Guarded Disjunctive Existential Rules

• The paper establishes that guarded rule sets preserve decidability by requiring a guard atom to cover all universally quantified variables, even when allowing disjunctive conclusions.
• Key methodology focuses on using diagrammatic compatibility and repairable direct products to precisely characterize finite axiomatizability and manage semantic branching.
• The work highlights algorithmic techniques such as bounded linearization, enabling the rewriting of guarded rules into equivalent linear forms to support efficient query answering.

Searching arXiv for recent and foundational papers on guarded disjunctive existential rules, DTGDs, and related finite axiomatizability/reasoning complexity. Guarded disjunctive existential rules are a syntactically restricted class of disjunctive existential rules in which the rule body is empty or contains an atom that mentions all universally quantified variables. In the database literature, disjunctive existential rules are also called disjunctive tuple-generating dependencies, and they extend existential-rule languages by allowing a disjunctive head rather than a single existentially quantified consequence. Recent work gives an exact model-theoretic characterization of finite axiomatizability for the guarded fragment in terms of criticality, closure under repairable direct products, and guarded-diagrammatic compatibility, and also studies when guarded rule sets can be rewritten into equivalent linear ones (Calautti et al., 16 Aug 2025). Earlier work on reasoning established that guardedness preserves decidability in the presence of head disjunction, but that query answering incurs a substantial complexity increase, reaching $2$EXPTIME-completeness in combined complexity for guarded disjunctive tuple-generating dependencies (Morak, 2014).

1. Formal definition and syntactic position

A disjunctive existential rule over a schema ${\mathcal S}$ is defined as a constant-free first-order sentence of the form

$$\forall \bar x \forall \bar y \left(\phi(\bar x,\bar y)\ \ \bigvee_{i=1}^{k} \exists \bar z_i\, \psi_i(\bar x_i,\bar z_i)\right),$$

where $k > 0$, $$\bar x,\bar y,\bar x_1,\ldots,\bar x_k,\bar z_1,\ldots,\bar z_k$$ are tuples of variables, each variable of $\bar x_i$ occurs in $\bar x$, each variable of $\bar x$ occurs in some $\bar x_i$, $\phi(\bar x,\bar y)$ is a possibly empty conjunction of atoms, and each ${\mathcal S}$0 is a non-empty conjunction of atoms (Calautti et al., 16 Aug 2025). The body is ${\mathcal S}$1, and the head is ${\mathcal S}$2 (Calautti et al., 16 Aug 2025).

Semantically, ${\mathcal S}$3 means that whenever the body is matched in ${\mathcal S}$4, at least one disjunct in the head can be satisfied by extending the match (Calautti et al., 16 Aug 2025). In the alternative notation used for disjunctive tuple-generating dependencies,

${\mathcal S}$5

the same idea is emphasized: one body match forces a choice among several existential consequences (Morak, 2014).

A disjunctive existential rule is guarded if its body is empty or has an atom that mentions all the universally quantified variables (Calautti et al., 16 Aug 2025). In the guarded-tuple-generating-dependency presentation, this body atom is called the guard, and guardedness is the standard structural condition requiring that the body contain an atom with all universally quantified variables of the rule (Morak, 2014). Linear rules form a closely related subclass: they are explicitly described as another main member of the guarded family of disjunctive existential rules (Calautti et al., 16 Aug 2025).

This guardedness condition is purely syntactic, but it has substantial semantic and algorithmic consequences. The 2025 characterization work treats guarded rules as a robust specialization of the general disjunctive existential-rule framework (Calautti et al., 16 Aug 2025), while the 2014 reasoning study treats guarded disjunctive tuple-generating dependencies as one of the central decidable Datalog${\mathcal S}$6 fragments after disjunction is added (Morak, 2014).

2. Semantics, model classes, and query answering

The model-theoretic viewpoint studies classes ${\mathcal S}$7 of relational structures and asks when such a class is exactly the set of models of a finite set of guarded disjunctive existential rules (Calautti et al., 16 Aug 2025). This shifts attention from derivational procedures to semantic invariants of model classes. The main result for the guarded setting is a finite-axiomatizability theorem parameterized by bounds on the numbers of universal variables, existential variables, and head disjuncts (Calautti et al., 16 Aug 2025).

