Linear Disjunctive Existential Rules

• Linear disjunctive existential rules are a subclass of existential rules with a single-atom body and disjunctive heads that enable choice-based uncertainty while retaining low data complexity.
• They extend linear existential rules by incorporating multiple alternatives in the head, impacting rewriting behavior and increasing combined complexity.
• Their study integrates decidability, refined rewriting techniques, and model-theoretic characterizations to advance ontology engineering and query answering.

Searching arXiv for recent and foundational papers on linear disjunctive existential rules. Linear disjunctive existential rules are existential rules whose body contains a single atom and whose head may contain a disjunction of existentially quantified conjunctions of atoms. In the terminology of the literature, they are a subclass of disjunctive tuple-generating dependencies (DTGDs), themselves a disjunctive extension of Datalog$^\pm$ and of existential-rule languages more broadly (Morak, 2014). This fragment occupies a distinctive position in rule-based knowledge representation and ontology-mediated query answering: it extends the representational scope of linear existential rules by allowing uncertain or choice-based consequences, while retaining favorable properties in some reasoning settings, notably decidability and low data complexity for several query classes (Morak, 2014). At the same time, disjunction sharply changes the behavior of rewriting, expressive power, and model-theoretic characterizations, so that linearity alone does not preserve all of the robust meta-properties known for the non-disjunctive case (Leclère et al., 2023).

1. Formal setting and basic definitions

A tuple-generating dependency is a rule of the form

$$\forall \vec{X} \left( \phi(\vec{X}) \rightarrow \exists \vec{Y} \, \psi(\vec{X}, \vec{Y}) \right),$$

while the disjunctive extension considered in the literature allows rules of the form

$$\forall \vec{X} \left( \varphi(\vec{X}) \rightarrow \bigvee_{i=1}^n \exists \vec{Y}_i \, \psi_i(\vec{X}, \vec{Y}_i) \right).$$

These are called disjunctive tuple-generating dependencies, or disjunctive existential rules (Morak, 2014). A linear disjunctive existential rule is the special case in which the body consists of a single atom (Leclère et al., 2023). In the broader dependency formalism of disjunctive embedded dependencies with inequalities (DEDs), a dependency is linear if $|PA(\varphi)| = 1$, that is, the body has exactly one predicate atom (Marconi et al., 2024).

This syntax allows direct expression of uncertain consequences. The motivating examples given in the literature include statements such as “every student is either an undergraduate or a graduate,” which cannot be represented in ordinary non-disjunctive existential rules without changing the semantics of the representation (Morak, 2014). The linear restriction limits how information is matched in premises, but does not constrain the multiplicity of alternatives in the head.

The fragment sits within a standard hierarchy. Linear rules are a subclass of guarded rules and generalize inclusion dependencies; inclusion dependencies impose additional restrictions such as the absence of repeated variables (Morak, 2014). In the model-theoretic literature on existential-rule languages, linear TGDs are characterized by preservation under unions, which distinguishes them from guarded and frontier-guarded classes (Zhang et al., 2020). A plausible implication is that linear disjunctive rules inherit a strong “local trigger” character from linearity, while disjunction contributes non-deterministic model generation and branching semantics.

2. Semantics and the disjunctive chase

Reasoning with disjunctive existential rules is typically based on the disjunctive chase, which generalizes the chase procedure for TGDs (Morak, 2014). When a disjunctive rule is applicable, the procedure branches on the alternative head disjuncts, thereby building a chase tree rather than a single chase sequence. The leaves of that tree represent models, and certain query answering reduces to checking whether the query holds at all leaves (Morak, 2014).

This branching semantics is central for linear disjunctive existential rules as well. Linearity restricts only the matching condition in the body; once a rule fires, existential variables and head disjunction create the same model branching as in the general DTGD setting. The 2026 formalization in Lean treats existential rules in a general form with disjunctive heads and constants,

$$\forall \vec{x}, \vec{y}.~ B[\vec{x}, \vec{y}] \to \bigvee_{i=1}^n \exists \vec{z}_i.~ H_i[\vec{y}_i, \vec{z}_i],$$

and models the disjunctive chase using possibly infinite trees (Gerlach, 24 Apr 2026). In that framework, the result in the presence of disjunction is not a single universal model but a universal model set, with one result per branch (Gerlach, 24 Apr 2026).

