Disjunctive Existential Rules Overview

• Disjunctive existential rules are first-order sentences with a conjunctive body and a disjunctive head that introduces alternative existential conclusions.
• They enable ontology-mediated query answering through chase-based semantics and query rewriting, impacting decidability and computational complexity.
• Practical applications include systems like ECOMPLETO in knowledge representation, balancing expressiveness with tractable reasoning in databases.

Searching arXiv for recent and foundational papers on disjunctive existential rules and related existential-rule languages. Disjunctive existential rules are first-order rules whose heads may contain both existential quantification and disjunction. In the database and knowledge-representation literature they are commonly presented as disjunctive tuple-generating dependencies (DTGDs) or, in the more general setting with equalities, disjunctive embedded dependencies (DEDs). They are used to represent domain knowledge for ontology-mediated query answering (OMQA), query rewriting, and rule-based reasoning with incomplete or choice-like information. A standard form is

$$\forall \mathbf{x}\,\forall \mathbf{y}\;\Bigl( B(\mathbf{x},\mathbf{y}) \;\longrightarrow\; \bigvee_{i=1}^n \exists \mathbf{z_i}\;H_i(\mathbf{x}_i,\mathbf{z_i}) \Bigr),$$

where the body is a conjunction of atoms and each head disjunct is itself a conjunction of atoms; conjunctive rules arise as the special case $n=1$ (Leclère et al., 2023). In OMQA, these rules induce ontologies via entailment of Boolean unions of conjunctive queries, and a central line of research studies their semantic behavior, expressive power, rewritability, decidable fragments, and complexity frontiers (Zhang, 2021).

1. Formalism and semantic interpretation

A disjunctive existential rule is a first-order sentence whose body is a conjunction of atoms and whose head is a disjunction of conjunctions, possibly with existential variables in each disjunct (Leclère et al., 2023). In the terminology of generalized dependencies, a disjunctive embedded dependency is a safe generalized dependency with $n \ge 1$, and a disjunctive tuple-generating dependency is a DED without equality in body or head (Zhang et al., 2020). When $k=1$, a DTGD is an ordinary TGD; when $k>1$, it is genuinely disjunctive (Zhang, 2021).

The Tarskian semantics is standard. A structure satisfies such a rule if every assignment satisfying the body can be extended so that at least one head disjunct is satisfied (Alfonso et al., 2023). Equivalently, whenever there is a homomorphism from the canonical body-instance into a structure, that homomorphism extends, for some head disjunct, to a homomorphism from that disjunct into the structure (Zhang et al., 2020).

In OMQA, one fixes disjoint data and query schemas and considers Boolean UCQs over the query schema. An ontology is represented as a set of pairs $(D,q)$, where $D$ is a finite database and $q$ is a Boolean UCQ, subject to closure under query conjunction, query implication, injective homomorphism, and constant renaming (Zhang, 2021). Given a finite set $\Sigma$ of DEDs, the induced ontology is

$$[\![\Sigma]\!]_{\mathscr{D},\mathscr{Q}} =\{(D,q)\mid D\cup\Sigma\models q\},$$

and an ontology is said to be defined by $n=1$0 when equality holds (Zhang, 2021).

A basic intuition behind the formalism is that disjunction models alternatives or uncertain choices while existential quantification introduces unnamed witnesses. One canonical example is the rule

$n=1$1

which entails $n=1$2 from $n=1$3, but entails neither $n=1$4 nor $n=1$5 alone (Zhang, 2021). Another standard illustration is

$n=1$6

used to encode a choice between two existentially witnessed alternatives (Morak, 2014).

2. Chase-based semantics and universal models

Reasoning with disjunctive existential rules is commonly based on a branching chase. For DTGDs, a nondeterministic chase step applies a trigger $n=1$7 when the body of $n=1$8 maps into the current fact base, then chooses one of the head disjuncts and adds the corresponding facts with fresh existential witnesses (Leclère et al., 2023). In a more explicit tree presentation, a disjunctive chase step transforms one instance into $n=1$9 children, one per disjunct (Morak, 2014).

A derivation tree is fair if every trigger that becomes enabled on a branch is eventually satisfied by choosing one of its disjuncts (Leclère et al., 2023). The chase result is the set of fact bases appearing along all branches of a fair derivation tree, possibly infinite (Leclère et al., 2023). In the universal-model view, the leaves of a fair disjunctive chase tree are universal models in the sense that each leaf satisfies the initial database and rule set, and every model of the knowledge base receives a homomorphism from some leaf (Morak, 2014).

