)=(\hat{\pi}_{A_{m}}\cdots\hat{\pi}_{A_{1}})(e)$ where $X=\mathcal{S}(e)\setminus\{A_{1},\ldots,A_{m}\}$ and the $A_{i}$’s are mutually distinct. Projecting away, allowing one to deal with one attribute at a time, is sometimes notationally more convenient.

3 Composition and Equivalence

In this section we define an operation called $k$-composition and show that augmenting $\mathsf{ARA}$ by composition allows one to reduce the required arity of the relations that are computed in subexpressions.

Definition 4**.**

Let $k$ be a nonnegative integer and let $l\in\{1,\ldots,k\}$. Let $r_{i}:\mathcal{T}_{D}(X_{i})\to K$ for $i\in\{1,\ldots,l\}$, let $X=X_{1}\cup\cdots\cup X_{l}$, and let $A\in X_{1}\cap\cdots\cap X_{l}$.

Define the $k$-composition $\zeta_{A,k}(r_{1},\ldots,r_{l}):\mathcal{T}_{D}(X\setminus\{A\})\to K$ as

[TABLE]

for all $t\in\mathcal{T}_{D}(X\setminus\{A\})$.

Note that $\zeta_{A,k}$ takes at most $k$ arguments.

We denote by $\mathsf{ARA}+\zeta_{k}$ the language obtained by extending $\mathsf{ARA}$ with $k$-composition. Consequently, if $e_{1},\ldots,e_{l}$ are $\mathsf{ARA}+\zeta_{k}$ expressions with $l\leq k$ and $A\in\mathcal{S}(e_{1})\cap\cdots\cap\mathcal{S}(e_{l})$, then $e=\zeta_{A,k}(e_{1},\ldots,e_{l})$ is an $\mathsf{ARA}+\zeta_{k}$ expression. Also, we let $\mathcal{S}(e):=(\mathcal{S}(e_{1})\cup\cdots\cup\mathcal{S}(e_{l}))\setminus\{A\}$.

Let $k$ be a nonnegative integer. We denote by $\mathsf{ARA}(k)$ the fragment of $\mathsf{ARA}$ in which the database schemas are restricted to arity at most $k$ and the relation schema of each subexpression is of cardinality at most $k$. In particular, join $e_{1}\Join e_{2}$ is only allowed if $|\mathcal{S}(e_{1}\Join e_{2})|\leq k$. The fragment $(\mathsf{ARA}+\zeta_{k})(k)$ is defined similarly.

From Definition 4 it is apparent that $(\mathsf{ARA}+\zeta_{k})(k)$ is subsumed by $\mathsf{ARA}(k+1)$. One of our main results (Corollary 8) provides the converse inclusion, when the database schemas and outputs are restricted to arity at most $k$. To this end, we establish a normal form for $\mathsf{ARA}$ expressions. First we prove the following technical but important lemma.

Lemma 5**.**

Let $r_{1},\ldots,r_{n}$ be relations with relation schemas $X_{1},\ldots,X_{n}$, respectively, and with respect to a domain assignment $D$. Assume that $A,B\in X_{1}\cup\cdots\cup X_{n}$ are distinct and compatible. Define, for $i\in\{1,\ldots,n\}$,

[TABLE]

where $A\to B$ denotes the one-to-one correspondence from $X_{i}$ to $(X_{i}\setminus\{A\})\cup\{B\}$ that assigns $A$ to $B$ and keeps the remaining attributes fixed.

Then

[TABLE]

Proof.

Let $X$ be a finite set of attributes with $A,B\in X$ distinct and compatible. Let $r:\mathcal{T}_{D}(X)\to K$ be a relation and $t\in\mathcal{T}_{D}(X\setminus\{A\})$.

We have

[TABLE]

where $\tilde{t}\in\mathcal{T}_{D}(X)$ is $\tilde{t}=t\cup\{(A,t(B))\}$. Thus, $\tilde{t}$ is obtained from $t$ by adding attribute $A$ with value $t(B)$. Indeed, the last summation of (1) is over a single tuple $u$, namely $u=\tilde{t}$.

In particular, applying (1) to $r_{1}\Join\cdots\Join r_{n}$, we obtain

[TABLE]

Denote the schemas of the relations $r^{\prime}_{1},\ldots,r^{\prime}_{n}$ by $X^{\prime}_{1},\ldots,X^{\prime}_{n}$, respectively. Let $i\in\{1,\ldots,n\}$. We distinguish three cases.

• •

If $A\notin X_{i}$, then $\tilde{t}|_{X_{i}}=t|_{X_{i}}$. Hence $r_{i}(\tilde{t}|_{X_{i}})=r^{\prime}_{i}(t|_{X^{\prime}_{i}})$.

• •

If $A\in X_{i}$ and $B\notin X_{i}$, then $\tilde{t}|_{X_{i}}=t|_{(X_{i}\setminus\{A\})\cup\{B\}}\circ\varphi=t|_{X^{\prime}_{i}}\circ\varphi$ with $\varphi=A\to B$. Hence, $r_{i}(\tilde{t}|_{X_{i}})=(\rho_{A\to B}(r_{i}))(t|_{X^{\prime}_{i}})=r^{\prime}_{i}(t|_{X^{\prime}_{i}})$.

• •

If $A,B\in X_{i}$, then, by (1) but now applied to $r_{i}$ and $t|_{X_{i}\setminus\{A\}}$, we have $r_{i}(\tilde{t}|_{X_{i}})=(\hat{\pi}_{A}(\sigma_{\{A,B\}}(r_{i})))(t|_{X_{i}\setminus\{A\}})=r^{\prime}_{i}(t|_{X^{\prime}_{i}})$.

