Optimal Convergence of Iterative Methods for Datalogo

2026-10-02

AI Summary

This study addresses the unknown convergence steps and the absence of theoretical bounds in the iterative evaluation of Datalog^circ. Drawing upon algebraic semantics and semiring theory, we systematically analyze the iterative convergence behavior of polynomial equation systems over p-stable semirings. We rigorously prove that the optimal tight bound on the number of convergence steps is O((p+1)n) and establish an upper bound of O((p+1)mn) on the total operation count. To our knowledge, this work provides the first optimal asymptotic bounds for both the convergence steps and the number of semiring operations, thereby filling a critical theoretical gap in the computational complexity analysis of Datalog^circ and offering a complete theoretical foundation for its evaluation framework.

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-na\"ive 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.

Problem

• Datalog^circ
• iterative methods
• convergence
• polynomial equations
• semiring

Innovation

• Datalog°
• iterative methods
• optimal convergence
• semiring
• polynomial equations