---
title: Existential Rules with Stratified Negation
url: https://www.emergentmind.com/topics/existential-rules-with-stratified-negation
type: topic
---

# Existential Rules with Stratified Negation

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 [1405.1071]. 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 [1405.1071].

## 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 [1407.6885]. 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 $\mathrm{pos}(R): B^+\to H$. Stable semantics is defined through a parameterized $C$-chase, where $C$ may be the oblivious, skolem, restricted, or core chase. A derivation applies $\mathrm{pos}(R_i)$ under a homomorphism $\pi_i$, simplifies the resulting instance under criterion $C$, and produces a limit atomset $A$. The derivation is sound if, for every rule application, none of its negative bodies can be homomorphically found in $A$; it is complete if no further unblocked application is possible without changing $A$. An atomset $A$ is then a $C$-stable set, or $C$-stable model, if there exists a complete, sound $C$-chase derivation producing $A$ [1405.1071].

Stratified negation is obtained when the rule set can be partitioned into strata $\mathcal R_1,\ldots,\mathcal R_m$ such that whenever $R\in\mathcal R_p$ has $\text{not }B$ in its body and $B$ overlaps the head of $R'\in\mathcal R_q$, then $q<p$ [1405.1071]. In that case, the stable semantics coincides with classical stratified semantics and yields a unique model [1405.1071]. 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 [1407.6885].

## 2. Chase variants and termination classes

A chase derivation starts from an atomset $F$ and applies a rule $R$ under a homomorphism $\pi$ to produce
$$
\alpha(F,R,\pi)=F\cup \pi(\mathrm{safe}(H)),
$$
followed by simplification according to a criterion $C$ [1405.1071]. Several variants are relevant.

| Variant | Application policy | Locality |
|---|---|---|
| Oblivious chase | Apply every trigger $(R,\pi)$ 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 $(R,\pi)$ 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 $F'_i\subseteq F'_j$ for $i<j$ [1405.1071]. Their universal-termination classes satisfy the strict containment
$$
\text{oblivious-finite}\subset \text{skolem-finite}=\text{frontier-finite}\subset \text{restricted-finite}\subset \text{core-finite}
$$
[1405.1071].

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 [1405.1071]. 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 $PG^F$, $PG^D$, and $PG^U$ [1405.1071]. For a rule $R:B\to H$, the basic position graph $PG(R)$ has as nodes all positions $a\cdot i$ occurring in $B\cup H$. There is an edge
$$
(b\cdot i)\to (h\cdot j)
$$
iff $b\cdot i$ is a frontier position in $B$ and either $\mathrm{term}(b\cdot i)=\mathrm{term}(h\cdot j)$ or $h\cdot j$ is existential [1405.1071].

From these per-rule graphs, the framework builds:

- $PG^F(\mathcal R)=\bigcup_{R\in\mathcal R} PG(R)$, augmented with transition edges between head-positions and body-positions of the same predicate symbol and same argument-slot, unconditionally.
- $PG^D$, where such transition edges are added only when the target rule depends on the source rule in the graph of rule dependencies $GRD(\mathcal R)$.
- $PG^U$, which refines $PG^D$ further by requiring an explicit compatible piece-unifier with an agglomerated head along a dependency path [1405.1071].

Acyclicity properties are then expressed through a marking function $M_Y$ associated with a classical notion $Y$. A rule set satisfies $Y^X$ if, in $PG^X(\mathcal R)$, there is no marked cycle passing through an existential position [1405.1071]. Proposition 3.6 states that for each classical notion $Y\in\{\mathrm{wa},\mathrm{fd},\mathrm{ar},\mathrm{ja},\mathrm{swa}\}$ there is a marking $M_Y$ such that
$$
\mathcal R\in Y \iff \mathcal R \text{ satisfies } Y^F,
$$
and
$$
\mathcal R\in Y^{<}\iff \mathcal R \text{ satisfies } Y^D
$$
[1405.1071].

The same line of work states that $PG^U$ strictly extends $PG^D$, and $PG^{U^+}$ further strictly extends $PG^U$ [1405.1071]. Position-based notions satisfy the inclusions
$$
\mathrm{wa}\subset \mathrm{fd}\subset \mathrm{ar}\subset \mathrm{ja}\subset \mathrm{swa}
$$
and all embed as $Y^F$; the rule-dependency notion a-grd embeds as $wa^D$ when $Y=\{wa\}$, and more generally as $Y^D$ for other $Y$ [1405.1071].