Therefore, in each case we obtain $r_{i}(\tilde{t}|_{X_{i}})=r^{\prime}_{i}(t|_{X^{\prime}_{i}})$. Consequently,

[TABLE]

We use the following terminology. Let $\mathcal{F}$ be any family of expressions. A selection of $\mathcal{F}$-expressions is an expression of the form $\sigma_{Y_{n}}\cdots\sigma_{Y_{1}}(f)$, where $f$ is an $\mathcal{F}$-expression and $n\geq 0$. Note the slight abuse of terminology as we allow multiple selection operations. Also, when we say that $e$ is a union of $\mathcal{F}$-expressions or a join of $\mathcal{F}$-expressions, we allow $e$ to be just a single expression in $\mathcal{F}$ (so union and join may be skipped).

We are now ready to formulate and prove a main result of this paper. This result is inspired by a result in [13, Theorem 3.4.5, Claim 2] which provides a proof of the classic equivalence of $\mathrm{FO}(3)$ and the algebra of binary relations.

Two $\mathsf{ARA}$ expressions $e_{1}$ and $e_{2}$ over the same database schema are called equivalent, naturally, if they yield the same output relation, for every domain assignment and every database instance respecting that domain assignment.

Theorem 6**.**

Let $\mathcal{S}$ be a database schema of arity at most $k$ and assume that $K$ is commutative. Every $\mathsf{ARA}(k+1)$ expression over $\mathcal{S}$ is equivalent to a union of selections of joins of $(\mathsf{ARA}+\zeta_{k})(k)$ expressions over $\mathcal{S}$.

Proof.

For brevity, if an expression $e$ is a union of selections of joins of $(\mathsf{ARA}+\zeta_{k})(k)$ expressions over $\mathcal{S}$, then we say that $e$ is in normal form.

The proof is by induction on the structure of $e$.

Relation names. Since relation names $R$ of $\mathcal{S}$ are of arity at most $k$, we have that $R$ is an $(\mathsf{ARA}+\zeta_{k})(k)$ expression over $\mathcal{S}$.

One. If $e$ is equivalent to a union of selections of joins of $(\mathsf{ARA}+\zeta_{k})(k)$ expressions $e_{1},\ldots,e_{n}$ over $\mathcal{S}$, then $\mathbf{1}(e)\equiv\mathbf{1}(e_{1}\Join\cdots\Join e_{n})\equiv\mathbf{1}(e_{1})\Join\cdots\Join\mathbf{1}(e_{n})$ and so $\mathbf{1}(e)$ is also equivalent to an expression in normal form.

Union. If both $e_{1}$ and $e_{2}$ are equivalent to expressions in normal form, then so is $e_{1}\cup e_{2}$.

Join. Since join distributes over union and since $\sigma_{Y}(e_{1})\Join e_{2}$ and $e_{1}\Join\sigma_{Y}(e_{2})$ are equivalent to $\sigma_{Y}(e_{1}\Join e_{2})$ (Proposition 2), we observe that if $e_{1}$ and $e_{2}$ are equivalent to expressions in normal form, then so is $e_{1}\Join e_{2}$.

Let $e$ be equivalent to an expression in normal form.

Selection. Since selection distributes over union (Proposition 2), $\sigma_{Y}(e)$ is also equivalent to an expression in normal form.

Renaming. Since renaming distributes over union and join and by the commutative property of renaming and selection (Lemma 3), $\rho_{\varphi}(e)$ is also equivalent to an expression in normal form.

Projection. In this case we additionally require that $|\mathcal{S}(e)|\leq k+1$ (which holds when $e$ is an $\mathsf{ARA}(k+1)$ expression). Since projection distributes over union, it suffices to assume that $e=\sigma_{Y_{m}}\cdots\sigma_{Y_{1}}(e^{\prime})$, where $e^{\prime}$ is a join of $(\mathsf{ARA}+\zeta_{k})(k)$ expressions. By Proposition 2 and Lemma 3 we may assume that the $Y_{i}$’s are mutually disjoint. We may also assume that the $Y_{i}$’s are all of cardinality at least $2$ (since $\sigma_{Y}$ on relations is the identity when $|Y|\leq 1$). Let $A\in\mathcal{S}(e)$. We prove that $\hat{\pi}_{A}(e)$ is a selection of a join of $(\mathsf{ARA}+\zeta_{k})(k)$ expressions.

If $A$ does not belong to any of the $Y_{i}$’s, then $\hat{\pi}_{A}(e)=\sigma_{Y_{m}}\cdots\sigma_{Y_{1}}(\hat{\pi}_{A}(e^{\prime}))$ by Proposition 2. If $A\in Y_{i}$ for some $i$, then this $i$ is unique since the $Y_{i}$’s are mutually disjoint. Since any two selections commute, we may assume that $A\in Y_{1}$. As before, $\hat{\pi}_{A}(e)=\sigma_{Y_{m}}\cdots\sigma_{Y_{2}}(\hat{\pi}_{A}(\sigma_{Y_{1}}(e^{\prime})))$. Since $Y_{1}$ is of cardinality at least $2$, there exists $B\in Y_{1}$ distinct from $A$. We have $\sigma_{Y_{1}}(e^{\prime})\equiv\sigma_{Y_{1}\setminus\{A\}}(\sigma_{\{A,B\}}(e^{\prime}))$. Therefore $\hat{\pi}_{A}(\sigma_{Y_{1}}(e^{\prime}))\equiv\sigma_{Y_{1}\setminus\{A\}}(\hat{\pi}_{A}(\sigma_{\{A,B\}}(e^{\prime})))$. By Lemma 5, we obtain that $\hat{\pi}_{A}(\sigma_{\{A,B\}}(e^{\prime}))$ is equivalent to a join of $(\mathsf{ARA}+\zeta_{k})(k)$ expressions. So, this case is settled.

It remains to show that $\hat{\pi}_{A}(e^{\prime})$ is a join of $(\mathsf{ARA}+\zeta_{k})(k)$ expressions. If $|\mathcal{S}(e^{\prime})|\leq k$, then $e^{\prime}$ itself is an $(\mathsf{ARA}+\zeta_{k})(k)$ expression and so is $\hat{\pi}_{A}(e^{\prime})$. So, assume that $|\mathcal{S}(e^{\prime})|=k+1$.

