Existential Rules with Stratified Negation

• Existential rules with stratified negation are ontology-oriented rule systems combining existential rule heads with nonmonotonic negation in rule bodies to produce unique stable models.
• The approach leverages various chase variants, such as oblivious, restricted, and core chases, to ensure termination and manage derivations efficiently.
• Unified acyclicity and self-blocking techniques refine rule dependencies, ensuring safe propagation and accurate handling of negation in query answering.

Searching arXiv for the cited paper and closely related work on existential rules with stratified negation. Existential rules with stratified negation are a class of ontology-oriented rule systems in which existential rule heads are combined with nonmonotonic negation in rule bodies, typically under stable-model semantics. In the formulation studied by Baget, Leclère, Mugnier, and Salvat, a nonmonotonic existential rule has the form

$$R:\; B^+,\; \text{not }B^-_1,\ldots,\text{not }B^-_k \to H$$

with the safety condition $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$ for all $i$, and a knowledge base is a pair $(F,\mathcal R)$ with $F$ a finite ground atomset and $\mathcal R$ a finite set of such rules (Baget et al., 2014). The central technical problem is that entailment with existential rules is undecidable, so practical reasoning relies on syntactic conditions that ensure termination of a breadth-first forward chaining procedure, the chase, and on semantics that remain well behaved when negation is introduced (Baget et al., 2014).

1. Formal setting and stable semantics

A positive existential rule is written as

$$\forall x_1\ldots x_n\forall y_1\ldots y_m\;\bigl(B(x,y)\to \exists z_1\ldots z_k\;H(x,z)\bigr),$$

where the frontier variables are the variables shared by body and head, the $y$ variables occur only in the body, and the $z$ variables occur only in the head (Baget et al., 2014). In practice, this is abbreviated as $B(x,y)\to \exists z\,H(x,z)$.

For nonmonotonic existential rules, the positive part of a rule is $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$0. Stable semantics is defined through a parameterized $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$1-chase, where $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$2 may be the oblivious, skolem, restricted, or core chase. A derivation applies $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$3 under a homomorphism $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$4, simplifies the resulting instance under criterion $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$5, and produces a limit atomset $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$6. The derivation is sound if, for every rule application, none of its negative bodies can be homomorphically found in $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$7; it is complete if no further unblocked application is possible without changing $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$8. An atomset $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$9 is then a $i$0-stable set, or $i$1-stable model, if there exists a complete, sound $i$2-chase derivation producing $i$3 (Baget et al., 2014).

Stratified negation is obtained when the rule set can be partitioned into strata $i$4 such that whenever $i$5 has $i$6 in its body and $i$7 overlaps the head of $i$8, then $i$9 (Baget et al., 2014). In that case, the stable semantics coincides with classical stratified semantics and yields a unique model (Baget et al., 2014). The long version further states that if an NME-program is stratified then it has a unique stable model and can be evaluated by a finite sequence of positive-only chases, one per stratum (Baget et al., 2014).

2. Chase variants and termination classes

A chase derivation starts from an atomset $(F,\mathcal R)$0 and applies a rule $(F,\mathcal R)$1 under a homomorphism $(F,\mathcal R)$2 to produce

$(F,\mathcal R)$3

followed by simplification according to a criterion $(F,\mathcal R)$4 (Baget et al., 2014). Several variants are relevant.

Variant	Application policy	Locality
Oblivious chase	Apply every trigger $(F,\mathcal R)$5 exactly once; no redundancy check	Local
Frontier chase	Block applications with identical restriction to the frontier	Local
Skolem chase	Skolemize existential variables, then run the oblivious chase	Local
Restricted chase	Apply $(F,\mathcal R)$6 only if it is useful	Local
Core chase	After each application, compute the core of the instance	Non-local

All chase variants listed above except the core chase are local, in the sense that $(F,\mathcal R)$7 for $(F,\mathcal R)$8 (Baget et al., 2014). Their universal-termination classes satisfy the strict containment

$(F,\mathcal R)$9

(Baget et al., 2014).

These distinctions matter because acyclicity conditions are formulated relative to a chosen chase variant. Theorems in the positive case are then transferred to the nonmonotonic case through the stable-chase construction (Baget et al., 2014). This suggests that the semantics of negation and the operational behavior of existential rule application cannot be separated cleanly: the relevant stable-model notion depends on the chase criterion that constructs it.