The query-answering viewpoint starts from a database ${\mathcal S}$8 and a theory ${\mathcal S}$9, and uses certain-answer semantics: $$\forall \bar x \forall \bar y \left(\phi(\bar x,\bar y)\ \ \bigvee_{i=1}^{k} \exists \bar z_i\, \psi_i(\bar x_i,\bar z_i)\right),$$0 For boolean conjunctive queries, the decision problem is whether $$\forall \bar x \forall \bar y \left(\phi(\bar x,\bar y)\ \ \bigvee_{i=1}^{k} \exists \bar z_i\, \psi_i(\bar x_i,\bar z_i)\right),$$1 (Morak, 2014). Under this semantics, disjunction in the rule head is significant because it creates branching in the model space: instead of a single existential consequence, at least one of several alternatives must hold (Morak, 2014).

To capture that branching operationally, the disjunctive setting uses a disjunctive chase. A disjunctive tuple-generating dependency $$\forall \bar x \forall \bar y \left(\phi(\bar x,\bar y)\ \ \bigvee_{i=1}^{k} \exists \bar z_i\, \psi_i(\bar x_i,\bar z_i)\right),$$2 is applicable to an instance $$\forall \bar x \forall \bar y \left(\phi(\bar x,\bar y)\ \ \bigvee_{i=1}^{k} \exists \bar z_i\, \psi_i(\bar x_i,\bar z_i)\right),$$3 if the body matches under some homomorphism $$\forall \bar x \forall \bar y \left(\phi(\bar x,\bar y)\ \ \bigvee_{i=1}^{k} \exists \bar z_i\, \psi_i(\bar x_i,\bar z_i)\right),$$4, but none of the disjuncts is already satisfied under an extension $$\forall \bar x \forall \bar y \left(\phi(\bar x,\bar y)\ \ \bigvee_{i=1}^{k} \exists \bar z_i\, \psi_i(\bar x_i,\bar z_i)\right),$$5. Applying $$\forall \bar x \forall \bar y \left(\phi(\bar x,\bar y)\ \ \bigvee_{i=1}^{k} \exists \bar z_i\, \psi_i(\bar x_i,\bar z_i)\right),$$6 yields a set of successor instances,

$$\forall \bar x \forall \bar y \left(\phi(\bar x,\bar y)\ \ \bigvee_{i=1}^{k} \exists \bar z_i\, \psi_i(\bar x_i,\bar z_i)\right),$$7

where each $$\forall \bar x \forall \bar y \left(\phi(\bar x,\bar y)\ \ \bigvee_{i=1}^{k} \exists \bar z_i\, \psi_i(\bar x_i,\bar z_i)\right),$$8 is obtained by adding the $$\forall \bar x \forall \bar y \left(\phi(\bar x,\bar y)\ \ \bigvee_{i=1}^{k} \exists \bar z_i\, \psi_i(\bar x_i,\bar z_i)\right),$$9-th disjunct with fresh nulls (Morak, 2014). Repeated fair applications generate a disjunctive chase tree whose leaves define $k > 0$0, and this set of leaves is a universal set model: for every model $k > 0$1, there exists $k > 0$2 and a homomorphism $k > 0$3 such that $k > 0$4 (Morak, 2014). Consequently,

$k > 0$5

for conjunctive query answering (Morak, 2014).

This division between semantic model classes and operational branching semantics is central to the topic. The former explains which classes can be axiomatized by finite guarded disjunctive rule sets; the latter explains why even guarded reasoning becomes substantially harder once head disjunction is present.

3. Finite axiomatizability and the guarded characterization

The central semantic theorem for guarded disjunctive existential rules states that for a collection $k > 0$6 of structures and $k > 0$7, with $k > 0$8, and $k > 0$9, the following are equivalent:

1. $$\bar x,\bar y,\bar x_1,\ldots,\bar x_k,\bar z_1,\ldots,\bar z_k$$0 is finitely axiomatizable by guarded $$\bar x,\bar y,\bar x_1,\ldots,\bar x_k,\bar z_1,\ldots,\bar z_k$$1-dexrs.
2. $$\bar x,\bar y,\bar x_1,\ldots,\bar x_k,\bar z_1,\ldots,\bar z_k$$2 is critical, closed under repairable direct products, and guarded-diagrammatically $$\bar x,\bar y,\bar x_1,\ldots,\bar x_k,\bar z_1,\ldots,\bar z_k$$3-compatible (Calautti et al., 16 Aug 2025).

This is an exact semantic characterization of finite axiomatizability by finite sets of guarded disjunctive existential rules with bounded numbers of universal variables, existential variables, and head disjuncts (Calautti et al., 16 Aug 2025). The theorem is a guarded refinement of a more general characterization for unrestricted disjunctive existential rules, and the paper presents it as evidence for the robustness of the diagrammatic-compatibility method (Calautti et al., 16 Aug 2025).