Soundness and completeness take the expected form. For DTGDs and Boolean UCQs,

$n \ge 1$0

under the nondeterministic chase presentation of (Zhang, 2021). In the derivation-tree presentation, for every UCQ $n \ge 1$1 and knowledge base $n \ge 1$2,

$n \ge 1$3

(Leclère et al., 2023). This equivalence is the semantic basis for both forward-chaining procedures and backward query rewriting.

The same semantic perspective also appears in Datalog with existential quantification and disjunction. There, an instantiation procedure produces a universal model set, and query answering reduces to entailment over that instantiated ground program (Alviano et al., 2012). This suggests a close methodological connection between disjunctive existential rules, disjunctive Datalog with existentials, and ontology-oriented chase semantics.

3. Expressive power in ontology-mediated query answering

A major result for OMQA concerns the program expressive power of disjunctive TGDs. An OMQA$n \ge 1$4-ontology is DTGD-definable iff it is closed under all database homomorphisms and under all constant substitutions (Zhang, 2021). More precisely, if $n \ge 1$5 and $n \ge 1$6 is a homomorphism fixing the query constants, then $n \ge 1$7; and if $n \ge 1$8 uniformly renames constants, then $n \ge 1$9 (Zhang, 2021). The characterization is purely model-theoretic and does not require tree automata, unlike the non-disjunctive linear-TGD case (Zhang, 2021).

This criterion identifies exactly the OMQA ontologies axiomatizable by finite DTGD sets (Zhang, 2021). The proof strategy has two directions. The “only if” part uses the nondeterministic chase and the homomorphism behavior of UCQs. The “if” part starts from a finite DED-ontology with the two closure properties and eliminates equalities through an auxiliary relation $k=1$0, forcing it to behave as an equivalence relation and propagating it onto every predicate; the result is a DTGD-only axiomatization (Zhang, 2021).

At a broader level, finite sets of DEDs are expressively complete for recursively enumerable OCQA ontologies (Zhang et al., 2016). This completeness result is formulated in a semantic framework where ontologies are sets of database-query pairs satisfying conjunction, implication, and injective-homomorphism closure. The proof uses a universal-model construction $k=1$1 and a simulation of a convergent, 2-bounded nondeterministic Turing machine by a finite DED set (Zhang et al., 2016). By contrast, neither finite DTGD sets nor finite ED sets are expressively complete for recursively enumerable OCQA ontologies (Zhang et al., 2016).

The contrast between DEDs and DTGDs is sharp. Any ontology defined by finite DTGDs is closed under arbitrary database homomorphisms, and this excludes ontologies that fail homomorphism closure (Zhang et al., 2016). Likewise, every ED-defined OCQA ontology is closed under direct products of databases, so EDs alone cannot capture all recursively enumerable OCQA ontologies (Zhang et al., 2016). This establishes disjunctive existential rules with equalities as a strictly more expressive rule language in the OCQA sense.

4. Model-theoretic characterizations beyond OMQA

The model theory of disjunctive existential rules has also been studied outside the OMQA setting. For arbitrary structures, a first-order sentence is equivalent to a finite set of DEDs iff it is preserved under globally-homomorphic preimages and admits both a trivial model and a sharp model (Zhang et al., 2020). Here, preservation under globally-homomorphic preimages refines ordinary homomorphism preservation, while the trivial and sharp model conditions eliminate negative constraints and enforce the safety conditions needed for DEDs (Zhang et al., 2020).

This characterization yields a polynomial-time test for deciding whether a finite set of generalized dependencies is equivalent to some finite DED set: it suffices to check the two small test structures corresponding to the trivial and sharp models (Zhang et al., 2020). For general FO-sentences, however, rewritability into finite DED sets is RE-complete, and coRE-complete over finite models (Zhang et al., 2020).

A more recent line of work studies finite axiomatizability of classes of relational structures by disjunctive existential rules. For a class $k=1$2, finite axiomatizability by $k=1$3-dexrs is characterized by three properties: criticality, closure under repairable direct products, and diagrammatic $k=1$4-compatibility (Calautti et al., 16 Aug 2025). Criticality requires arbitrarily large complete finite structures; repairable direct products generalize product closure to accommodate disjunction; diagrammatic compatibility is a locality-style condition based on diagrams and bounded forbidden patterns (Calautti et al., 16 Aug 2025).

