Computing Stable Models for Nonmonotonic Existential Rules

1 Introduction

Logic-based knowledge representation (KR) languages are widely used to model complex, structured information, e.g., in biology [Gkoutos et al., 2012] and chemistry [Hastings et al., 2012]. Structured knowledge models, such as the ChEBI database and ontology of chemical compounds [de Matos et al., 2010], serve as shared reference terminologies. Reasoning supports a wide range of tasks including quality assur-

plement statistical and machine learning approaches, e.g., in classifying chemical structures [Ferreira and Couto, 2010].

Many ontologies, including ChEBI, are based on description logics (DLs); DLs are, however, severely limited in their ability to model structures that are not tree-shaped. This explains, e.g., why ChEBI does not model molecular structures in its ontology, thus excluding its main content from logical reasoning. Numerous extension of DLs, such as description graphs [Motik et al., 2009], provide carefully restricted kinds of rule-based or graph-based modelling, but remain largely unrealised in tools and applications. Moreover, a form of closed-world assumption is often needed to reason about the absence of structural features, e.g., to conclude that a molecule is inorganic if it does not contain carbon. This can be naturally modelled using a nonmonotonic DL, but such DLs currently lack tool support [Motik and Rosati, 2010].

This motivates the use of (nonmonotonic) rule languages for modelling ontologies. Existential rules-function-free Horn rules with existential quantifiers in rule heads-have been proposed as an ontology and data integration language [Calì et al., 2010; Baget et al., 2011a], and can be viewed as a restricted kind of logic programs with function symbols. Recent works have considered nonmonotonic rule-based ontology languages using stratified negation [Calì et al., 2009; Magka et al., 2012], stable model semantics [Eiter et al., 2012], and well-founded semantics [Gottlob et al., 2012]. If we additionally remove the stratification requirement, then the resulting language allows for the accurate modelling of complex finite structures such as those found in ChEBI.

Unfortunately, reasoning in these formalisms is computationally challenging. If negation is stratified, then all of these semantics agree, and programs have uniquely determined sta-

too restrictive for many applications. Moreover, even without negation, satisfiability, fact entailment, query answering, and the existence of finite models are all undecidable; and, while many non-stratified programs also have unique stable models, this property, too, is undecidable in general. As most ontologies are concerned with finite, uniquely determined structures, these problems raise serious doubts about the use of such formalisms in ontological modelling.

We address this issue by presenting new conditions that are computationally feasible to check, and that identify a large class of programs having finite and unique stable models. These conditions are based on an analysis of whether one rule relies on another, in the sense that it might either be 'triggered' or 'inhibited' by the other rule's application. These relationships allow us to define R-acyclicity and Rstratification. Specifically, our contributions are as follows:

• We define R-acyclic and R-stratified logic programs, and show that recognising such programs is coNP-complete.
• We show that R-acyclic programs have finite stable models, and that reasoning is coN2EXPTIME-complete (NPcomplete for data complexity).
• We show that R-stratified programs have unique stable models, so that reasoning becomes deterministic, and that if programs are also R-acyclic, reasoning becomes 2EXPTIME-complete (P-complete for data complexity).
• We extend reliances to exploit constraints, and show that this strictly generalises our earlier criteria. Reasoning complexities carry over, but deciding R-acyclicity and R-stratification under constraints is complete for ΠP2 .

• We conduct a case study with ChEBI, which demonstrates that our conditions do not preclude suitable modelling, that R-stratification can be exploited to allow the DLV reasoner [Leone et al., 2006] to scale to the large number of rules in our experiments, and that DLV can then be used to discover missing relationships in ChEBI.

We first introduce basic notions (Section 2) and discuss the use of nonmonotonic existential rules in ontological modelling (Section 3). Next, we study positive reliances and Racyclicity (Section 4), negative reliances and R-stratification (Section 5), and the extension of these notions with constraints (Section 6). We then present the ChEBI case study (Section 7), discuss related works (Section 8), and conclude (Section 9). A companion report contains details and proofs that were omitted for reasons of space [Magka et al., 2013].