Three ingredients are indispensable. First, criticality is inherited from the unrestricted setting. A structure $$\bar x,\bar y,\bar x_1,\ldots,\bar x_k,\bar z_1,\ldots,\bar z_k$$4 is $$\bar x,\bar y,\bar x_1,\ldots,\bar x_k,\bar z_1,\ldots,\bar z_k$$5-critical if $$\bar x,\bar y,\bar x_1,\ldots,\bar x_k,\bar z_1,\ldots,\bar z_k$$6 and every relation is full, i.e. $$\bar x,\bar y,\bar x_1,\ldots,\bar x_k,\bar z_1,\ldots,\bar z_k$$7. A collection $$\bar x,\bar y,\bar x_1,\ldots,\bar x_k,\bar z_1,\ldots,\bar z_k$$8 is critical if it contains a $$\bar x,\bar y,\bar x_1,\ldots,\bar x_k,\bar z_1,\ldots,\bar z_k$$9-critical structure for every $\bar x_i$0. The relevant lemma states that a collection of structures finitely axiomatizable by disjunctive existential rules is critical (Calautti et al., 16 Aug 2025).

Second, the appropriate product closure is not ordinary direct-product closure but closure under repairable direct products. Given $\bar x_i$1, their direct product $\bar x_i$2 is defined as usual. A structure $\bar x_i$3 is a repairable direct product of $\bar x_i$4 and $\bar x_i$5 if $\bar x_i$6 and the projective homomorphism $\bar x_i$7 extends to a homomorphism $\bar x_i$8 (Calautti et al., 16 Aug 2025). A corresponding lemma states that any class finitely axiomatizable by disjunctive existential rules is closed under repairable direct products (Calautti et al., 16 Aug 2025).

Third, the new ingredient is guarded-diagrammatic compatibility. The general notion of diagrammatic compatibility is based on $\bar x_i$9-diagrams $\bar x$0 for finite substructures $\bar x$1, and roughly ensures that if a structure $\bar x$2 violates a bounded disjunctive existential rule, then the collection $\bar x$3 contains some structure witnessing the same violation (Calautti et al., 16 Aug 2025). In the guarded case, the notion is specialized to guarded local configurations. A structure $\bar x$4 is guarded if either $\bar x$5, or there is $\bar x$6 such that $\bar x$7 (Calautti et al., 16 Aug 2025). Then $\bar x$8 is guarded-diagrammatically $\bar x$9-compatible with $\bar x$0 if, for every guarded substructure $\bar x$1 of $\bar x$2 with $\bar x$3 and $\bar x$4, and every $\bar x$5-diagram $\bar x$6 of $\bar x$7 relative to $\bar x$8, there exists $\bar x$9 such that $\bar x_i$0 (Calautti et al., 16 Aug 2025).

The paper explicitly notes that guarded-diagrammatic $\bar x_i$1-compatibility implies diagrammatic $\bar x_i$2-compatibility (Calautti et al., 16 Aug 2025). The guarded variant is therefore stronger and more specialized: it restricts the local tests to substructures whose active domain is covered by a single fact, matching the syntactic role of the guard in rule bodies (Calautti et al., 16 Aug 2025). This suggests that guardedness is reflected semantically not through a different global closure principle, but through a filtered notion of local configurational simulation.

4. Closure phenomena, locality, and why repairable products are needed

Ordinary direct products are insufficient for disjunctive existential rules. The paper gives a counterexample using the disjunctive existential rule

$\bar x_i$3

Let $\bar x_i$4 have facts $\bar x_i$5, and $\bar x_i$6 have facts $\bar x_i$7. Then both $\bar x_i$8 and $\bar x_i$9, but

$\phi(\bar x,\bar y)$0

and $\phi(\bar x,\bar y)$1 (Calautti et al., 16 Aug 2025). This example motivates the replacement of standard direct-product closure by closure under repairable direct products (Calautti et al., 16 Aug 2025).

The same example also clarifies the distinctive semantic burden introduced by disjunction. Each factor satisfies the rule by realizing a different disjunct, but the direct product need not realize any one disjunct coherently. A plausible implication is that the model-theoretic behavior of guarded disjunctive existential rules is governed not just by local guardedness, but also by how alternative head realizations interact under products.