The same framework extends to linear and guarded disjunctive existential rules through linear-diagrammatic and guarded-diagrammatic compatibility (Calautti et al., 16 Aug 2025). The paper also gives a guarded-to-linear rewriting criterion via a bounded linearization lemma and an elementary-time decision procedure, stated to be 3-EXPTIME (Calautti et al., 16 Aug 2025). This suggests that the model-theoretic analysis of disjunctive existential rules supports not only expressiveness classification but also normalization and fragment-reduction questions.

5. Query rewriting and the finite-unification frontier

While chase-based query answering with disjunctive existential rules had been well studied, query rewriting within UCQs was addressed directly by a sound and complete rewriting operator that mirrors chase steps by rewriting steps (Leclère et al., 2023). The construction starts from piece-unifiers in the conjunctive case and generalizes them to disjunctive piece-unifiers, one per head disjunct, with jointly admissible partitions (Leclère et al., 2023). The resulting one-step rewriting operator $k=1$5 supports a correspondence between fact-level chase steps and query-level rewriting steps (Leclère et al., 2023).

A breadth-first rewriting algorithm computes level by level, uses a minimal-cover operator to remove subsumed queries, and returns a minimal UCQ-rewriting when one exists (Leclère et al., 2023). The theorem states that for every UCQ $k=1$6, fact base $k=1$7, and disjunctive rule set $k=1$8,

$k=1$9

iff there is a finite rewriting $k>1$0 produced by iterated $k>1$1-steps such that $k>1$2 (Leclère et al., 2023).

A central negative result is that “truly disjunctive” nonrecursive rules need not admit finite UCQ rewritings. Already for

$k>1$3

and

$k>1$4

any UCQ-rewriting must contain one CQ for each natural number $k>1$5, namely the paths of length $k>1$6 from a $k>1$7 to a $k>1$8; these CQs are pairwise incomparable by homomorphism, so no finite UCQ-rewriting exists (Leclère et al., 2023). More generally, for any connected disjunctive rule $k>1$9 not equivalent to a single-disjunct rule, there exists a Boolean CQ $(D,q)$0 such that $(D,q)$1 is not UCQ-rewritable (Leclère et al., 2023).

This undermines the usefulness of finite unification sets in the unrestricted disjunctive setting. The notion of fus, which in the conjunctive setting denotes rule classes under which every UCQ admits a finite UCQ-rewriting, “seems to have little relevance in this setting” (Leclère et al., 2023). Nevertheless, decidable islands remain, including atomic or full CQs under single-atom disjunctive rules and source-to-target mappings, where each piece-unifier strictly reduces target-predicate atoms (Leclère et al., 2023).

Subsequent work introduces syntactic classes designed to recover the fus property. Connected domain-restricted rules and connected linear rules are fus in the non-disjunctive case (Alfonso et al., 2023). For the disjunctive case, the extension uses disconnected disjunction: distinct disjuncts may not reference variables from the same body-connected component (Alfonso et al., 2023). If $(D,q)$2 is any fus set of non-disjunctive rules and $(D,q)$3 is a finite set of D-disjunctive rules, then $(D,q)$4 is fus even in the presence of disjunction (Alfonso et al., 2023).

6. Decidability, complexity, and restricted fragments

The computational effect of adding disjunction depends strongly on the syntactic fragment. For guarded disjunctive TGDs, Boolean CQ answering is 2EXPTIME-hard in combined complexity, already with a fixed set of binary DIDs, and lies in coNP in data complexity (Morak, 2014). Membership in 2EXPTIME is obtained by a tree-automata construction over the disjunctive chase, while coNP data membership follows by nondeterministically guessing a finite chase branch and checking the query in polynomial time (Morak, 2014).

For sticky disjunctive TGDs, Boolean CQ answering becomes undecidable even in data complexity, already for a single sticky DTGD with two disjuncts (Morak, 2014). The proof encodes a Turing machine with an infinite tape, using disjunction to guess successor tape symbols and nontermination of the chase to encode infinitary computation (Morak, 2014).

For weakly-acyclic DTGDs, the disjunctive chase always terminates in finitely many steps on any database (Morak, 2014). In this fragment, Boolean CQ answering is P-complete in data complexity and 2EXPTIME-hard in combined complexity; an upper bound in 2EXPTIME is obtained by constructing the entire chase tree and checking the leaves (Morak, 2014).

