eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-03-11
6:1
6:20
10.4230/LIPIcs.ICDT.2020.6
article
On the Expressiveness of LARA: A Unified Language for Linear and Relational Algebra
Barceló, Pablo
1
2
https://orcid.org/0000-0003-2293-2653
Higuera, Nelson
3
2
Pérez, Jorge
3
2
Subercaseaux, Bernardo
3
2
IMC, Pontificia Universidad Católica de Chile, Santiago, Chile
IMFD, Santiago, Chile
Department of Computer Science, University of Chile, Santiago, Chile
We study the expressive power of the Lara language - a recently proposed unified model for expressing relational and linear algebra operations - both in terms of traditional database query languages and some analytic tasks often performed in machine learning pipelines. We start by showing Lara to be expressive complete with respect to first-order logic with aggregation. Since Lara is parameterized by a set of user-defined functions which allow to transform values in tables, the exact expressive power of the language depends on how these functions are defined. We distinguish two main cases depending on the level of genericity queries are enforced to satisfy. Under strong genericity assumptions the language cannot express matrix convolution, a very important operation in current machine learning operations. This language is also local, and thus cannot express operations such as matrix inverse that exhibit a recursive behavior. For expressing convolution, one can relax the genericity requirement by adding an underlying linear order on the domain. This, however, destroys locality and turns the expressive power of the language much more difficult to understand. In particular, although under complexity assumptions the resulting language can still not express matrix inverse, a proof of this fact without such assumptions seems challenging to obtain.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol155-icdt2020/LIPIcs.ICDT.2020.6/LIPIcs.ICDT.2020.6.pdf
languages for linear and relational algebra
expressive power
first order logic with aggregation
matrix convolution
matrix inverse
query genericity
locality of queries
safety