The 2025 paper also situates diagrammatic compatibility with respect to older locality notions. It proves that diagrammatic $\phi(\bar x,\bar y)$2-compatibility implies $\phi(\bar x,\bar y)$3-locality (Calautti et al., 16 Aug 2025). The inclusion is strict in informational terms, because diagrammatic compatibility tracks the number of head disjuncts $\phi(\bar x,\bar y)$4, while locality does not (Calautti et al., 16 Aug 2025). The same rule $\phi(\bar x,\bar y)$5 is used to illustrate the distinction: if $\phi(\bar x,\bar y)$6 is its model class, then $\phi(\bar x,\bar y)$7 is finitely axiomatizable by $\phi(\bar x,\bar y)$8-disjunctive existential rules, hence diagrammatically $\phi(\bar x,\bar y)$9-compatible and ${\mathcal S}$00-local, but it is not finitely axiomatizable by ${\mathcal S}$01-disjunctive existential rules (Calautti et al., 16 Aug 2025). This shows that locality alone cannot distinguish one-disjunct from two-disjunct classes, whereas diagrammatic compatibility can (Calautti et al., 16 Aug 2025).

For the guarded fragment, the same lesson persists with the restriction to guarded substructures. The guarded characterization does not merely inherit unrestricted semantic criteria; it sharpens the local compatibility condition so that it respects the guard atom’s coverage of universal variables (Calautti et al., 16 Aug 2025).

5. Reasoning complexity for guarded disjunctive tuple-generating dependencies

The main complexity result for guarded disjunctive tuple-generating dependencies is that disjunction does not destroy decidability for guarded theories, but it causes a significant complexity jump (Morak, 2014). For guarded theories with disjunction, the paper states a strong ${\mathcal S}$02EXPTIME lower bound in combined complexity, even for very restricted formalisms such as fixed sets of disjunctive inclusion dependencies (Morak, 2014). Its later summary reports for guarded DTGDs that combined complexity is ${\mathcal S}$03EXPTIME-complete and data complexity is co-complete for arbitrary queries; for atomic queries, guarded DTGDs remain ${\mathcal S}$04EXPTIME-complete in combined complexity and co-complete in data complexity (Morak, 2014).

The guardedness notion used there is the standard one: a rule is guarded if the body contains an atom mentioning all universally quantified variables, and that atom is the guard (Morak, 2014). Linear TGDs and inclusion dependencies are noted to be trivially guarded because they have only one body atom (Morak, 2014). Guarded DTGDs therefore inherit the same structural discipline as ordinary guarded existential rules, but the presence of disjunctive heads changes the complexity landscape dramatically.

The lower bound is robust under strong syntactic simplification. The paper states that the ${\mathcal S}$05EXPTIME lower bound already holds for fixed sets of disjunctive inclusion dependencies, with either fixed sets of DIDs of arity at most ${\mathcal S}$06, or non-fixed sets of DIDs of arity at most ${\mathcal S}$07 (Morak, 2014). This shows that the complexity blow-up is not due to elaborate body syntax. Rather, it is caused by branching choice in the rule head (Morak, 2014).

The proof ideas described in the research summary align with this interpretation. For arbitrary queries, the ${\mathcal S}$08EXPTIME lower bound is obtained via a Büchi tree automaton simulation (Morak, 2014). The upper bounds for guarded DTGDs are connected to expressive fragments of first-order logic such as the Guarded Fragment and Guarded-Negation First-Order Logic (Morak, 2014). This suggests that guarded disjunctive existential rules occupy a boundary where guarded tree-likeness is still sufficient for decidability, but no longer sufficient for moderate combined complexity.

6. Rewriting guarded rules into equivalent linear rules

The 2025 paper studies a specific transformation problem for guarded disjunctive existential rules: whether a finite guarded rule set can be rewritten into an equivalent finite set of linear disjunctive existential rules (Calautti et al., 16 Aug 2025). It defines the problem ${\mathcal S}$09 as follows: the input is a finite set ${\mathcal S}$10 of guarded dexrs, and the output is a finite set ${\mathcal S}$11 of linear dexrs such that ${\mathcal S}$12, if one exists; otherwise ${\mathcal S}$13 (Calautti et al., 16 Aug 2025).

The main theorem states that ${\mathcal S}$14 is computable in elementary time (Calautti et al., 16 Aug 2025). The key technical tool is the Bounded Linearization Lemma. If a collection ${\mathcal S}$15 of structures over a schema ${\mathcal S}$16 is finitely axiomatizable by ${\mathcal S}$17-dexrs and

${\mathcal S}$18

then the following are equivalent:

1. ${\mathcal S}$19 is finitely axiomatizable by linear dexrs.
2. ${\mathcal S}$20 is linear-diagrammatically ${\mathcal S}$21-compatible (Calautti et al., 16 Aug 2025).