2 Preliminaries

We consider a standard first-order language. We use the letters a, b for constants, f, g for functions, x, y, z, for variables, and t for terms. Lists of terms 〈t1, . . . , tn〉 are abbreviated as t, similarly for lists of variables x. We treat lists as sets when order is irrelevant. A special nullary predicate symbol ⊥ is used to denote falsity. We use Pred(ϵ), Var(ϵ), and Const(ϵ) to denote the predicates, variables, and constants, respectively, that occur in an expression ϵ. Atoms, i.e., formulae without operators, are written α, β, γ. When used like a formula, sets of atoms always denote the conjunction of their members. Nonmonotonic negation is denoted not. For a set A of atoms, we define not A := {not α | α ∈ A}. A nonmonotonic existential rule (or simply rule) is of the form

where the positive body B+, negative body B-, and head H are sets (or conjunctions) of atoms without function symbols, such that Var(B+) = x ∪ z, Var(B-) ⊆ x ∪ z, and Var(H) ⊆ x∪y. We abbreviate r as (B+,B-,H). When writing rules as in (1), universal quantifiers are usually omitted. Sets of rules are called (logic) programs.

The skolemisation sk(r) of a rule r as in (1) is obtained by replacing each variable y ∈ y in H by a skolem term fy(x), where fy is a fresh skolem function symbol of arity |x|. Given a program P, we set sk(P) := {sk(r) | r ∈ P}. Assuming a fixed choice of skolem functions, sk is a bijection between rules and their skolemisations, which allows us to use the term rule liberally without risk of confusion. Our results refer to rules (or their skolemisations), and do not generally hold for arbitrary logic programming rules with function symbols.

A term or formula is ground if it contains no variables. Ground atoms are called facts. The Herbrand universe HU(P) of a program P is the set of all ground terms formed with constants and function symbols from sk(P) (using an auxiliary constant if Const(sk(P)) = ∅). The grounding ground(P) of P is the set of all rules that can be obtained from rules in sk(P) by uniformly replacing variables with terms fromHU(P).

An (Herbrand) interpretation M is a set of facts with ⊥ /∈ M. Satisfaction is defined as usual: M |= B+, not Bholds if B+ ⊆M and B- ∩M = ∅; M |= (B+, B-, H)

if M̸|= B+, not B- or M |= H; and M |= P if M |= r for all r ∈ P. The Gelfond-Lifschitz reduct of P w.r.t. M is GL(P,M) := {(B+, ∅,H) | (B+,B-,H) ∈ ground(P) and B- ∩M = ∅}. M is a stable model of P, written M |=SM P, if M |= GL(P,M) and there is no smaller model M′ ⊊ M with M′ |= GL(P,M). We consider cautious entailment: for a program P and a fact α, P |= α if α ∈ M for all stable models M of P. Consequences of programs can be computed with the TP operator:

Definition 1. Consider a program P and set of facts F. For a rule r ∈ P with sk(r) = (B+, B-, H), define

Given a program P , a sequence of disjoint programs P = P1, . . . , Pn is a stratification of P if P = ⋃ni=1 Pi and, for all programs Pi, Pj ∈ P, rules (B+ 1 ,B1 ,H1) ∈ Pi and (B+ 2 ,B2 ,H2) ∈ Pj, and every predicate R ∈ Pred(H1), we have: (i) if R ∈ Pred(B+ 2 ) then i ≤ j, and (ii) if R ∈ Pred(B2 ) then i < j. The elements of P are called strata. P is stratified if it has a stratification. The TP operator can be used to characterise stable models; for stratified programs, we even obtain a deterministic computation procedure [Apt and Bol, 1994].

