{"id":1191061,"date":"2026-10-26T16:22:39","date_gmt":"2026-10-26T23:22:39","guid":{"rendered":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/publication\/optimal-convergence-of-iterative-methods-for-datalogo\/"},"modified":"2026-10-30T11:50:29","modified_gmt":"2026-10-30T18:50:29","slug":"optimal-convergence-of-iterative-methods-for-datalogo","status":"publish","type":"msr-research-item","link":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/publication\/optimal-convergence-of-iterative-methods-for-datalogo\/","title":{"rendered":"Optimal Convergence of Iterative Methods for Datalogo"},"content":{"rendered":"\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math><semantics><mrow><mtext>mathsf{Datalog}^circ<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">mathsf{Datalog}^circ<\/annotation><\/semantics><\/math> 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&#8221;ive evaluation and demand transformation. <math><semantics><mrow><mtext>mathsf{Datalog}^circ<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">mathsf{Datalog}^circ<\/annotation><\/semantics><\/math> 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 <math><semantics><mrow><mtext>mathsf{Datalog}^circ<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">mathsf{Datalog}^circ<\/annotation><\/semantics><\/math> 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 <math><semantics><mrow><mtext>mathsf{Datalog}^circ<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">mathsf{Datalog}^circ<\/annotation><\/semantics><\/math>, 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 <math><semantics><mrow><mtext>mathsf{Datalog}^circ<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">mathsf{Datalog}^circ<\/annotation><\/semantics><\/math>: 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 <math><mi>p<\/mi><\/math>-stable semiring. In particular, we show that the number of steps until convergence is <math><mrow><mi>O<\/mi><mo stretchy=\"false\">(<\/mo><mo stretchy=\"false\">(<\/mo><mi>p<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/math> where <math><mi>n<\/mi><\/math> is the output size of the <math><semantics><mrow><mtext>mathsf{Datalog}^circ<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">mathsf{Datalog}^circ<\/annotation><\/semantics><\/math> program. As the number of steps until convergence is only a proxy for the runtime of <math><semantics><mrow><mtext>mathsf{Datalog}^circ<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">mathsf{Datalog}^circ<\/annotation><\/semantics><\/math> evaluation, we also consider the number of semiring operations used and show that, for a natural class of algorithms, <math><mrow><mi>O<\/mi><mo stretchy=\"false\">(<\/mo><mo stretchy=\"false\">(<\/mo><mi>p<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mi>m<\/mi><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/math> is a tight upper bound, where <math><mi>m<\/mi><\/math> is the maximum number of semiring operations per iteration.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>mathsf{Datalog}^circmathsf{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&#8221;ive evaluation and demand transformation. mathsf{Datalog}^circmathsf{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 [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":[{"type":"text","value":"Simon Frisk","user_id":0},{"type":"text","value":"Sungjin Im","user_id":0},{"type":"text","value":"Paraschos Koutris","user_id":0},{"type":"text","value":"Benjamin Moseley","user_id":0},{"type":"text","value":"Hung Q. Ngo","user_id":0},{"type":"text","value":"K. 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