Since join is commutative (because $K$ is) and associative, we can regard $e^{\prime}$ as a join of a multiset $F$ of $(\mathsf{ARA}+\zeta_{k})(k)$ expressions. By Lemma 3, for expressions $e_{1}$ and $e_{2}$, if $A\notin\mathcal{S}(e_{1})$, then $\hat{\pi}_{A}(e_{1}\Join e_{2})\equiv e_{1}\Join\hat{\pi}_{A}(e_{2})$. Therefore, we may assume that for every $f\in F$, we have $A\in\mathcal{S}(f)$. Hence, with $\mathcal{P}$ the set of all $k$-element subsets of $\mathcal{S}(e^{\prime})$ containing $A$, there exists a function $p$ that assigns to each $f\in F$ a set $S\in\mathcal{P}$ with $\mathcal{S}(f)\subseteq S$. Let $\mathcal{R}$ be the range of $p$. Thus $|\mathcal{R}|\leq|\mathcal{P}|=k$. Let, for $S\in\mathcal{R}$, $e_{S}:=\,\Join_{f\in F,p(f)=S}f$. Note that each $e_{S}$ is an $(\mathsf{ARA}+\zeta_{k})(k)$ expression since $|\mathcal{S}(e_{S})|\leq|S|=k$. We have $e^{\prime}\equiv\,\,\Join_{S\in\mathcal{R}}e_{S}$ whence $\hat{\pi}_{A}(e^{\prime})\equiv\hat{\pi}_{A}(\Join_{S\in\mathcal{R}}e_{S})$, which coincides with $\zeta_{A,k}((e_{S})_{S\in\mathcal{R}})$ by Definition 4. We thus obtain an $(\mathsf{ARA}+\zeta_{k})(k)$ expression as desired. ∎

Example 7**.**

Assume that $K$ is commutative and consider the $\mathsf{ARA}(3)$ expression

[TABLE]

where $\mathcal{S}(R)=\{A,B\}$, $\mathcal{S}(S)=\{B,C\}$, $\mathcal{S}(T)=\{A,C\}$ ($A,B,C$ are mutually distinct), and $\varphi$ sends $A$ to $B$ and $C$ to itself. The proof of Theorem 6 obtains an equivalent expression in normal form as follows.

[TABLE]

The last expression is in the normal form since $S$, $\rho_{\varphi}(T)$, $\zeta_{A,2}(R\Join R,T)$, $\hat{\pi}_{A}(\sigma_{\{A,B\}}(R))$, and $\rho_{A\to B}(T)$ are all $(\mathsf{ARA}+\zeta_{2})(2)$ expressions.

Note that we likely cannot omit the “selections of” in the above theorem. For example, for $k=2$ consider $\sigma_{\{A,C\}}(R\Join S)$ where $R$ and $S$ are relation names with $\mathcal{S}(R)=\{A,B\}$ and $\mathcal{S}(S)=\{B,C\}$.

Remark*.*

Theorem 6 still holds if the $\mathbf{1}$ operator is omitted from the definition of $\mathsf{ARA}$. Indeed, in the proof we can simply omit the case for the $\mathbf{1}$ operator, which is not used anywhere else.

Since union, selection, and join do not decrease the number of attributes of relations, we have the following corollary to Theorem 6, which establishes the main result announced in the Introduction.

Corollary 8**.**

Let $\mathcal{S}$ be a database schema of arity at most $k$ and assume that $K$ commutative. Every $\mathsf{ARA}(k+1)$ expression $e$ over $\mathcal{S}$ with $|\mathcal{S}(e)|\leq k$ is equivalent to an $(\mathsf{ARA}+\zeta_{k})(k)$ expression over $\mathcal{S}$.

Remark*.*

We remark that transforming an expression into the normal form of Theorem 6 may lead to an exponential increase in expression length. The reason is that the proof uses distributivity of join over union. Indeed, each time we replace an expression of the form $(e_{1}\cup e_{2})\Join e_{3}$ by $(e_{1}\Join e_{3})\cup(e_{2}\Join e_{3})$ there is a duplication of $e_{3}$. The proof of the classic translation of $\mathrm{FO}(3)$ to the algebra of binary relations also induces an exponential increase of expression length for similar reasons. A proof that this blowup is unavoidable remains open, both for our result and for the classical result (to the best of our knowledge).

4 Matrices

In this section we show that $(\mathsf{ARA}+\zeta_{2})(2)$ is equivalent to a natural version of $\mathsf{MATLANG}$ [2]. As a consequence of Corollary 8, we then obtain that also $\mathsf{ARA}(3)$, with database schemas and output relations restricted to arity at most $2$, is equivalent to $\mathsf{MATLANG}$. We begin by recalling the definition of this language.

4.1 $\mathsf{MATLANG}$

Let us fix the countable infinite sets $\mathbf{matvar}$ and $\mathbf{size}$, where the latter has a distinguished element $1\in\mathbf{size}$. The elements of $\mathbf{matvar}$ are called matrix variables and the elements of $\mathbf{size}$ are called size terms.

A matrix schema is a function $\mathcal{S}:V\to\mathbf{size}\times\mathbf{size}$ with $V\subseteq\mathbf{matvar}$ both finite and nonempty. We write $(\alpha,\beta)\in\mathbf{size}\times\mathbf{size}$ also as $\alpha\times\beta$.

$\mathsf{MATLANG}$ expressions are recursively defined as follows. At the same time we assign a matrix schema to each $\mathsf{MATLANG}$ expression by extending $\mathcal{S}$ from matrix variables to arbitrary $\mathsf{MATLANG}$ expressions.