This semantics clarifies an important point. Disjunction does not merely add a syntactic convenience; it changes the semantic object used for reasoning from a universal model to a family of models. That shift explains why complexity rises relative to the non-disjunctive linear fragment and why rewriting behavior differs substantially from the ordinary finite-unification-set picture familiar from linear TGDs (Morak, 2014, Leclère et al., 2023).

3. Complexity and decidability

The doctoral-summary paper on the impact of disjunction in Datalog$^\pm$ gives the main complexity classification for guarded-based fragments and identifies the linear case explicitly (Morak, 2014). For linear DTGDs, query answering is decidable. The combined complexity is EXPTIME-complete, while the data complexity is $AC^0$ (Morak, 2014). The same summary reports that for atomic queries, combined complexity remains EXPTIME-complete and data complexity remains aligned with the general linear case, namely $AC^0$ (Morak, 2014).

The contrast with neighboring fragments is instructive. Guarded DTGDs are 2EXPTIME-complete in combined complexity and coNP-complete in data complexity, while sticky TGDs become undecidable once disjunction is added, even in data complexity (Morak, 2014). Weakly-acyclic rule sets remain decidable with a “reasonable and expected increase in complexity,” but linear DTGDs preserve the most favorable data-complexity classification among the classes explicitly discussed there (Morak, 2014).

The following table summarizes the complexity figures stated in the literature for the linear fragment and selected neighboring classes.

Fragment	Combined complexity	Data complexity
Linear DTGDs	EXPTIME-complete	$AC^0$
Guarded DTGDs	2EXPTIME-complete	coNP-complete
Disjunctive IDs/DIDs	2EXPTIME-complete	coNP-complete
Sticky DTGDs	Undecidable	Undecidable

These bounds establish that linearity meaningfully controls the effect of disjunction on data complexity, but not on combined complexity. The literature states that combined complexity increases from PSPACE in the non-disjunctive linear setting to EXPTIME with disjunction (Morak, 2014). This suggests that head branching is computationally expensive even when premise matching remains syntactically minimal.

A common misconception is that linearity automatically guarantees broad tractability. The complexity results show a more precise picture: tractability is robust at the data-complexity level, but the full reasoning problem remains EXPTIME-complete once rules, data, and query are all part of the input (Morak, 2014).

4. Query rewriting and the limits of UCQ-rewritability

Query rewriting for disjunctive existential rules has been studied much less than chase-based reasoning, and the 2023 work on rewriting with disjunctive existential rules provides the main framework currently available (Leclère et al., 2023). That paper defines a sound and complete rewriting operator based on disjunctive piece-unifiers and establishes a tight correspondence between chase steps and rewriting steps. For a rule $R = B \rightarrow H_1 \lor \dots \lor H_n$ and a UCQ $$\forall \vec{X} \left( \phi(\vec{X}) \rightarrow \exists \vec{Y} \, \psi(\vec{X}, \vec{Y}) \right),$$0, the rewriting step is

$$\forall \vec{X} \left( \phi(\vec{X}) \rightarrow \exists \vec{Y} \, \psi(\vec{X}, \vec{Y}) \right),$$1

and the central theorem states: $$\forall \vec{X} \left( \phi(\vec{X}) \rightarrow \exists \vec{Y} \, \psi(\vec{X}, \vec{Y}) \right),$$2 This gives a sound and complete rewriting semantics for disjunctive rules (Leclère et al., 2023).

For linear disjunctive existential rules, the rewriting picture is sharply query-sensitive. If the rule set is linear and the query is an atomic CQ, UCQ-rewritability is guaranteed because only atomic rewritings are generated (Leclère et al., 2023). By contrast, for arbitrary UCQs, even with linear rules, rewriting may fail to terminate because the number and structure of generated queries may be unbounded (Leclère et al., 2023). The paper further shows that for any “truly disjunctive” nonrecursive rule, there exists a conjunctive query with no UCQ-rewriting, undermining the relevance of finite unification sets in the disjunctive setting (Leclère et al., 2023).