3. Unified acyclicity via position graphs

The 2014 framework unifies position-based and rule-dependency approaches through three increasingly refined graphs, denoted $F$0, $F$1, and $F$2 (Baget et al., 2014). For a rule $F$3, the basic position graph $F$4 has as nodes all positions $F$5 occurring in $F$6. There is an edge

$F$7

iff $F$8 is a frontier position in $F$9 and either $\mathcal R$0 or $\mathcal R$1 is existential (Baget et al., 2014).

From these per-rule graphs, the framework builds:

• $\mathcal R$2, augmented with transition edges between head-positions and body-positions of the same predicate symbol and same argument-slot, unconditionally.
• $\mathcal R$3, where such transition edges are added only when the target rule depends on the source rule in the graph of rule dependencies $\mathcal R$4.
• $\mathcal R$5, which refines $\mathcal R$6 further by requiring an explicit compatible piece-unifier with an agglomerated head along a dependency path (Baget et al., 2014).

Acyclicity properties are then expressed through a marking function $\mathcal R$7 associated with a classical notion $\mathcal R$8. A rule set satisfies $\mathcal R$9 if, in $$\forall x_1\ldots x_n\forall y_1\ldots y_m\;\bigl(B(x,y)\to \exists z_1\ldots z_k\;H(x,z)\bigr),$$0, there is no marked cycle passing through an existential position (Baget et al., 2014). Proposition 3.6 states that for each classical notion $$\forall x_1\ldots x_n\forall y_1\ldots y_m\;\bigl(B(x,y)\to \exists z_1\ldots z_k\;H(x,z)\bigr),$$1 there is a marking $$\forall x_1\ldots x_n\forall y_1\ldots y_m\;\bigl(B(x,y)\to \exists z_1\ldots z_k\;H(x,z)\bigr),$$2 such that

$$\forall x_1\ldots x_n\forall y_1\ldots y_m\;\bigl(B(x,y)\to \exists z_1\ldots z_k\;H(x,z)\bigr),$$3

and

$$\forall x_1\ldots x_n\forall y_1\ldots y_m\;\bigl(B(x,y)\to \exists z_1\ldots z_k\;H(x,z)\bigr),$$4

(Baget et al., 2014).

The same line of work states that $$\forall x_1\ldots x_n\forall y_1\ldots y_m\;\bigl(B(x,y)\to \exists z_1\ldots z_k\;H(x,z)\bigr),$$5 strictly extends $$\forall x_1\ldots x_n\forall y_1\ldots y_m\;\bigl(B(x,y)\to \exists z_1\ldots z_k\;H(x,z)\bigr),$$6, and $$\forall x_1\ldots x_n\forall y_1\ldots y_m\;\bigl(B(x,y)\to \exists z_1\ldots z_k\;H(x,z)\bigr),$$7 further strictly extends $$\forall x_1\ldots x_n\forall y_1\ldots y_m\;\bigl(B(x,y)\to \exists z_1\ldots z_k\;H(x,z)\bigr),$$8 (Baget et al., 2014). Position-based notions satisfy the inclusions

$$\forall x_1\ldots x_n\forall y_1\ldots y_m\;\bigl(B(x,y)\to \exists z_1\ldots z_k\;H(x,z)\bigr),$$9

and all embed as $y$0; the rule-dependency notion a-grd embeds as $y$1 when $y$2, and more generally as $y$3 for other $y$4 (Baget et al., 2014).

4. Stratified negation, self-blocking, and refined acyclicity

Negation changes the interpretation of cycles because some syntactic cycles are operationally harmless. The first refinement is the notion of a self-blocking rule: an NME-rule

$y$5

is self-blocking if for some negative body $y$6 one already has

$y$7

Such rules never fire soundly and can be dropped (Baget et al., 2014).

The same idea is lifted to dependencies. A unifier of $y$8 with $y$9 is self-blocking if composing $z$0 and $z$1 along that unifier yields a self-blocking rule. In $z$2 and $z$3, edges corresponding only to self-blocking unifiers are omitted (Baget et al., 2014). More generally, even when $z$4 contains a cycle through an existential position, the cycle may still be harmless if every compatible sequence of unifiers along that cycle is itself self-blocking. The refined notion $z$5 declares a rule set acyclic when every marked cycle for an existential position in $z$6 is self-blocking in that sense (Baget et al., 2014).