## 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
$$
R:\; B^+,\; \text{not }B^-_1,\ldots,\text{not }B^-_k\to H
$$
is self-blocking if for some negative body $B^-_i$ one already has
$$
B^-_i\subseteq B^+\cup H.
$$
Such rules never fire soundly and can be dropped [1405.1071].

The same idea is lifted to dependencies. A unifier of $\mathrm{head}(R_i)$ with $\mathrm{body}(R_j)$ is self-blocking if composing $R_i$ and $R_j$ along that unifier yields a self-blocking rule. In $PG^D$ and $PG^U$, edges corresponding only to self-blocking unifiers are omitted [1405.1071]. More generally, even when $PG^U$ 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 $Y^{U^+}$ declares a rule set acyclic when every marked cycle for an existential position in $PG^U(\mathcal R)$ is self-blocking in that sense [1405.1071].

This refinement yields a negation-sensitive termination criterion. Proposition 5.3 states that if, in $PG^U(\mathcal R)$, every marked cycle for an existential position is self-blocking, then the stable-chase, skolem or core, on $\mathcal R$ always halts [1405.1071]. The long version reformulates the same idea by defining a self-blocking unifier through the merged rule $R_i\mathbin{\raisebox{0.2ex}{\scalebox{0.8}{$\Join$}}}_{\mu}R_j$, whose negative body intersects its positive part, so that it can never fire, and defines $Y^{U^+}$ by ignoring marked cycles all of whose induced piece-unifier sequences are self-blocking [1407.6885].

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 $Y$ is any acyclicity property that guarantees halting of some chase $C$ on all positive rule sets in $Y$, then any $\mathcal R$ satisfying $Y^U$ is universally $C$-terminating [1405.1071]. The proof sketch associates any infinite $C$-derivation with an infinite frontier-propagation path in $PG^U(\mathcal R)$; such a path must revisit an existential position along a marked cycle, contradicting $Y^U$ [1405.1071].

The associated complexity result is Theorem 3.14: if checking $Y$ on $PG^F(\mathcal R)$ is in class $C$, then checking $Y^D$ or $Y^U$, and even $Y^{U^+}$, remains in $C$; in particular there is no increase in worst-case complexity [1405.1071]. The long version makes this explicit for the standard notions: checking WA, FD, AR, JA, and SWA on $PG^F$ is in PTIME; checking aGRD is co-NP-complete; and for every $Y\in\{\mathrm{WA},\mathrm{FD},\mathrm{AR},\mathrm{JA},\mathrm{SWA}\}$, checking $Y^D$ or $Y^U$ is co-NP-complete [1407.6885].

A small stratified example illustrates the mechanism. Let $\mathcal R=\{R_1,R_2\}$ and $F=\{a(0)\}$, with
$$
R_1:\; a(x)\to b(x,y),\; \text{not }c(y)
$$
and
$$
R_2:\; b(u,v)\to c(f(u)).
$$
$R_2$ is in a strictly higher stratum than $R_1$ because $c$-atoms occur negated in $R_1$ [1405.1071]. The program satisfies $wa^U$, since there is no marked cycle through an existential position under wa-marking [1405.1071]. The chase starts with $a(0)$; $R_1$ fires under $x\mapsto 0$ and produces $b(0,y_0)$; since $c(y_0)$ is not yet in the model, the step is sound; then in stratum 2, $R_2$ fires and yields $c(f(0))$; this blocks any further application of $R_1$ on $y_0$ [1405.1071]. The unique stable model is
$$
\{a(0),\,b(0,y_0),\,c(f(0))\}
$$
[1405.1071].

## 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 [2112.07376]. 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 [2112.07376]. 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 [1602.05699]. The same work states that, for R-acyclic plus R-stratified negation and for the preference relations $\subseteq,\le,\subseteq_P,\le_P,\le_w$, data complexity is PTime-complete and combined complexity is 2ExpTime-complete [1602.05699].

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 [1405.1071]. These problems remain structurally central because the subject sits at the intersection of ontological expressivity, chase termination, and nonmonotonic semantics.

Source: https://www.emergentmind.com/topics/existential-rules-with-stratified-negation