# 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.

## Key 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.