Fact 1. Given a program P, a set of facts F, and a stable model M |=SM P ∪ F , we have M = T ∞ GL(P,M) (F ).

3 Modelling with Nonmonotonic Rules

Rule-based formalisms are well suited for modelling relational structures, irrespective of whether these structures are tree-shaped or cyclic. We consider practical examples related to the modelling of chemical compounds in ChEBI. The structure of molecules can be readily represented as a logical structure. For example, the formula MH2O(x, y, z) := o(x) ∧ bond(x, y) ∧ bond(x, z) ∧ h(y) ∧ h(z) could represent a water molecule (using unidirectional bonds for simplicity). We model molecules as members of a unary predicate mol, related to their constituting atoms by the predicate hA (has atom). The following rule infers the structure of the six atoms of methanol (CH3OH), described by the formula MCH3OH(y):

Molecules can also be classified by their structure, e.g., to identify molecules that contain oxygen, or organic hydroxy molecules (those with a substructure C-O-H):

It is not hard to express syntactic identity with a predicate =, predefined in most rule engines; see [Magka et al., 2013] for details. Using not we can express syntactic inequality and define, e.g., molecules with exactly one carbon atom:

The fact methanol(a) and the rules (2)-(6) have a unique stable model (using skolem functions f1, . . . , f6 for (2)):

We can thus conclude, e.g., that methanol is an organic hydroxy molecule. To obtain such inferences for organic hydroxy molecules in general, we can use another rule:

The fact orgHydroxy(b) and the rules (3)-(7) have a unique stable model (using skolem functions g1, . . . , g3 for (7)):

Hence, organic hydroxy molecules are structures with oxygen, as expected. However, if we consider all of the above rules and facts together, then rather thanM1 ∪M2 we obtain M1 ∪ M2 ∪ {hA(a, gi(a))3i=1 , MCOH(g1(a), g2(a), g3(a)), multiC(a)}\{oneC(a)} as the unique stable model, since rule (7) is applicable to orgHydroxy(a). Thus, the stable model is no longer a faithful representation of the molecule a, which is wrongly classified as a multi-carbon molecule.

Nonmonotonic negation can be used to overcome this problem. We replace rules (4) and (7) by the following, where we abbreviate orgHydroxy by oH:

The predicates r ('recognised') and n ('new') ensure that only one of these rules is applicable to a given structure. The above facts with rules (2), (3), (5), (6), (8), and (9) have the unique stable model M1 ∪M2 ∪ {r(a), n(g1(b)), n(g2(b)), n(g3(b))}, as desired. However, the resulting set of rules is not stratified, which causes various problems. First, we cannot be sure that the stable model will be unique for other sets of facts. Second, rule engines may need to apply more complex algorithms to find the stable model. Our experiments in Section 7 suggest that this may cause performance issues that prevent rule engines from computing entailments at all. The goal of this work is to overcome these issues.

4 Positive Reliances and R-Acyclicity

As recalled in Fact 1, every stable model of a logic program can be obtained from a (possibly infinite) sequence of consecutive rule applications. Insights about the semantics of a program can thus be gained by analysing, for all pairs of rules r1 and r2, whether an application of r1 can potentially enable a later application of r2. In this section, we formalise this idea of positive reliance between rules and define R-acyclic programs, which have stable models of bounded size.

Definition 2 (Positive Reliance). Let r1 and r2 be rules such that sk(r1) = (B+ 1 ,B1 ,H1) and sk(r2) = (B+ 2 ,B2 ,H2); w.l.o.g. assume that Var(r1) ∩Var(r2) = ∅. Rule r2 positively relies on r1 (written r1 + -→ r2) if there exists a set of facts F that contains no skolem terms and a substitution θ such that:

Thus, r1 + -→ r2 holds if there is a situation (defined by F) where r1 is applicable (P1)/(P2), r2 is not applicable (P5), and applying r1 allows r2 to derive something new (P3)/(P4)/(P6).

