IterLara: A Turing Complete Algebra for Big Data, AI, Scientific Computing, and Database

TL;DR

IterLara extends the Lara algebra with iterative operators, achieving Turing completeness and capable of representing complex computations like matrix inversion, unifying operations across big data, AI, scientific computing, and databases.

Contribution

It introduces IterLara, a Turing complete algebra that unifies various computational operations, including iterative processes, in a single formal framework.

Findings

IterLara can represent matrix inversion and determinant.

IterLara with iteration is proven to be Turing complete.

Operation Count (OP) metric aligns with existing computation metrics.

Abstract

\textsc{Lara} is a key-value algebra that aims at unifying linear and relational algebra with three types of operation abstraction. The study of \textsc{Lara}'s expressive ability reports that it can represent relational algebra and most linear algebra operations. However, several essential computations, such as matrix inversion and determinant, cannot be expressed in \textsc{Lara}. \textsc{Lara} cannot represent global and iterative computation, either. This article proposes \textsc{IterLara}, extending \textsc{Lara} with iterative operators, to provide an algebraic model that unifies operations in general-purpose computing, like big data, AI, scientific computing, and database. We study the expressive ability of \textsc{Lara} and \textsc{IterLara} and prove that \textsc{IterLara} with aggregation functions can represent matrix inversion, determinant. Besides, we demonstrate that \textsc{IterLara} with no limitation of function utility is Turing complete. We also propose the Operation Count (OP) as a metric of computation amount for \textsc{IterLara} and ensure that the OP metric is in accordance with the existing computation metrics.