This bounded characterization makes an exhaustive search possible. The algorithm ${\mathcal S}$22 computes the bound ${\mathcal S}$23, enumerates all linear ${\mathcal S}$24-dexrs entailed by the input guarded set, and returns them if they are non-empty and entail the input; otherwise it returns ${\mathcal S}$25 (Calautti et al., 16 Aug 2025). The paper states that the algorithm runs in elementary time and refines this to triple-exponential time; it also remarks that an optimal double-exponential bound would follow from a stronger polynomial bound on the number of disjuncts, but this remains open (Calautti et al., 16 Aug 2025).

The significance of this result is methodological. The guarded fragment is strictly more general syntactically than the linear fragment, but not every guarded theory needs genuinely guarded interaction in its body. The rewriting theorem provides an exact algorithmic route for detecting when guardedness is eliminable in favor of linearity (Calautti et al., 16 Aug 2025). A plausible implication is that the semantic notion of guarded-diagrammatic compatibility is not only classificatory, but also constructive enough to support rule synthesis.

7. Related semantics with negation and inconsistency handling

A related line of work studies guarded existential rules under stable model semantics with inconsistency handled by repairing the rule set rather than the database (Wan et al., 2016). This setting does not use head disjunction in the same formal sense as disjunctive tuple-generating dependencies; the rules are normal existential rules with possibly multiple head atoms, but not head disjunction (Wan et al., 2016). Nevertheless, it is relevant to the broader guarded existential-rule landscape because it shows how guardedness continues to control complexity when nonmonotonicity and inconsistency management are added.

In that framework, a normal rule is guarded if one positive body atom contains all universally quantified variables of the rule (Wan et al., 2016). A preferred rule repair is a subset ${\mathcal S}$26 such that ${\mathcal S}$27 has at least one stable model and every strictly preferred superset has none (Wan et al., 2016). Query answering under repair semantics requires a query to hold under all preferred rule repairs (Wan et al., 2016).

For guarded and stratified rule sets, and preferences ${\mathcal S}$28, deciding whether ${\mathcal S}$29 for a covered NBCQ ${\mathcal S}$30 is PTime-complete for data complexity and ${\mathcal S}$31ExpTime-complete for combined complexity (Wan et al., 2016). If ${\mathcal S}$32 is guarded with full negation, then for a covered NBCQ ${\mathcal S}$33, query answering is in BH for data complexity and ${\mathcal S}$34ExpTime-complete for combined complexity (Wan et al., 2016). The paper’s stated message is that rule repair semantics does not increase complexity beyond the baseline guarded query-answering problem in these cases (Wan et al., 2016).

This line of work should not be conflated with guarded disjunctive existential rules in the strict sense. The formal language differs, because the “disjunctive” aspect there is not explicit head disjunction (Wan et al., 2016). Even so, the comparison is instructive. It suggests that guardedness is a robust structural discipline across several extensions of existential rules, but the source of hardness differs: in guarded DTGDs, branching is induced directly by head disjunction (Morak, 2014), whereas in stable-model repair semantics it is induced by nonmonotonicity and repair selection (Wan et al., 2016).

8. Conceptual significance and common points of confusion

A recurring misconception is that guardedness by itself keeps reasoning low-complexity. The guarded-disjunctive results show otherwise. Guardedness preserves decidability in the presence of disjunction, but combined complexity still rises to ${\mathcal S}$35EXPTIME, and this hardness already appears in fixed sets of disjunctive inclusion dependencies (Morak, 2014). The correct conclusion is therefore more limited: guardedness is a safeguard against undecidability in this setting, not a guarantee of modest complexity.

A second point of confusion concerns products. For many rule languages, closure under direct products is a natural semantic invariant. For disjunctive existential rules, ordinary direct products fail, and the correct invariant is closure under repairable direct products (Calautti et al., 16 Aug 2025). This refinement is not technical ornamentation; it is forced by the semantics of disjunctive heads.

A third point concerns the relationship between guardedness and locality. The 2025 characterization makes clear that locality is too coarse because it does not encode the number of head disjuncts, whereas diagrammatic compatibility does (Calautti et al., 16 Aug 2025). In the guarded case, the appropriate notion is not general diagrammatic compatibility but guarded-diagrammatic compatibility, which restricts attention to guarded substructures (Calautti et al., 16 Aug 2025).

Taken together, these results place guarded disjunctive existential rules at the intersection of finite axiomatizability theory, guarded-fragment model theory, and high-complexity ontology-based reasoning. Their defining syntactic restriction is simple: one body atom must guard all universal variables, or the body is empty (Calautti et al., 16 Aug 2025). Yet the resulting theory is rich enough to require specialized semantic invariants, a branching chase semantics, exact finite-axiomatizability theorems, and nontrivial rewriting algorithms [(Calautti et al., 16 Aug 2025); (Morak, 2014)].