Example 1. Consider rule r(4) of (4), and rule r′ (7) obtained from (7) by replacing variable x with x′ . We find that r(4) + -→ r′ (7) since F := {MCOH(b)} ∪ {hA(a, bi)}3i=1 and θ := {x ↦→ a, y ↦→ b, x′ ↦→ a} satisfy (P1)-(P6).

In contrast, r′ (7) ̸+ -→ r(4). Intuitively, r(4) can only derive facts that are already necessary to apply r′ (7) in the first place, thus violating (P6). More formally, suppose that r′ (7) + -→ r(4) could be shown using F ′ and θ′ . By (P1) and (P6), θ′ (x) ̸= θ′ (x′ ). Thus, by (P3), hA(x, yi)θ′ ∈ F ′ for all i ∈ {1, 2, 3}. Since F ′ must not contain skolem terms, θ′ (yi) ̸= gi(θ′ (x′ )), so MCOH(y)θ′ ⊆ F ′ , again by (P3). Thus (P5) would be violated.

Various previous works consider similar notions. The activation relation by Greco et al. [2012] is most similar to Definition 2, but allows F to contain function terms to accommodate arbitrary disjunctive logic programs with functions. Our stronger restriction is needed to show r′ (7) ̸+ -→ r(4) in Example 1. This illustrates how we can take advantage of the specific structure of existential rules to discard certain potential interactions. Other similar notions are the ≺ relation by Deutsch et al. [2008] and the rule dependency by Baget et al. [2011a], neither of which cover negation. Baget et al. omit condition (P6), needed to show r′ (7) ̸+ -→ r(4) in Example 1.

If a finite program has an infinite stable model, some rule with an existential quantifier must be applicable an infinite number of times. This, however, requires that there is a cycle in rule reliances, motivating the following definition.

Definition 3 (R-Acyclic). A program P is R-acyclic if there is no cycle of positive reliances r1 + -→ . . . + -→ rn + -→ r1 that involves a rule with an existential quantifier.

Example 2. The complete list of positive reliances for the rules r(2), . . . , r(7) is r(2) + -→ r(3), r(2) + -→ r(4), r(2) + -→ r(5), r(2) + -→ r(6), r(4) + -→ r(7), r(7) + -→ r(3), r(7) + -→ r(5), and r(7) + -→ r(6). Thus the program is R-acyclic. To model =, we assume that yi = yi is derived for all existential variables yi.

We prove that checking positive reliance for two rules is NP-complete. Similar results are shown by Deutsch et al. [2008] and by Baget et al. [2011b] for rules without negation. The complexity refers to the size of the two involved rules rather than to the size of the whole program: in practice, positive reliances can be checked efficiently by checking the applicability of one of the rules to a linear number of facts.

Theorem 1. Given rules r1 and r2, the problem of deciding whether r1 + -→ r2 is NP-complete. Checking whether a program P is R-acyclic is coNP-complete.

The main result of this section shows that entailment under stable model semantics is decidable for R-acyclic programs. Hardness for coN2EXPTIME can be shown by reducing the word problem of 2EXPTIME-bounded non-deterministic Turing machines to cautious entailment, adapting constructions by Calì et al. [2012] and Krötzsch and Rudolph [2011].

Theorem 2. Let P be an R-acyclic program and let F ∪ {α} be a set of facts. Every stable model of P ∪F has size doubly exponential in the size of P and polynomial in the size of F. Deciding P ∪ F |= α is coN2EXPTIME-complete w.r.t. program complexity and coNP-complete w.r.t. data complexity.

5 Negative Reliances and R-Stratification

While positive reliances allow us to estimate if one rule can 'trigger' another rule, the use of nonmonotonic negation may also give rise to the opposite interaction where one rule 'inhibits' another. In this section, we formalise this by defining negative reliances between rules. This suggests a new kind of stratification, which generalises the classical notion but can still be decided efficiently.