This is one of the main conceptual discontinuities between linear TGDs and linear DTGDs. In the non-disjunctive case, linearity is classically associated with finite rewriting behavior. In the disjunctive case, linearity still supports UCQ-rewritability for atomic queries, but no longer suffices to guarantee finite complete rewritings for arbitrary UCQs (Leclère et al., 2023). The breadth-first rewriting algorithm introduced there halts and outputs a minimal UCQ-rewriting if one exists; otherwise, the resulting sound and complete rewriting can be infinite (Leclère et al., 2023).

A further negative result concerns mappings. The same work shows that deciding whether a UCQ admits a UCQ-rewriting through a disjunctive mapping is undecidable (Leclère et al., 2023). This does not target linear rules alone, but it reinforces the broader point that disjunction invalidates several of the classical rewriting intuitions inherited from existential-rule fragments without disjunctive heads.

5. Structural refinements and finite unification phenomena

The loss of general finite-rewriting guarantees has motivated the study of more restrictive subclasses. A 2023 paper introduces connected linear rules and connected domain restricted rules, and then extends the analysis to disjunctive existential rules via the notion of disconnected disjunction (Alfonso et al., 2023). In that framework, connected linear existential rules form a finite unification set, but naively extending them to the disjunctive case does not preserve that property (Alfonso et al., 2023).

The proposed remedy is disconnected disjunction: a disjunctive existential rule has disconnected disjunction if, in its head, each disjunct does not share variables with the same connected component of the body (Alfonso et al., 2023). The paper states that if all rules are connected linear rules with disconnected disjunction, all rewritings produced remain within the class, and the rule set is again a finite unification set (Alfonso et al., 2023).

For ordinary linear disjunctive existential rules, this result is significant as a boundary statement. It indicates that the linear condition alone is not the right structural invariant for finite rewriting once disjunction is present; additional connectivity constraints on how head disjuncts relate to body variables are needed (Alfonso et al., 2023). A plausible implication is that disconnectedness conditions may play, for disjunctive rules, a role analogous to acyclicity or guardedness conditions in non-disjunctive rewriting theory.

The same work also presents ECOMPLETO, a system for query rewriting with disjunctive existential rules that can handle UCQs with universally quantified negation (Alfonso et al., 2023). The experimental statement in the provided data is limited to the claim that the system consistently produces finite UCQ-rewritings and reports performance on different ontologies and queries (Alfonso et al., 2023).

6. Expressive power and model-theoretic characterizations

The expressive behavior of disjunctive existential rules has been studied from both ontology-mediated and axiomatizability perspectives. In the program-expressive-power setting, DTGD ontologies over UCQs are characterized by closure under database homomorphisms and constant substitutions (Zhang, 2021). For constant-free queries, closure under database homomorphisms suffices (Zhang, 2021). The same paper shows that for CQ answering, every DTGD ontology can be encoded by TGDs, so disjunction does not increase program expressive power there; for UCQ answering, however, TGDs match DTGDs exactly when the ontology is query constructive (Zhang, 2021).

This distinction is relevant for linear disjunctive existential rules because it shows that disjunction alters expressive power in a query-language-sensitive way. The simple example

$$\forall \vec{X} \left( \phi(\vec{X}) \rightarrow \exists \vec{Y} \, \psi(\vec{X}, \vec{Y}) \right),$$3

illustrates a DTGD ontology in which $$\forall \vec{X} \left( \phi(\vec{X}) \rightarrow \exists \vec{Y} \, \psi(\vec{X}, \vec{Y}) \right),$$4 may be entailed without either disjunct being entailed alone, violating query constructivity and preventing TGD definability under full UCQ answering (Zhang, 2021).

A complementary line of work addresses finite axiomatizability of classes of structures by disjunctive existential rules. A 2025 paper characterizes finite axiomatizability by finite sets of disjunctive existential rules using criticality, closure under repairable direct products, and a new property called diagrammatic compatibility (Calautti et al., 16 Aug 2025). It then specializes this analysis to linear disjunctive existential rules by introducing linear-diagrammatic compatibility and proves an equivalence: a collection of structures is finitely axiomatizable by linear $$\forall \vec{X} \left( \phi(\vec{X}) \rightarrow \exists \vec{Y} \, \psi(\vec{X}, \vec{Y}) \right),$$5-dexrs iff it is critical, closed under repairable direct products, and linear-diagrammatically $$\forall \vec{X} \left( \phi(\vec{X}) \rightarrow \exists \vec{Y} \, \psi(\vec{X}, \vec{Y}) \right),$$6-compatible (Calautti et al., 16 Aug 2025).

