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

$\text{textsc}Lara$ is a key-value algebra that aims at unifying linear and relational algebra with three types of operation abstraction. The study of $\text{textsc}Lara$'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 $\text{textsc}Lara$. $\text{textsc}Lara$ cannot represent global and iterative computation, either. This article proposes $\text{textsc}IterLara$, extending $\text{textsc}Lara$ 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 $\text{textsc}Lara$ and $\text{textsc}IterLara$ and prove that $\text{textsc}IterLara$ with aggregation functions can represent matrix inversion, determinant. Besides, we demonstrate that $\text{textsc}IterLara$ with no limitation of function utility is Turing complete. We also propose the Operation Count (OP) as a metric of computation amount for $\text{textsc}IterLara$ and ensure that the OP metric is in accordance with the existing computation metrics.