A $\mathsf{MATLANG}$ expression $e$ over a matrix schema $\mathcal{S}$ is equal to

Variable

a matrix variable $M$ of $\mathcal{S}$;

Transpose

$(e^{\prime})^{T}$, where $e^{\prime}$ is a $\mathsf{MATLANG}$ expression, and $\mathcal{S}(e):=\beta\times\alpha$ if $\mathcal{S}(e^{\prime})=\alpha\times\beta$;

One-vector

$\mathbf{1}(e^{\prime})$, where $e^{\prime}$ is a $\mathsf{MATLANG}$ expression, and $\mathcal{S}(e):=\alpha\times 1$ if $\mathcal{S}(e^{\prime})=\alpha\times\beta$;

Diagonalization

$\mathrm{diag}(e^{\prime})$, where $e^{\prime}$ is a $\mathsf{MATLANG}$ expression with $\mathcal{S}(e^{\prime})=\alpha\times 1$, and $\mathcal{S}(e):=\alpha\times\alpha$;

Multiplication

$e_{1}\cdot e_{2}$, where $e_{1}$ and $e_{2}$ are $\mathsf{MATLANG}$ expressions with $\mathcal{S}(e_{1})=\alpha\times\beta$ and $\mathcal{S}(e_{2})=\beta\times\gamma$, and $\mathcal{S}(e):=\alpha\times\gamma$;

Addition

$e_{1}+e_{2}$, where $e_{1}$ and $e_{2}$ are $\mathsf{MATLANG}$ expressions with $\mathcal{S}(e_{1})=\mathcal{S}(e_{2})$, and $\mathcal{S}(e):=\mathcal{S}(e_{1})$; or

Hadamard product

$e_{1}\circ e_{2}$, where $e_{1}$ and $e_{2}$ are $\mathsf{MATLANG}$ expressions with $\mathcal{S}(e_{1})=\mathcal{S}(e_{2})$, and $\mathcal{S}(e):=\mathcal{S}(e_{1})$.

A size assignment is a function $\sigma$ that assigns to each size term a strictly positive integer with $\sigma(1)=1$. Let $\mathcal{M}$ be the set of all matrices over $K$. We say that $M\in\mathcal{M}$ conforms to $\alpha\times\beta\in\mathbf{size}\times\mathbf{size}$ by $\sigma$ if $M$ is a $\sigma(\alpha)\times\sigma(\beta)$-matrix.

If $\mathcal{S}:V\to\mathbf{size}\times\mathbf{size}$ is a matrix schema, then an instance of $\mathcal{S}$ with respect to $\sigma$ is a function $\mathcal{I}:V\to\mathcal{M}$ such that, for each $M\in V$, the matrix $\mathcal{I}(M)$ conforms to $\mathcal{S}(M)$ by $\sigma$.

Remark*.*

In practice, a size assignment need only be defined on the size terms that are used in the schema. Thus, it can be finitely specified. While here we have chosen to keep the notion of size assignment and instance separate, it may perhaps be more natural to think of the size assignment as being part of the instance.

Example 9**.**

This example is similar to Example 1. Let $K$ be the set of integers and let $\mathcal{S}$ be a matrix schema on $\{\text{no{\_}courses},\allowbreak\text{course{\_}fee}\}$ with $\mathcal{S}(\text{no{\_}courses})=\text{student}\times\text{dptm}$ and $\mathcal{S}(\text{course{\_}fee})=\text{dptm}\times 1$. Let $\sigma$ be a size assignment with $\sigma(\text{student})=2$ and $\sigma(\text{dptm})=3$. An instance $\mathcal{I}$ of $\mathcal{S}$ with respect to $\sigma$ is shown in Figure 2.

The semantics for $\mathsf{MATLANG}$ is given by the following matrix operations. Let $A$ be an $m\times n$-matrix over $K$. We define $\mathbf{1}(A)$ to be the $m\times 1$-matrix (i.e., column vector) with $\mathbf{1}(A)_{i,1}=1$. If $n=1$ (i.e., $A$ is a column vector), then $\mathrm{diag}(A)$ is the $m\times m$-matrix with $\mathrm{diag}(A)_{i,j}$ equal to $A_{i,1}$ if $i=j$ and to [math] otherwise. If $B$ is an $m\times n$-matrix, then $A\circ B$ denotes the Hadamard product of $A$ and $B$. In other words, $(A\circ B)_{i,j}=A_{i,j}*B_{i,j}$. Matrix addition and matrix multiplication are as usual denoted by $+$ and $\cdot$, respectively.

Formally, let $\mathcal{S}$ be a matrix schema, let $e$ be a $\mathsf{MATLANG}$ expression over $\mathcal{S}$, and let $\mathcal{I}$ be a matrix instance of $\mathcal{S}$. Then the output matrix $e(\mathcal{I})$ of $e$ under $\mathcal{I}$ is defined in the obvious way, given the operations just defined. If $e=M$ with $M$ a matrix variable of $\mathcal{S}$, then $e(\mathcal{I})$ is naturally defined to be equal to $\mathcal{I}(M)$.

Remark*.*

Matrix addition and the Hadamard product are the pointwise applications of addition and product, respectively. The original definition of $\mathsf{MATLANG}$ [2] is more generally defined in terms of an arbitrary set $\Omega$ of allowed pointwise functions. So, $\mathsf{MATLANG}$ as defined above fixes $\Omega$ to $\{+,\cdot\}$. This restriction was also considered by Geerts [3] (who also allows multiplication by constant scalars, but this is not essential).

Also, the original definition of $\mathsf{MATLANG}$ fixes $K$ to the field of complex numbers and complex transpose is considered instead of (ordinary) transpose. Of course, transpose can be expressed using complex transpose and pointwise application of conjugation.

In the following subsections we provide simulations between $\mathsf{MATLANG}$ and $(\mathsf{ARA}+\zeta_{2})(2)$. The notations for the different translations that will be given are summarized in Table 1.

4.2 Simulating $\mathsf{MATLANG}$ in $(\mathsf{ARA}+\zeta_{2})(2)$

For notational convenience, instead of fixing a one-to-one correspondence between $\mathbf{rel}$ and $\mathbf{matvar}$, we assume that $\mathbf{rel}=\mathbf{matvar}$.