This result places linear disjunctive existential rules within a precise model-theoretic framework. In the non-disjunctive setting, linear TGDs are associated with preservation under unions (Zhang et al., 2020). In the disjunctive setting, finite axiomatizability requires more refined locality and product-style conditions (Calautti et al., 16 Aug 2025). That shift reflects the semantic branching introduced by disjunction and the fact that direct-product behavior must be repaired rather than assumed outright.

The same 2025 work also states that diagrammatic compatibility can be exploited to rewrite guarded disjunctive existential rules into equivalent linear disjunctive existential rules, if such a rewriting exists (Calautti et al., 16 Aug 2025). This makes linearity not only a source of complexity reductions, but also a target normal form in definability analysis.

7. Variants, applications, and related boundaries

The notion of a linear disjunctive existential rule appears in several neighboring settings. In the broad Datalog$$\forall \vec{X} \left( \phi(\vec{X}) \rightarrow \exists \vec{Y} \, \psi(\vec{X}, \vec{Y}) \right),$$7 formalism, linear programs are guarded rules with no side atoms, and the 2012 work on disjunctive Datalog with existential quantifiers identifies many decidable fragments by extending known guardedness paradigms to the disjunctive case (Alviano et al., 2012). That paper provides the general language backdrop in which linear disjunctive rules emerge as one of the simplest decidable classes (Alviano et al., 2012).

In inconsistency-tolerant reasoning under closed predicates, the 2024 study of consistent query answering considers linear DEDs, where the body has exactly one predicate atom, possibly together with inequalities (Marconi et al., 2024). Under the closed-predicate assumption, repairs are obtained only by tuple deletion, and for linear dependencies AR and IAR semantics coincide for unions of conjunctive queries (Marconi et al., 2024). The paper reports PTIME-completeness for repair checking, instance checking, and Boolean UCQ entailment for linear dependencies, and $$\forall \vec{X} \left( \phi(\vec{X}) \rightarrow \exists \vec{Y} \, \psi(\vec{X}, \vec{Y}) \right),$$8 for the acyclic+linear case (Marconi et al., 2024). These results concern repair semantics rather than standard open-world entailment, but they show that the linear syntactic discipline continues to enforce tractable behavior in a substantially different semantic regime.

Another boundary result comes from conservative extension. For linear TGDs, deciding CQ-conservative extension and hom-conservative extension is undecidable (Jung et al., 2022). This concerns non-disjunctive linear rules, not linear DTGDs directly, but it is relevant as a caution against overgeneralizing from query-answering decidability. Even very tame existential-rule fragments can support undecidable meta-problems about ontology evolution and equivalence (Jung et al., 2022). A plausible implication is that adding disjunction would not improve this picture and may worsen it.

Finally, transitivity interacts delicately with linearity. The 2015 paper on combining existential rules and transitivity shows that transitivity can be safely added to linear rules for atomic CQs, and for general CQs under a safety condition, using a pattern-based rewriting algorithm (Baget et al., 2015). Although this result concerns linear existential rules rather than disjunctive linear rules, it clarifies that “linear” is not a uniform guarantee of good behavior under arbitrary language extensions; the precise interaction depends on the extension under study.

In sum, linear disjunctive existential rules form a minimal syntactic environment in which existential quantification and disjunctive uncertainty coexist. Their significance lies in the combination of three facts established across the literature: they preserve decidability and $$\forall \vec{X} \left( \phi(\vec{X}) \rightarrow \exists \vec{Y} \, \psi(\vec{X}, \vec{Y}) \right),$$9 data complexity for standard query answering (Morak, 2014); they admit sound and complete rewriting procedures, but only query-specific finite rewritability, with atomic queries as the clean positive case (Leclère et al., 2023); and they possess a now well-developed model theory, including finite axiomatizability criteria and linearization results for guarded disjunctive theories (Calautti et al., 16 Aug 2025). These features make the fragment both a practical target for ontology engineering and a sharp boundary object for complexity, expressivity, and rewriting theory in the study of existential rules.