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Monotone Rewritability and the Analysis of Queries, Views, and Rules

Michael Benedikt, Stanislav Kikot, Johannes Marti, Piotr Ostropolski-Nalewaja

TL;DR

The paper investigates monotone determinacy and rewritability of queries from views under existential rules, focusing on decidability and the expressive power of rewritings. It develops a forward-backward framework and automata-theoretic tools to analyze how views, queries, and rules interact, distinguishing finite and unrestricted models and leveraging bounded treewidth properties. The authors establish decidability results for several guarded and linear regimes and provide various rewriting guarantees (UCQ, Datalog, MDL, PosLFP), while also revealing surprising undecidability boundaries via reductions to tiling and cellular automata. These results deepen our understanding of when view-based query answering is tractable and how rewriting languages must be chosen in the presence of rules, with implications for data integration and ontology-based access. The work also introduces robust methodological tools (bounded treewidth view images, compactness, and automata-based entailment witnesses) that can be applied to broader problems in knowledge representation and database theory.

Abstract

We study the interaction of views, queries, and background knowledge in the form of existential rules. The motivating questions concern monotonic determinacy of a query using views w.r.t. rules, which refers to the ability to recover the query answer from the views via a monotone function. We study the decidability of monotonic determinacy, and compare with variations that require the ``recovery function'' to be in a well-known monotone query language, such as conjunctive queries or Datalog. Surprisingly, we find that even in the presence of basic existential rules, the borderline between well-behaved and badly-behaved answerability differs radically from the unconstrained case. In order to understand this boundary, we require new results concerning entailment problems involving views and rules.

Monotone Rewritability and the Analysis of Queries, Views, and Rules

TL;DR

The paper investigates monotone determinacy and rewritability of queries from views under existential rules, focusing on decidability and the expressive power of rewritings. It develops a forward-backward framework and automata-theoretic tools to analyze how views, queries, and rules interact, distinguishing finite and unrestricted models and leveraging bounded treewidth properties. The authors establish decidability results for several guarded and linear regimes and provide various rewriting guarantees (UCQ, Datalog, MDL, PosLFP), while also revealing surprising undecidability boundaries via reductions to tiling and cellular automata. These results deepen our understanding of when view-based query answering is tractable and how rewriting languages must be chosen in the presence of rules, with implications for data integration and ontology-based access. The work also introduces robust methodological tools (bounded treewidth view images, compactness, and automata-based entailment witnesses) that can be applied to broader problems in knowledge representation and database theory.

Abstract

We study the interaction of views, queries, and background knowledge in the form of existential rules. The motivating questions concern monotonic determinacy of a query using views w.r.t. rules, which refers to the ability to recover the query answer from the views via a monotone function. We study the decidability of monotonic determinacy, and compare with variations that require the ``recovery function'' to be in a well-known monotone query language, such as conjunctive queries or Datalog. Surprisingly, we find that even in the presence of basic existential rules, the borderline between well-behaved and badly-behaved answerability differs radically from the unconstrained case. In order to understand this boundary, we require new results concerning entailment problems involving views and rules.
Paper Structure (33 sections, 69 theorems, 60 equations, 18 figures)

This paper contains 33 sections, 69 theorems, 60 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tools for the positive results
  4. Applying the tools for decidability and rewritability
  5. Surprising Undecidability
  6. Conclusions
  7. Additional details for codes, automata, Datalog, and the characterization of monotonic determinacy, Section \ref{['sec:prelims']}
  8. Basics of tree codes, tree automata, and Monadic Second Order Logic
  9. More details on approximating Datalog with CQs
  10. Tree-like structure for the chase of instances, and the chase of approximations
  11. Proof of Proposition \ref{['prop:pipeline']}: the process of Figure \ref{['alg:query-mondet-constraints']} is correct
  12. Additional material related to tools for positive results, Section \ref{['sec:tools']}
  13. More details concerning the forward-backward method
  14. Proofs of Certain Answer Rewriting results
  15. Proof of Theorem \ref{['thm:fcecdatalog']}: Finite controllability of EC Datalog under Frontier One TGDs
  16. ...and 18 more sections

Key Result

Proposition 1

$Q$ is monotonically determined over $\mathbf{V}$ w.r.t. $\Sigma$ if and only if the process of Figure alg:query-mondet-constraints returns true.

Figures (18)

  • Figure 1: Decidability/Complexity without rules
  • Figure 2: Rewritability without rules
  • Figure 3: Process for checking monotonic determinacy.
  • Figure 4: Decidability with Rules
  • Figure 5: Rewritability with Rules; † thebook
  • ...and 13 more figures

Theorems & Definitions (153)

  • Example 1
  • Example 2
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 4
  • Theorem 4
  • ...and 143 more