A closely related complexity picture appears in Datalog$(D,q)$5. Query answering is undecidable in general; guarded Datalog$(D,q)$6 is decidable and in 2ExpTime with a matching 2ExpTime lower bound; weakly-guarded Datalog$(D,q)$7 is also in 2ExpTime and 2ExpTime-hard; and data complexity ranges from $(D,q)$8 to ExpTime depending on the fragment and query shape (Alviano et al., 2012). These results indicate that disjunctive existential rules inherit the expressivity-versus-tractability pattern characteristic of existential-rule languages, but with significantly sharper worst-case behavior once disjunction is admitted.

Under closed predicates and repair-based semantics, DEDs with inequalities yield a further complexity landscape for consistent query answering. Repair checking is coNP-complete for arbitrary DEDs, PTIME-complete for linear DEDs, PTIME for full DEDs, and in $(D,q)$9 for acyclic DEDs (Marconi et al., 2024). For IAR- and AR-query answering, the general problem is $D$0-complete, while full or acyclic DEDs give coNP-completeness, linear DEDs give PTIME, and several combined fragments, including acyclic+linear and full+linear, are in $D$1 (Marconi et al., 2024). This suggests that disjunctive existential rules also play a role in inconsistency-tolerant semantics, not only in open-world OMQA.

7. Systems, applications, and recurring misconceptions

A concrete implementation of backward-chaining rewriting for disjunctive existential rules is ECOMPLETO, implemented in Elixir and targeting Boolean conjunctive queries with universally quantified negation (Alfonso et al., 2023). Its parser accepts DLGP$D$2 ontologies with disjunctive heads written as square-bracket lists and minus-prefixed atoms for negation; its rewriting engine alternates non-disjunctive and disjunctive rewriting phases; and, for fus fragments such as CDR/CLR plus disconnected disjunction, the outer fixpoint loop halts (Alfonso et al., 2023).

The reported evaluation used two enriched ontologies—LUBM with 70 extra disjoint class axioms and Travel with 10 disjoint class axioms—and 500 random Boolean CQs per ontology, each with 3 atoms, 2 negated, and 1 answer variable in the frontier (Alfonso et al., 2023). ECOMPLETO “succeeded in producing finite UCQ-rewritings for all 1 000 queries,” with the following reported means and standard deviations (Alfonso et al., 2023):

Metric	LUBM	Travel
mean time	18.59 m	0.035 m
std time	1.67 m	0.150 m
mean memory	370 MB	104 MB
std memory	37 MB	16.7 MB

The paper further states that Travel queries rewrote on average 500× faster and used $D$3 of the RAM compared to LUBM (Alfonso et al., 2023).

Several misconceptions recur in the literature. One is that disjunctive existential rules can be understood purely as ordinary TGDs plus a syntactic head disjunction. The formal and complexity results show otherwise: disjunction changes rewritability, breaks many classical fus arguments, and can raise complexity to 2EXPTIME-hardness or undecidability depending on the fragment (Leclère et al., 2023). A second misconception is that tree-automata machinery is intrinsic to all disjunctive-TGD expressive-power results. For the OMQA program expressive power of DTGDs, the key characterization is purely model-theoretic; tree automata are needed only in the non-disjunctive linear-TGD case (Zhang, 2021). A third misconception is that direct-product closure is the right closure notion for disjunctive existential rules in general. The finite-axiomatizability results show that direct products may fail under disjunction, motivating repairable direct products instead (Calautti et al., 16 Aug 2025).

Disjunctive existential rules therefore occupy a distinctive position in database theory and knowledge representation. They generalize classical existential rules by admitting alternative existential consequences, they support a precise chase semantics via branching universal models, and they admit exact model-theoretic characterizations in both OMQA and axiomatizability settings (Zhang, 2021). At the same time, their algorithmic behavior is markedly more delicate than that of their non-disjunctive counterparts, which has driven the design of restricted fragments such as guarded, weakly-acyclic, connected linear, connected domain-restricted, and disconnected-disjunction classes (Morak, 2014). A plausible implication is that the long-term development of the area will continue to balance three interacting goals already visible in the current literature: semantic expressiveness, finite rewritability, and controlled combinatorics of branching reasoning.