This refinement yields a negation-sensitive termination criterion. Proposition 5.3 states that if, in $z$7, every marked cycle for an existential position is self-blocking, then the stable-chase, skolem or core, on $z$8 always halts (Baget et al., 2014). The long version reformulates the same idea by defining a self-blocking unifier through the merged rule $z$9\Join$B(x,y)\to \exists z\,H(x,z)$0, whose negative body intersects its positive part, so that it can never fire, and defines $B(x,y)\to \exists z\,H(x,z)$1 by ignoring marked cycles all of whose induced piece-unifier sequences are self-blocking (Baget et al., 2014).

A plausible implication is that stratified negation is not merely an ordering discipline over predicates; in existential-rule systems it also acts as a filter on which propagation cycles are semantically realizable during chase construction.

5. Termination theorems, complexity, and a worked stratified example

The main positive-case result is Theorem 3.12: if $B(x,y)\to \exists z\,H(x,z)$2 is any acyclicity property that guarantees halting of some chase $B(x,y)\to \exists z\,H(x,z)$3 on all positive rule sets in $B(x,y)\to \exists z\,H(x,z)$4, then any $B(x,y)\to \exists z\,H(x,z)$5 satisfying $B(x,y)\to \exists z\,H(x,z)$6 is universally $B(x,y)\to \exists z\,H(x,z)$7-terminating (Baget et al., 2014). The proof sketch associates any infinite $B(x,y)\to \exists z\,H(x,z)$8-derivation with an infinite frontier-propagation path in $B(x,y)\to \exists z\,H(x,z)$9; such a path must revisit an existential position along a marked cycle, contradicting $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$00 (Baget et al., 2014).

The associated complexity result is Theorem 3.14: if checking $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$01 on $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$02 is in class $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$03, then checking $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$04 or $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$05, and even $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$06, remains in $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$07; in particular there is no increase in worst-case complexity (Baget et al., 2014). The long version makes this explicit for the standard notions: checking WA, FD, AR, JA, and SWA on $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$08 is in PTIME; checking aGRD is co-NP-complete; and for every $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$09, checking $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$10 or $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$11 is co-NP-complete (Baget et al., 2014).

A small stratified example illustrates the mechanism. Let $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$12 and $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$13, with

$\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$14

and

$\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$15

$\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$16 is in a strictly higher stratum than $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$17 because $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$18-atoms occur negated in $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$19 (Baget et al., 2014). The program satisfies $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$20, since there is no marked cycle through an existential position under wa-marking (Baget et al., 2014). The chase starts with $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$21; $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$22 fires under $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$23 and produces $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$24; since $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$25 is not yet in the model, the step is sound; then in stratum 2, $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$26 fires and yields $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$27; this blocks any further application of $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$28 on $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$29 (Baget et al., 2014). The unique stable model is

$\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$30

(Baget et al., 2014).

6. Later developments: negative queries, rule repairs, and open problems

Later work extended the discussion in two directions. First, for query answering with negation, the 2021 study observes that stratification, as used for Datalog, is not enough for existential rules, since existential rules still admit multiple universal models that can differ on negative queries (Ellmauthaler et al., 2021). That work therefore proposes universal core models as a basis for a meaningful non-monotonic semantics for queries with negation, and identifies affection-safe, core-safe, and effectively core-safe classes of queries that can be evaluated over restricted or oblivious chase results instead of explicit cores (Ellmauthaler et al., 2021). A plausible implication is that rule-level stratified negation and query-level negation over universal models raise distinct semantic issues, even when both are formulated over the same existential-rule base.

Second, for inconsistent ontologies under stable-model semantics, rule repair semantics selects preferred subsets of rules that still admit stable models. For R-acyclic existential rules with R-stratified or guarded existential rules with stratified negations, both the data complexity and combined complexity of query answering under rule repair semantics remain the same as under the conventional query answering semantics (Wan et al., 2016). The same work states that, for R-acyclic plus R-stratified negation and for the preference relations $\mathrm{vars}(B^-_i)\subseteq \mathrm{vars}(B^+)$31, data complexity is PTime-complete and combined complexity is 2ExpTime-complete (Wan et al., 2016).

The original 2014 framework also identifies several open problems: tight complexity classification of query answering under the new acyclic classes; logical foundations relating core-stable sets to classical first-order or circumscription-based semantics for general NME-rules; and possible further refinements combining unifier-blocking with argument-restrictions or chasing under equality (Baget et al., 2014). These problems remain structurally central because the subject sits at the intersection of ontological expressivity, chase termination, and nonmonotonic semantics.