Proof shows efficient steps needed to solve extended Datalog programs
Optimal Convergence of Iterative Methods for Datalogo
Data Structures and Algorithms
Summary
Finding solutions to certain computer problems can be like solving equations with repeated guessing until the answer settles down. The authors studied a version of a programming language called Datalog that is more flexible but still lets you understand how it works mathematically. They proved exactly how many steps it takes for these guessing methods to find the final answers, giving a clear limit that matches what was already known in simpler cases. This helps people know how efficient algorithms based on this language can be.
What this means in practice
- •For database system builders: Optimize query evaluation by understanding exact iteration bounds for advanced Datalog extensions used in recursive queries.
- •For program analysis tool developers: Improve static analysis scalability by applying tight convergence bounds in sound fixpoint computation methods over semirings.
A theory result. No direct application yet.
Authors
Simon Frisk, Sungjin Im, Paraschos Koutris, Benjamin Moseley, Hung Ngo, Kirk Pruhs, Hangdong Zhao
Abstract
$\mathsf{Datalog}^\circ$ has been introduced as an extension to Datalog that increases expressiveness, yet retains simple least fixpoint semantics and admits optimization techniques such as semi-naïve evaluation and demand transformation. $\mathsf{Datalog}^\circ$ accomplishes this by generalizing the {\em or} and {\em and} operators of Datalog to addition and multiplication over a semiring. Finding a (minimal) fixpoint of a $\mathsf{Datalog}^\circ$ program is equivalent to finding a solution to a system of polynomial equations over the underlying semiring. Solving these systems of polynomial equations is not only a fundamental problem in the theory of $\mathsf{Datalog}^\circ$, but also has many applications in computer science, such as in databases, program analysis, and optimization. This paper resolves a key open problem in the theory of $\mathsf{Datalog}^\circ$: we prove a tight upper bound on the number of steps until convergence of iterative methods for solving these polynomial equation systems over a commutative $p$-stable semiring. In particular, we show that the number of steps until convergence is $O((p+1)n)$ where $n$ is the output size of the $\mathsf{Datalog}^\circ$ program. As the number of steps until convergence is only a proxy for the runtime of $\mathsf{Datalog}^\circ$ evaluation, we also consider the number of semiring operations used and show that, for a natural class of algorithms, $O((p+1)mn)$ is a tight upper bound, where $m$ is the maximum number of semiring operations per iteration.