On the Expressive Power of Query Languages for Matrices

@article{Brijder2018OnTE,
  title={On the Expressive Power of Query Languages for Matrices},
  author={Robert Brijder and Floris Geerts and Jan Van den Bussche and Timmy Weerwag},
  journal={ACM Transactions on Database Systems (TODS)},
  year={2018},
  volume={44},
  pages={1 - 31}
}
We investigate the expressive power of MATLANG, a formal language for matrix manipulation based on common matrix operations and linear algebra. The language can be extended with the operation inv for inverting a matrix. In MATLANG + inv, we can compute the transitive closure of directed graphs, whereas we show that this is not possible without inversion. Indeed, we show that the basic language can be simulated in the relational algebra with arithmetic operations, grouping, and summation. We… 

Figures from this paper

MATLANG: Matrix operations and their expressive power

The expressive power of MATLANG, a formal language for matrix manipulation based on common matrix operations and linear algebra, is investigated and it is shown that the basic language can be simulated in the relational algebra with arithmetic operations, grouping, and summation.

On the Expressive Power of Linear Algebra on Graphs

  • Floris Geerts
  • Mathematics, Computer Science
    Theory of Computing Systems
  • 2020
This paper considers MATLANG, a matrix query language recently introduced, in which some basic linear algebra functionality is supported, and investigates the problem of characterising the equivalence of graphs, represented by their adjacency matrices, for various fragments ofMATLANG.

On the Expressive Power of Linear Algebra on Graphs

This paper considers M A T L A N G, a matrix query language recently introduced, in which some basic linear algebra functionality is supported, and investigates the problem of characterising the equivalence of graphs, represented by their adjacency matrices, for various fragments of M AT L AN G.

When Can Matrix Query Languages Discern Matrices?

We investigate when two graphs, represented by their adjacency matrices, can be distinguished by means of sentences formed in MATLANG, a matrix query language which supports a number of elementary

Expressive Power of Linear Algebra Query Languages

The matrix query language MATLANG is extended with this type of recursion, and it is shown that this suffices to express classical linear algebra algorithms.

On the Expressiveness of LARA: A Unified Language for Linear and Relational Algebra

This work shows LARA to be expressive complete with respect to first-order logic with aggregation, and distinguishes two main cases depending on the level of genericity queries are enforced to satisfy.

On matrices and K-relations

The matrix query language can express all matrix queries expressible in the positive relational algebra on $K-relations, when intermediate arities are restricted to three, and is offered an analogue to the situation in classical logic, where the algebra of binary relations is equivalent to first-order logic with three variables.

Functional collection programming with semi-ring dictionaries

SDQL is developed, a statically typed language that can express relational algebra with aggregations, linear algebra, and functional collections over data such as relations and matrices using semi-ring dictionaries, and unifies a wide range of optimizations commonly used in databases (DB) and linear algebra (LA).

Expressiveness of Matrix and Tensor Query Languages in terms of ML Operators

This short paper studies a matrix and a tensor query language that have been recently proposed in the database literature and shows, by using examples, how these proposals are in line with the practical interest in rethinking tensor abstractions.

Recursive SPARQL for Graph Analytics

A minimalistic extension of SPARQL to allow for expressing analytical tasks is proposed, and it is shown that this language can express key analytical tasks on graphs, offering a more declarative alternative to existing frameworks and languages.

References

SHOWING 1-10 OF 75 REFERENCES

On the Expressive Power of Linear Algebra on Graphs

This paper considers M A T L A N G, a matrix query language recently introduced, in which some basic linear algebra functionality is supported, and investigates the problem of characterising the equivalence of graphs, represented by their adjacency matrices, for various fragments of M AT L AN G.

On matrices and K-relations

The matrix query language can express all matrix queries expressible in the positive relational algebra on $K-relations, when intermediate arities are restricted to three, and is offered an analogue to the situation in classical logic, where the algebra of binary relations is equivalent to first-order logic with three variables.

Expressive power of SQL

  • L. Libkin
  • Computer Science
    Theor. Comput. Sci.
  • 2001

Embedding Tarskian Semantics in Vector Spaces

An unprecedented way of computing the least models defined by Datalog programs in linear spaces via matrix equations is proposed and empirically show its effectiveness compared to state-of-the-art approaches.

LaraDB: A Minimalist Kernel for Linear and Relational Algebra Computation

The LARADB implementation outperforms Accumulo's native MapReduce integration on a core task involving join and aggregation in the form of matrix multiply, especially at smaller scales that are typically a poor fit for scale-out approaches.

Logics with aggregate operators

This work considers a database query language that expresses all the standard aggregates found in commercial query languages, and shows how it can be translated into the aggregate logic, thereby providing a number of expressivity bounds that do not depend on a particular class of arithmetic functions.

Logics with Rank Operators

This work introduces extensions of first-order logic (FO) and fixed-point logic (FP) with operators that compute the rank of a definable matrix and shows that FO+rk_p can define deterministic and symmetric transitive closure and captures the complexity class MOD_pL, for all prime values of p.

The complexity of relational query languages (Extended Abstract)

The pattern which will be shown is that the expression complexity of the investigated languages is one exponential higher then their data complexity, and for both types of complexity the authors show completeness in some complexity class.

A linear algebraic approach to datalog evaluation

  • Taisuke Sato
  • Computer Science
    Theory and Practice of Logic Programming
  • 2017
The proposed linear algebraic approach to Datalog evaluation is inspired by the emergence of big knowledge graphs and expected to contribute to the realization of rich and scalable logical inference for knowledge graphs.

On the Descriptive Complexity of Linear Algebra

The boundary of definability in $\ensuremath{FP}+\textsf{C}}} is explored with respect to problems from linear algebra and suggestions on how the logic might be extended are looked at.
...