Let us now fix injective functions $\mathrm{row}:\mathbf{size}\setminus\{1\}\to\mathbf{att}$ and $\mathrm{col}:\mathbf{size}\setminus\{1\}\to\mathbf{att}$ such that (1) $\mathrm{row}(\alpha)$ and $\mathrm{col}(\alpha)$ are compatible for all $\alpha\in\mathbf{size}\setminus\{1\}$ and (2) the range of $\mathrm{row}$ is disjoint from the range of $\mathrm{col}$. To reduce clutter, we also write, for $\alpha\in\mathbf{size}\setminus\{1\}$, $\mathrm{row}(\alpha)$ as $\mathrm{row}_{\alpha}$ and $\mathrm{col}(\alpha)$ as $\mathrm{col}_{\alpha}$.

Let $s\in\mathbf{size}\times\mathbf{size}$. We associate to $s$ a relation schema $\Gamma(s)$ with $|\Gamma(s)|\leq 2$ as follows.

[TABLE]

where $\alpha\neq 1\neq\beta$.

Let $\mathcal{S}$ be a matrix schema on a set of matrix variables $V$. We associate to $\mathcal{S}$ a database schema $\Gamma(\mathcal{S})$ on $V$ as follows. For $M\in V$, we set $(\Gamma(\mathcal{S}))(M):=\Gamma(\mathcal{S}(M))$.

Let $\sigma$ be a size assignment. We associate to $\sigma$ a domain assignment $D(\sigma)$ where, for $\alpha\in\mathbf{size}$, $(D(\sigma))(\mathrm{row}_{\alpha}):=(D(\sigma))(\mathrm{col}_{\alpha}):=\{1,\ldots,\sigma(\alpha)\}$.

Let $M\in\mathcal{M}$ conform to $s=\alpha\times\beta$ by $\sigma$. We associate to $M$ a relation $\mathrm{Rel}_{s,\sigma}(M):\mathcal{T}_{D(\sigma)}(\Gamma(s))\to K$ as follows. We have $(\mathrm{Rel}_{s,\sigma}(M))(t):=M_{i,j}$, where (1) $i$ is equal to $t(\mathrm{row}_{\alpha})$ if $\alpha\neq 1$ and equal to $1$ if $\alpha=1$; and (2) $j$ is equal to $t(\mathrm{col}_{\beta})$ if $\beta\neq 1$ and equal to $1$ if $\beta=1$.

Let $\mathcal{S}:V\to\mathbf{size}\times\mathbf{size}$ be a matrix schema and let $\mathcal{I}$ be a matrix instance of $\mathcal{S}$ with respect to $\sigma$. We associate to $\mathcal{I}$ an instance $\mathrm{Rel}_{\mathcal{S},\sigma}(\mathcal{I})$ of database schema $\Gamma(\mathcal{S})$ with respect to $D(\sigma)$ as follows. For $M\in V$, we set $(\mathrm{Rel}_{\mathcal{S},\sigma}(\mathcal{I}))(M):=\mathrm{Rel}_{\mathcal{S}(M),\sigma}(\mathcal{I}(M))$.

Example 10**.**

Recall $\mathcal{I}$, $\mathcal{S}$, and $\sigma$ from Example 9. We have that $(\Gamma(\mathcal{S}))(\text{no{\_}courses})=\{\mathrm{row}_{\text{student}},\mathrm{col}_{\text{dptm}}\}$ and $(\Gamma(\mathcal{S}))(\text{course{\_}fee})=\{\mathrm{row}_{\text{dptm}}\}$. The database instance $\mathrm{Rel}_{\mathcal{S},\sigma}(\mathcal{I})$ is shown in Figure 3.

We now show that every $\mathsf{MATLANG}$ expression can be simulated by an $(\mathsf{ARA}+\zeta_{2})(2)$ expression.

Lemma 11**.**

For each $\mathsf{MATLANG}$ expression $e$ over a matrix schema $\mathcal{S}$, there exists an $(\mathsf{ARA}+\zeta_{2})(2)$ expression $\Upsilon(e)$ over database schema $\Gamma(\mathcal{S})$ such that (1) $\Gamma(\mathcal{S}(e))=(\Gamma(\mathcal{S}))(\Upsilon(e))$ and (2) for all size assignments $\sigma$ and matrix instances $\mathcal{I}$ of $\mathcal{S}$ with respect to $\sigma$, we have $\mathrm{Rel}_{\mathcal{S}(e),\sigma}(e(\mathcal{I}))=(\Upsilon(e))(\mathrm{Rel}_{\mathcal{S},\sigma}(\mathcal{I}))$.

Proof.

Assume that $\mathcal{S}(e)=\alpha\times\beta$, where $\alpha,\beta\in\mathbf{size}$. We define $\Upsilon(e)$ explicitly.

• •

If $e=M$ is a matrix variable of $\mathcal{S}$, then $\Upsilon(e):=M$.

• •

If $e=(e^{\prime})^{T}$, then

[TABLE]

where $\varphi$ maps $\mathrm{col}_{\alpha}$ to $\mathrm{row}_{\alpha}$ and $\mathrm{row}_{\beta}$ to $\mathrm{col}_{\beta}$.

• •

If $e=\mathbf{1}(e^{\prime})$, then

[TABLE]

• •

If $e=\mathrm{diag}(e^{\prime})$, then

[TABLE]

• •

If $e=e_{1}\cdot e_{2}$ where $\mathcal{S}(e_{1})=\alpha\times\gamma$ and $\mathcal{S}(e_{2})=\gamma\times\beta$, then we consider two cases. If $\gamma=1$, then $\Upsilon(e):=\Upsilon(e_{1})\Join\Upsilon(e_{2})$. If $\gamma\neq 1$, then $\Upsilon(e):=\zeta_{C,2}(\rho_{\varphi_{1}}(\Upsilon(e_{1})),\rho_{\varphi_{2}}(\Upsilon(e_{2})))$, where $\varphi_{1}(\mathrm{col}_{\gamma})=\varphi_{2}(\mathrm{row}_{\gamma})=C\notin\{\mathrm{row}_{\alpha},\mathrm{col}_{\beta}\}$ and $\varphi_{1}$ and $\varphi_{2}$ are the identity otherwise.