Definition 4 (Negative Reliance). Let r1 and r2 be rules such that sk(r1) = (B+ 1 ,B1 ,H1) and sk(r2) = (B+ 2 ,B2 ,H2); w.l.o.g. assume that Var(r1) ∩ Var(r2) = ∅. Rule r2 negatively relies on r1 (written r1 - -→ r2) if there exists a set of facts F that contains no skolem terms

and a substitution θ such that: B1 θ ∩F = ∅ (N2)

Example 3. Consider rule r(8) of (8), and rule r′ (9) obtained from (9) by variable x with x′ . We can show r(8) - -→ r′ (9) using F := {oH(a),MCOH(b)} ∪ {hA(a, bi)}3i=1 and θ := {x ↦→ a, y ↦→ b, x′ ↦→ a}. Conversely, r′ (9) ̸- -→ r(8) follows from a similar argument as in Example 1, since F is not allowed to contain skolem terms.

The following definition is inspired by the classical notion of stratification in logic programming.

Definition 5 (R-Stratification). A sequence of disjoint programs P = P1, . . . , Pn is an R-stratification of a program P if P = ⋃ni=1 Pi and, for every two programs Pi, Pj ∈ P and rules r1 ∈ Pi and r2 ∈ Pj , we have:

Example 4. For P consisting of rules r(2), r(3), r(5), r(6), r(8), and r(9) we obtain the reliances r(2) + -→ r(8) - -→ r(9) + -→ r(3) , r(2) + -→ r(3), r(2) + -→ r(6), r(2) + -→ r(5) - -→ r(6), r(9) + -→ r(5), and r(9) + -→ r(6). An R-stratification of P is therefore given by P1 := {r(2), r(8)}, P2 := {r(3), r(5), r(9)}, and P3 := {r(6)}. In contrast, P is not stratified due to rules r(8) and r(9).

Together with the previous example, the next result shows that R-stratification properly generalises stratification.

The graph structure that is induced by reliances, defined next, can be used to decide R-stratification in practice, as shown in Proposition 2 below.

Definition 6 (Graph of Reliances). For a program P, the graph of reliances GoR(P) is a directed graph that has the rules of P as its vertices and two sets of edges: positive edges that correspond to the positive reliances of P and negative edges that correspond to the negative reliances of P.

Proposition 2. P is R-stratified iff its graph of reliances GoR(P ) contains no directed cycle with a negative edge.

From the previous result it is clear that, given the graph of reliances, R-stratification can be decided in polynomial time. The overall complexity is therefore dominated by the complexity of checking individual reliances-in this sense, it is polynomial in the total number of rules, and coNP-complete only in the maximal size of a rule. Moreover, in contrast to the NP-completeness of checking positive reliances (Theorem 1), negative reliances can be detected in polynomial time.

Theorem 3. Given rules r1 and r2, it can be decided in polynomial time whether r1 - -→ r2. Checking whether a program P is R-stratified is coNP-complete.

It remains to show that R-stratified programs have at most one stable model, and that this model can always be obtained by repeated application of rules according to their stratification. This leads to a semi-decision procedure for entailment. If the program is also R-acyclic, we obtain a decision procedure and tight complexity bounds.

Note that Definition 4 does not include a condition that corresponds to (P6) fromDefinition 2. Indeed, as the next example shows, such a condition would not lead to a notion of R-stratification that ensures unique stable models.

Example 5. Given the rules r1 : not p → q and r2 : q → p, we find that r1 + -→ r2 and r2 - -→ r1, so that the program is not R-stratified. Indeed, it has no stable models for the empty set of facts. Yet, if we required that H2θ ̸⊆ F in Definition 4, then r2 - -→ r1 would not hold, and the program would be Rstratified. Intuitively speaking, negative reliances do not just consider the case where r2 could derive something new, but also the case where r2 has already been used in a derivation that is no longer justified after applying r1.

