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On the Convergence Rate of Linear Datalog ^∘ over Stable Semirings

Authors Sungjin Im, Benjamin Moseley, Hung Ngo, Kirk Pruhs

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Author Details

Sungjin Im
  • University of California, Merced, CA, USA
Benjamin Moseley
  • Carnegie Mellon University, Pittsburgh, PA, USA
Hung Ngo
  • RelationalAi, Berkeley, CA, USA
Kirk Pruhs
  • University of Pittsburgh, Pittsburgh, PA, USA

Funding

Moseley was supported in part by a Google Research Award, an Inform Research Award, a Carnegie Bosch Junior Faculty Chair, and NSF grants CCF-2121744 and CCF-1845146. Pruhs was supported in part by NSF grants CCF-1907673, CCF-2036077, CCF-2209654 and an IBM Faculty Award. Im was supported in part by NSF grants CCF-1844939 and CCF-2121745.

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Sungjin Im, Benjamin Moseley, Hung Ngo, and Kirk Pruhs. On the Convergence Rate of Linear Datalog ^∘ over Stable Semirings. In 27th International Conference on Database Theory (ICDT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 290, pp. 11:1-11:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ICDT.2024.11

Abstract

Datalog^∘ is an extension of Datalog, where instead of a program being a collection of union of conjunctive queries over the standard Boolean semiring, a program may now be a collection of sum-product queries over an arbitrary commutative partially ordered pre-semiring. Datalog^∘ is more powerful than Datalog in that its additional algebraic structure alows for supporting recursion with aggregation. At the same time, Datalog^∘ retains the syntactic and semantic simplicity of Datalog: Datalog^∘ has declarative least fixpoint semantics. The least fixpoint can be found via the naïve evaluation algorithm that repeatedly applies the immediate consequence operator until no further change is possible.
It was shown in [Mahmoud Abo Khamis et al., 2022] that, when the underlying semiring is p-stable, then the naïve evaluation of any Datalog^∘ program over the semiring converges in a finite number of steps. However, the upper bounds on the rate of convergence were exponential in the number n of ground IDB atoms.
This paper establishes polynomial upper bounds on the convergence rate of the naïve algorithm on linear Datalog^∘ programs, which is quite common in practice. In particular, the main result of this paper is that the convergence rate of linear Datalog^∘ programs under any p-stable semiring is O(pn³). Furthermore, we show a matching lower bound by constructing a p-stable semiring and a linear Datalog^∘ program that requires Ω(pn³) iterations for the naïve iteration algorithm to converge. Next, we study the convergence rate in terms of the number of elements in the semiring for linear Datalog^∘ programs. When L is the number of elements, the convergence rate is bounded by O(pn log L). This significantly improves the convergence rate for small L. We show a nearly matching lower bound as well.

Subject Classification

ACM Subject Classification
  • Theory of computation → Database query languages (principles)
Keywords
  • Datalog
  • convergence rate
  • semiring

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