• •

If $e=e_{1}+e_{2}$, then $\Upsilon(e):=\Upsilon(e_{1})\cup\Upsilon(e_{2})$.

• •

If $e=e_{1}\circ e_{2}$, then $\Upsilon(e):=\Upsilon(e_{1})\Join\Upsilon(e_{2})$.

It is straightforward to verify by induction on the structure of $e$ that $\Upsilon(e)$ satisfies the given properties. ∎

4.3 Simulating $(\mathsf{ARA}+\zeta_{2})(2)$ in $\mathsf{MATLANG}$

In order to simulate $(\mathsf{ARA}+\zeta_{2})(2)$ in $\mathsf{MATLANG}$, we equip $\mathbf{att}$ with some linear ordering $<$.

Again we assume that $\mathbf{rel}=\mathbf{matvar}$. Let us fix an injective function $\Psi:\mathbf{att}\to\mathbf{size}\setminus\{1\}$.

Let $X\subseteq\{A_{1},A_{2}\}$ be a relation schema for some $A_{1}$ and $A_{2}$ with $A_{1}<A_{2}$. We associate to $X$ an element $\Theta(X)\in\mathbf{size}\times\mathbf{size}$ as follows. We have

[TABLE]

Let $\mathcal{S}$ a database schema on a set $N$ of relation names of arities at most $2$. We associate to $\mathcal{S}$ a matrix schema $\Theta(\mathcal{S})$ on $N$ as follows. For $R\in N$, we set $(\Theta(\mathcal{S}))(R):=\Theta(\mathcal{S}(R))$.

Let $D$ be a domain assignment. We associate to $D$ a size assignment $\sigma(D)$ where, for $A\in\mathbf{att}$, $(\sigma(D))(D(A))=|D(A)|$. If every element in the range of a domain assignment $D$ is of the form $\{1,\ldots,n\}$ for some $n$, then we say that $D$ is consecutive.

Let $D$ be a consecutive domain assignment. Given a relation $r:\mathcal{T}_{D}(X)\to K$ with $X\subseteq\{A_{1},A_{2}\}$ and $A_{1}<A_{2}$, we associate a matrix $\mathrm{Mat}_{D}(r)$ conforming to $\Theta(X)$ by $\sigma(D)$ as follows. We define $(\mathrm{Mat}_{D}(r))_{i,j}:=r(t)$, where $t$ is (1) the tuple with $t(A_{1})=i$ and $t(A_{2})=j$ if $|X|=2$; (2) the tuple with $t(A)=i$ and $j=1$ if $X=\{A\}$ for some $A$; and (3) the unique tuple of $\mathcal{T}_{D}(X)$ if $X=\emptyset$.

Let $\mathcal{S}$ a database schema on a set $N$ of relation names of arities at most $2$, and let $\mathcal{I}$ be a database of $\mathcal{S}$ instance with respect to $D$. We associate to $\mathcal{I}$ a matrix instance $\mathrm{Mat}_{D}(\mathcal{I})$ of $\mathrm{Mat}(\mathcal{S})$ with respect to $\sigma(D)$ as follows. For $R\in N$, we set $(\mathrm{Mat}_{D}(\mathcal{I}))(R):=\mathrm{Mat}_{D}(\mathcal{I}(R))$.

Example 12**.**

Recall $\mathcal{I}$, $\mathcal{S}$, and $D$ from Example 1. To reduce clutter, assume that $\mathbf{att}=\mathbf{size}\setminus\{1\}$ and that $\Psi$ is the identity function. Take $\text{student}<\text{dptm}$. We have that $(\Theta(\mathcal{S}))(\text{no{\_}courses})=\text{student}\times\text{dptm}$ and $(\Theta(\mathcal{S}))(\text{course{\_}fee})=\text{dptm}\times 1$. Consider domain assignment $D^{\prime}$ and database instance $\mathcal{I}^{\prime}$ obtained from $D$ and $\mathcal{I}$, respectively, by replacing Alice by $1$, Bob by $2$, CS by $1$, Math by $2$, and Bio by $3$. Note that $D^{\prime}$ is consecutive. The instance $\mathrm{Mat}_{D^{\prime}}(\mathcal{I}^{\prime})$ is shown in Figure 2.

We now show that every $(\mathsf{ARA}+\zeta_{2})(2)$ expression can be simulated by an $\mathsf{MATLANG}$ expression.

Lemma 13**.**

For each $(\mathsf{ARA}+\zeta_{2})(2)$ expression $e$ over a database schema $\mathcal{S}$ of arity at most $2$, there exists a $\mathsf{MATLANG}$ expression $\Phi(e)$ over matrix schema $\Theta(\mathcal{S})$ such that (1) $\Theta(\mathcal{S}(e))=(\Theta(\mathcal{S}))(\Phi(e))$ and (2) for all consecutive domain assignments $D$ and database instances $\mathcal{I}$ with respect to $D$, we have $\mathrm{Mat}_{D}(e(\mathcal{I}))=(\Phi(e))(\mathrm{Mat}_{D}(\mathcal{I}))$.

Proof.

Assume that $\mathcal{S}(e)\subseteq\{A_{1},A_{2}\}$. We explicitly define $\Phi(e)$.

• •

If $e=R$ is a relation name of $\mathcal{S}$, then $\Phi(e):=R$.

• •

If $e=e_{1}\cup e_{2}$, then $\Phi(e):=\Phi(e_{1})+\Phi(e_{2})$.

• •

If $e=\hat{\pi}_{A}(e^{\prime})$ and $A_{1}<A_{2}$, then

[TABLE]

• •

If $e=\sigma_{Y}(e^{\prime})$, then

[TABLE]

• •

If $e=\rho_{\varphi}(e^{\prime})$ and $A_{1}<A_{2}$, then

[TABLE]

• •

If $e=\mathbf{1}(e^{\prime})$, then

[TABLE]

• •

If $e=e_{1}\Join e_{2}$ and $A_{1}<A_{2}$, then

[TABLE]

where $s(e,e^{\prime})$ denotes $\mathbf{1}(\Phi(e^{\prime}))\cdot\Phi(e)\cdot\mathbf{1}(\Phi(e^{\prime})^{T})^{T}$.

