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The Sticky Path to Expressive Querying: Decidability of Navigational Queries under Existential Rules

Piotr Ostropolski-Nalewaja, Sebastian Rudolph

TL;DR

The paper investigates whether regular path queries ($RPQ$) retain decidability under existential rules, revealing a sharp contrast between forward-chaining $\mathtt{fcs}$-style fragments and backward-chaining $\mathtt{fus}$ fragments. It proves undecidability of Boolean $RPQ$ entailment over arbitrary $\mathtt{fus}$ via a grid-encoding reduction of two-counter machines, while establishing decidability for sticky rulesets in the plain $RPQ$ setting by reducing to finite RPQ-controllability and providing both RE and co-RE procedures. A brittle boundary emerges: slightly extending $RPQ$ (to HRPQs) or relaxing stickiness again yields undecidability, showing the decidability result for sticky $RPQ$ is highly sensitive to the exact query language. The work thus maps precise frontiers for navigational querying in ontology-based data access and informs which rule-fragment and query-language combinations permit practical reasoning.

Abstract

Extensive research in the field of ontology-based query answering has led to the identification of numerous fragments of existential rules (also known as tuple-generating dependencies) that exhibit decidable answering of atomic and conjunctive queries. Motivated by the increased theoretical and practical interest in navigational queries, this paper considers the question for which of these fragments decidability of querying extends to regular path queries (RPQs). In fact, decidability of RPQs has recently been shown to generally hold for the comprehensive family of all fragments that come with the guarantee of universal models being reasonably well-shaped (that is, being of finite cliquewidth). Yet, for the second major family of fragments, known as finite unification sets (short: fus), which are based on first-order-rewritability, corresponding results have been largely elusive so far. We complete the picture by showing that RPQ answering over arbitrary fus rulesets is undecidable. On the positive side, we establish that the problem is decidable for the prominent fus subclass of sticky rulesets, with the caveat that a very mild extension of the RPQ formalism turns the problem undecidable again.

The Sticky Path to Expressive Querying: Decidability of Navigational Queries under Existential Rules

TL;DR

The paper investigates whether regular path queries () retain decidability under existential rules, revealing a sharp contrast between forward-chaining -style fragments and backward-chaining fragments. It proves undecidability of Boolean entailment over arbitrary via a grid-encoding reduction of two-counter machines, while establishing decidability for sticky rulesets in the plain setting by reducing to finite RPQ-controllability and providing both RE and co-RE procedures. A brittle boundary emerges: slightly extending (to HRPQs) or relaxing stickiness again yields undecidability, showing the decidability result for sticky is highly sensitive to the exact query language. The work thus maps precise frontiers for navigational querying in ontology-based data access and informs which rule-fragment and query-language combinations permit practical reasoning.

Abstract

Extensive research in the field of ontology-based query answering has led to the identification of numerous fragments of existential rules (also known as tuple-generating dependencies) that exhibit decidable answering of atomic and conjunctive queries. Motivated by the increased theoretical and practical interest in navigational queries, this paper considers the question for which of these fragments decidability of querying extends to regular path queries (RPQs). In fact, decidability of RPQs has recently been shown to generally hold for the comprehensive family of all fragments that come with the guarantee of universal models being reasonably well-shaped (that is, being of finite cliquewidth). Yet, for the second major family of fragments, known as finite unification sets (short: fus), which are based on first-order-rewritability, corresponding results have been largely elusive so far. We complete the picture by showing that RPQ answering over arbitrary fus rulesets is undecidable. On the positive side, we establish that the problem is decidable for the prominent fus subclass of sticky rulesets, with the caveat that a very mild extension of the RPQ formalism turns the problem undecidable again.
Paper Structure (24 sections, 22 theorems, 22 equations, 4 figures)

This paper contains 24 sections, 22 theorems, 22 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Chase and Existential Rules
  4. Query Rewritability
  5. Sticky Rulesets
  6. On the Single-Head Assumption
  7. Undecidability of RPQs over FUS
  8. Database $\mathcal{D}_{\mathrm{grid}}$ and Ruleset $\mathcal{R}_{\mathrm{grid}}$
  9. Grid $\mathcal{G}$
  10. RPQ $\mathcal{Q}_{\mathcal{M}}$
  11. Correspondence between $\mathcal{Q}_{\mathcal{M}}$ and $\mathcal{M}$
  12. Deciding RPQs over Sticky Rulesets
  13. Recursive Enumerability
  14. Co-Recursive Enumerability
  15. Overview of the Proof of \ref{['lem:sticky-co-semi-decision']}
  16. ...and 9 more sections

Key Result

Corollary 1

Let $\mathcal{R}$ be a ruleset such that for every database $\mathcal{D}$, there exists a universal model of $\mathcal{D},\mathcal{R}$ having finite cliquewidth. Then C2RPQ answering wrt. $\mathcal{R}$ is decidable.

Figures (4)

  • Figure 1: Depiction of $Ch(\mathcal{D}_{\mathrm{grid}}, \mathcal{R}_{\mathrm{grid}})$ restricted to predicates $\mathtt{IncX}$, $\mathtt{DecX}$, $\mathtt{IncY}$, $\mathtt{DecY}$, $\mathtt{XZero}$, $\mathtt{YZero}$, henceforth denoted $\mathcal{G}$.
  • Figure 2: States other than $\mathtt{q}$, $\mathtt{q_f}$, and $\mathtt{q_t}$ are part of $\mathbb{Q}_{aux}$ and are marked with diamonds in the figure. Label $\Sigma(\mathtt{C} \texttt{==} 0)$ evaluates to $\mathtt{XZero}$ or $\mathtt{YZero}$ depending whether $\mathtt{C}$ is $\mathtt{C}_{\mathtt{x}}$ or $\mathtt{C}_{\mathtt{y}}$; other labels evaluate to $\mathtt{IncX}, \mathtt{IncY}, \mathtt{DecX}$, or $\mathtt{DecY}$. The top branch of the figure corresponds to the "then" branch of the instruction assigned to the state $\mathtt{q}$ of $\mathcal{M}$. Note that $\Sigma(\mathtt{C} \texttt{==} 0)$ appears twice on the top branch. This redundancy is added to ensure an equal number of states on each branch toward establishing \ref{['obs:threesteps']}; it is by no means critical to the proof.
  • Figure 3: Instruction assigned to state $\mathtt{q}$ of TCA $\mathcal{M}$.
  • Figure :

Theorems & Definitions (52)

  • Corollary 1: following from ICDT2023, ICDT2023
  • Definition 2
  • Lemma 3
  • Definition 4: Sticky
  • proof
  • Definition 6
  • Theorem 7
  • Definition 8
  • proof
  • Lemma 10
  • ...and 42 more