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Evaluating Datalog over Semirings: A Grounding-based Approach

Hangdong Zhao, Shaleen Deep, Paraschos Koutris, Sudeepa Roy, Val Tannen

TL;DR

This work tackles the problem of obtaining tight data-time bounds for evaluating Datalog programs over naturally-ordered semirings by introducing a two-phase grounding-based framework: (i) grounding the program into a σ-equivalent polynomial system, and (ii) computing the least fixpoint of this grounding over the semiring. It develops structure-aware grounding methods and semiring-aware fixpoint algorithms, achieving state-of-the-art bounds for practical fragments (e.g., CFL reachability, APSP, linear Datalog) and proving matching lower bounds in certain regimes. The framework unifies and extends previous results, enabling efficient evaluation across a broad class of semirings, including finite-rank and absorptive semirings with total order, and provides a general grounding algorithm based on tree decompositions and the PANDA technique. These results have practical impact on optimizing recursive, semiring-annotated queries in data analytics and program analysis while clarifying the trade-offs between grounding size and semiring operations.

Abstract

Datalog is a powerful yet elegant language that allows expressing recursive computation. Although Datalog evaluation has been extensively studied in the literature, so far, only loose upper bounds are known on how fast a Datalog program can be evaluated. In this work, we ask the following question: given a Datalog program over a naturally-ordered semiring $σ$, what is the tightest possible runtime? To this end, our main contribution is a general two-phase framework for analyzing the data complexity of Datalog over $σ$: first ground the program into an equivalent system of polynomial equations (i.e. grounding) and then find the least fixpoint of the grounding over $σ$. We present algorithms that use structure-aware query evaluation techniques to obtain the smallest possible groundings. Next, efficient algorithms for fixpoint evaluation are introduced over two classes of semirings: (1) finite-rank semirings and (2) absorptive semirings of total order. Combining both phases, we obtain state-of-the-art and new algorithmic results. Finally, we complement our results with a matching fine-grained lower bound.

Evaluating Datalog over Semirings: A Grounding-based Approach

TL;DR

This work tackles the problem of obtaining tight data-time bounds for evaluating Datalog programs over naturally-ordered semirings by introducing a two-phase grounding-based framework: (i) grounding the program into a σ-equivalent polynomial system, and (ii) computing the least fixpoint of this grounding over the semiring. It develops structure-aware grounding methods and semiring-aware fixpoint algorithms, achieving state-of-the-art bounds for practical fragments (e.g., CFL reachability, APSP, linear Datalog) and proving matching lower bounds in certain regimes. The framework unifies and extends previous results, enabling efficient evaluation across a broad class of semirings, including finite-rank and absorptive semirings with total order, and provides a general grounding algorithm based on tree decompositions and the PANDA technique. These results have practical impact on optimizing recursive, semiring-annotated queries in data analytics and program analysis while clarifying the trade-offs between grounding size and semiring operations.

Abstract

Datalog is a powerful yet elegant language that allows expressing recursive computation. Although Datalog evaluation has been extensively studied in the literature, so far, only loose upper bounds are known on how fast a Datalog program can be evaluated. In this work, we ask the following question: given a Datalog program over a naturally-ordered semiring , what is the tightest possible runtime? To this end, our main contribution is a general two-phase framework for analyzing the data complexity of Datalog over : first ground the program into an equivalent system of polynomial equations (i.e. grounding) and then find the least fixpoint of the grounding over . We present algorithms that use structure-aware query evaluation techniques to obtain the smallest possible groundings. Next, efficient algorithms for fixpoint evaluation are introduced over two classes of semirings: (1) finite-rank semirings and (2) absorptive semirings of total order. Combining both phases, we obtain state-of-the-art and new algorithmic results. Finally, we complement our results with a matching fine-grained lower bound.
Paper Structure (35 sections, 23 theorems, 37 equations, 2 figures, 2 tables, 4 algorithms)

This paper contains 35 sections, 23 theorems, 37 equations, 2 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Datalog
  4. Semirings and Their Properties
  5. Datalog over Semirings
  6. Framework and Main Results
  7. Grounding Generation
  8. Grounding Evaluation
  9. Applications
  10. Grounding Evaluation
  11. 2-canonical Grounding
  12. Finite-rank Semirings
  13. Absorptive Semirings with Total Order
  14. Grounding of Acyclic Datalog
  15. Grounding of General Datalog
  16. ...and 20 more sections

Key Result

theorem 1

Let $P$ be a rulewise-acyclic Datalog program over some semiring $\sigma$, with input size $m$, active domain size $n$, and $\mathsf{arity}(P) \leq k$. Then, we can construct a $\sigma$-equivalent grounding in time (and has size) $O(n^{k-1} \cdot (m + n^{k}))$.

Figures (2)

  • Figure 1: A join tree with its corresponding rewriting.
  • Figure 2: A join tree with its corresponding rewriting for \ref{['ex:nonbinary']} when using Algorithm \ref{['grounded-acyclicprogram']}.

Theorems & Definitions (24)

  • theorem 1
  • theorem 2
  • theorem 3
  • theorem 4
  • theorem 5
  • corollary 1
  • lemma 1
  • proposition 1
  • proposition 2
  • proposition 3
  • ...and 14 more