• •

If $e=\zeta_{A_{3},2}(e_{1},e_{2})$ with $\mathcal{S}(e_{1})=\{A_{1},A_{3}\}$ and $\mathcal{S}(e_{2})=\{A_{2},A_{3}\}$, then

[TABLE]

where $(\cdot)^{T(A,B)}$, for attributes $A$ and $B$, denotes identity if $A<B$ and transpose if $A>B$.

The last bullet covers the case where $|\mathcal{S}(e_{1})\operatorname{\triangle}\mathcal{S}(e_{2})|=2$, where $\operatorname{\triangle}$ denotes symmetric difference. If $|\mathcal{S}(e_{1})\operatorname{\triangle}\mathcal{S}(e_{2})|\leq 1$, then $\zeta_{A_{3},2}(e_{1},e_{2})\equiv\hat{\pi}_{A_{3}}(e_{1}\Join e_{2})$ is expressible in $\mathsf{ARA}(2)$ (since then $|\mathcal{S}(e_{1}\Join e_{2})|\leq 2$) and so $\Phi$ can be extended to cover this case as well.

It is straightforward to verify by induction on the structure of $e$ that $\Phi(e)$ satisfies the given properties. ∎

We remark that the number of cases in the expression for $\Phi(e)$ with $e=e_{1}\Join e_{2}$ in the above proof can be significantly reduced if we assume that $K$ is commutative (i.e., join is commutative).

4.4 Relationship with $\mathsf{ARA}(3)$ and complexity

Corollary 8, Lemma 11, and Lemma 13 together establish the equivalence of $\mathsf{MATLANG}$ with the language $\mathsf{ARA}(3)$ restricted to database schemas and output relations of arity at most $2$.

Theorem 14**.**

For each $\mathsf{ARA}(3)$ expression $e$ over a database schema $\mathcal{S}$ of arity at most $2$ and with $|\mathcal{S}(e)|\leq 2$, there exists a $\mathsf{MATLANG}$ expression $e^{\prime}$ such that $\mathrm{Mat}_{D}(e(\mathcal{I}))=e^{\prime}(\mathrm{Mat}_{D}(\mathcal{I}))$ for all consecutive domain assignments $D$ and instances $\mathcal{I}$ with respect to $\mathcal{S}$ over $D$.

Conversely, for each $\mathsf{MATLANG}$ expression $e$ over a matrix schema $\mathcal{S}$, there exists an $\mathsf{ARA}(3)$ expression $e^{\prime}$ such that $\mathrm{Rel}_{\mathcal{S}(e),\sigma}(e(\mathcal{I}))=e^{\prime}(\mathrm{Rel}_{\mathcal{S},\sigma}(\mathcal{I}))$ for all size assignments $\sigma$ and matrix instances $\mathcal{I}$ of $\mathcal{S}$ with respect to $\sigma$.

As to complexity, we note that the translations $\Upsilon$ and $\Phi$ given by Lemmas 11 and 13, taken at face value, are exponential. They can, however, be readily adapted to become linear (for fixed schemas; quadratic when the schema is part of the input). The adaptations that need to be done are as follows.

For a $\mathsf{MATLANG}$ expression $e^{\prime}$ with $\mathcal{S}(e^{\prime})=\alpha\times 1$ with $\alpha\neq 1$, there is a constant-length expression $\mathrm{Tp}_{\alpha}$ with $\mathcal{S}(\mathrm{Tp}_{\alpha})=\alpha\times 1$. Indeed, since $\alpha$ is a size term of $\mathcal{S}(e^{\prime})$ distinct from $1$, there is a matrix variable $M$ with $\mathcal{S}(M)$ equal to either $\alpha\times\gamma$ or $\gamma\times\alpha$ for some $\gamma$. Taking $\mathrm{Tp}_{\alpha}:=\mathbf{1}(M)$ in the former case and $\mathrm{Tp}_{\alpha}:=\mathbf{1}(M^{T})$ in the latter case, we have $\mathcal{S}(\mathrm{Tp}_{\alpha})=\alpha\times 1$ as desired. The only source of exponential growth in Lemma 11 is the expression $\sigma_{\{\mathrm{row}_{\alpha},\mathrm{col}_{\alpha}\}}(\Upsilon(e^{\prime})\Join\mathbf{1}(\rho_{\mathrm{row}_{\alpha}\to\mathrm{col}_{\alpha}}(\Upsilon(e^{\prime}))))$ appearing in the $\mathrm{diag}(e^{\prime})$ case, which is equivalent to $\sigma_{\{\mathrm{row}_{\alpha},\mathrm{col}_{\alpha}\}}(\Upsilon(e^{\prime})\Join\rho_{\mathrm{row}_{\alpha}\to\mathrm{col}_{\alpha}}(\Upsilon(\mathrm{Tp}_{\alpha})))$.

For the converse translation, we observe that, for an $\mathsf{ARA}$ expression $e^{\prime}$ with $X:=\mathcal{S}(e^{\prime})\subseteq\{A_{1},A_{2}\}$, there is a constant-length expression $\mathrm{Tp}_{X}$ with $\mathcal{S}(\mathrm{Tp}_{X})=X$. Indeed, if $A\in\mathcal{S}(e^{\prime})$, then there exists $A^{\prime}\in\mathcal{S}(R_{A^{\prime}})$ for some relation name $R_{A^{\prime}}$ such that $A^{\prime}$ is compatible with $A$. Taking $\mathrm{Tp}_{X}:=\Join_{A\in X}\rho_{A^{\prime}\to A}(\pi_{\{A^{\prime}\}}(R_{A^{\prime}}))$ if $X\neq\emptyset$ and $\mathrm{Tp}_{\emptyset}:=\pi_{\emptyset}(R)$ for some relation name $R$, we have $\mathcal{S}(\mathrm{Tp}_{X})=X$ as desired. Replacing each occurrence of $\mathbf{1}(\Phi(e^{\prime}))$ by the equivalent expression $\mathbf{1}(\Phi(\mathrm{Tp}_{\mathcal{S}(e^{\prime})}))$ and each occurrence of $\mathbf{1}(\Phi(e^{\prime})^{T})$ by the equivalent expression $\mathbf{1}(\Phi(\mathrm{Tp}_{\mathcal{S}(e^{\prime})})^{T})$ in the proof of Lemma 13 avoids exponential growth.

4.5 Indistinguishability

Using a recent result by Geerts on indistinguishability in $\mathsf{MATLANG}$ [3], we can also relate $\mathsf{ARA}(3)$ to $\mathsf{C}^{3}$, the three-variable fragment of first-order logic with counting [14]. Let $A_{1}$ and $A_{2}$ be matrices of the same dimensions $m\times n$. We view $A_{1}$ and $A_{2}$ as instances of a schema $\mathcal{S}$ on a single matrix name $M$ with $\mathcal{S}(M)=\alpha\times\beta$, with respect to the size assignment $\sigma$ that maps $\alpha$ to $m$ and $\beta$ to $n$. We say that $A_{1}$ and $A_{2}$ are indistinguishable in $\mathsf{MATLANG}$, denoted by $A_{1}\equiv_{\mathsf{MATLANG}}A_{2}$, if for each $\mathsf{MATLANG}$ expression $e$ over $\mathcal{S}$ with $\mathcal{S}(e)=\emptyset$, we have $e(A_{1})=e(A_{2})$. Similarly, one can define indistinguishability of binary $K$-relations $r_{1}$ and $r_{2}$ in $\mathsf{ARA}(3)$, denoted by $r_{1}\equiv_{\mathsf{ARA}(3)}r_{2}$. This leads to the following corollary to Theorem 14. Let $s=\alpha\times\beta$.

Corollary 15**.**

$A_{1}\equiv_{\mathsf{MATLANG}}A_{2}$* if and only if $\mathrm{Rel}_{s,\sigma}(A_{1})\equiv_{\mathsf{ARA}(3)}\mathrm{Rel}_{s,\sigma}(A_{2})$.*

Geerts’s result concerns finite undirected graphs $G_{1}$ and $G_{2}$ with the same number of nodes. Recall that $G_{1}$ and $G_{2}$ are called indistinguishable in $\mathsf{C}^{3}$, denoted by $G_{1}\equiv_{\mathsf{C}^{3}}G_{2}$, if each $\mathsf{C}^{3}$-sentence over a single binary relation variable has the same truth value on $G_{1}$ and $G_{2}$. Denote the adjacency matrix of $G$ by $\mathsf{Adj}(G)$.

Theorem 16** ([3]).**

Fix $K$ to be the field of complex numbers. Then $\mathsf{Adj}(G_{1})\equiv_{\mathsf{MATLANG}}\mathsf{Adj}(G_{2})$ if and only if $G_{1}\equiv_{\mathsf{C}^{3}}G_{2}$.

We can immediately conclude the following, still fixing $K$ to be the field of complex numbers.

Corollary 17**.**

$G_{1}\equiv_{\mathsf{C}^{3}}G_{2}$* if and only if $\mathrm{Rel}_{s,\sigma}(\mathsf{Adj}(G_{1}))\equiv_{\mathsf{ARA}(3)}\mathrm{Rel}_{s,\sigma}(\mathsf{Adj}(G_{2}))$ (for suitable $s$ and $\sigma$).*

5 Conclusion

In related work, Yan, Tannen, and Ives consider provenance for linear algebra operators [18]. In that approach, provenance tokens represent not the matrix entries (as in our work), but the matrices themselves. Polynomial expressions (with matrix addition and matrix multiplication) are derived to show the provenance of linear algebra operations applied to these matrices.

Our result that every matrix query expressible in $\mathsf{ARA}(3)$ is also expressible in $\mathsf{MATLANG}$ provides a partial converse to the observation already made in the original paper [2], to the effect that $\mathsf{MATLANG}$ can be expressed in $\mathcal{L}_{\rm Aggr}(3)$: the relational calculus with summation and numerical functions [8], restricted to three base variables.333$\mathcal{L}_{\rm Aggr}$ is a two-sorted logic with base variables and numerical variables. This observation was made in the extended setting of $\mathsf{MATLANG}$ that allows arbitrary pointwise functions (Remark Remark). For the language considered here, $\mathsf{ARA}(3)$ provides a more appropriate upper bound for comparison, and $\mathsf{ARA}(3)$ is still a natural fragment of $\mathcal{L}_{\rm Aggr}(3)$.

When allowing arbitrary pointwise functions in $\mathsf{MATLANG}$, we actually move beyond the positive relational algebra, as queries involving negation can be expressed. For example, applying the function $x\land\lnot y$ pointwise to the entries of two $n\times n$ boolean matrices representing two binary relations $R$ and $S$ on $\{1,\dots,n\}$, we obtain the set difference $R-S$. It is an interesting research question to explore expressibility of queries in $\mathsf{MATLANG}$ in this setting. For example, consider the following $\mathcal{L}_{\rm Aggr}(3)$ query on two matrices $M$ and $N$:

[TABLE]

Here, $M(i,k,x)$ means that $M_{i,k}=x$, and similarly for $N(j,i,x)$.

The above query, which does not even use summation, reuses the base variable $i$ and checks whether each row of $M$, viewed as a set of entries, is included in some row of $N$, again viewed as a set of entries. We conjecture that the query is not expressible in $\mathsf{MATLANG}$ with arbitrary pointwise functions. Developing techniques for showing this is an interesting direction for further research.

Finally, recall that our main result Corollary 8 assumes that $K$ is commutative. It should be investigated whether or not this result still holds in the noncommutative case.

Acknowledgements

We thank Floris Geerts for inspiring discussions. R.B. is a postdoctoral fellow of the Research Foundation – Flanders (FWO).