We now define a computation scheme that can be used to obtain the unique stable model of R-stratified programs, or to derive a contradiction ⊥ if no such model exists.

Definition 7. For a set of facts F and a program P with Rstratification P = P1, . . . , Pn, define S0P(F) := F and

For the remainder of this section, let P denote an Rstratified program with R-stratification P = P1, . . . , Pn, let F denote a set of facts, and define SiP := SiP(F).

We first show that SnP is a (not necessarily unique) stable model of F ∪ P, provided that ⊥ /∈ SnP. The next two lemmas are key ingredients to this proof. Intuitively speaking, Lemma 1 asserts that, if the body of a rule r ∈ Pi is satisfied at some point while computing SiP, then it will remain satisfied in all later stages of the computation. The crucial claim is that the negative part of the rule will not be derived at any later stage. The proof of Lemma 1 relies on the definition of - -→.

Lemma 1. Consider numbers 1 ≤ i ≤ j ≤ k ≤ n and ℓ ≥ 0, a rule r ∈ Pi with skolemisation sk(r) = (B+, B-, H), and a substitution θ. Then T ℓPj (Sj-1 P ) |= B+θ, not B-θ implies SkP |= B+θ, not B-θ.

Lemma 2 complements the previous result. Intuitively speaking, it states that a rule r ∈ Pi, which is clearly satisfied after computing SiP, will remain satisfied in all later stages of the computation. The key part of this claim concerns the case that r is satisfied because its positive body is not satisfied. In this case, the positive body will never become satisfied later on, unless the head of the rule becomes satisfied as well. This argument hinges upon the definition of + -→.

Lemma 2. Consider numbers 1 ≤ i < j ≤ k ≤ n, a rule r ∈ Pi, and a substitution θ. Then SjP |= sk(r)θ implies SkP |= sk(r)θ.

Using Lemmas 1 and 2, we can show the following result.

The main result of this section is that stable models of Rstratified programs are unique. Its proof is obtained by first showing that M |=SM P ∪ F implies SnP = M, which in turn is established by showing inductively that, for all k ∈ {0, . . . , n}, SkP = T ∞ GL(⋃ki=1 Pi,M) (F) [Magka et al., 2013].

Theorem 4. If ⊥ /∈ SnP, then SnP is the unique stable model of F ∪ P . Otherwise F ∪ P does not have a stable model.

We can further improve the complexity results of Theorem 2 for programs that are both R-acyclic and R-stratified. The Turing machine reduction used to show Theorem 2 can directly be used to show hardness: the constructed program is R-stratified precisely if the Turing machine is deterministic.

Theorem 5. Let P be an R-acyclic R-stratified program, let F be a set of facts, and let α be a fact. Deciding P ∪ F |= α is 2EXPTIME-complete w.r.t. program complexity and Pcomplete w.r.t. data complexity.

6 Reliances under Constraints

To widen the classes of logic programs with unique stable models, it has been proposed to study stratification for a particular set of facts [Bidoit and Froidevaux, 1991]. Indeed, it might be that a program that does not have a unique stable model for all sets of facts still has a unique stable model for all sets of facts that arise in the context of a given application. On the other hand, notions that depend on a particular set of facts do not easily capture a wider class of relevant sets of facts, making it hard to develop logic programs that are robust to changing inputs.

In this section, we therefore propose a generalisation of Racyclicity and R-stratification that considers constraints, that is, rules of the form B+ → ⊥ where B+ is a set of atoms. As illustrated by the following example, constraints restrict the possible types of input so that more programs are stratified.

Example 6. Organic molecules are those containing carbon and each inorganic entity is a molecule of geological origin:

r1 : mol(x) ∧ hA(x, y) ∧ c(y) → organic(x) r2 : mol(x) ∧not organic(x) → inorganic(x) r3 : inorganic(x) → mol(x) ∧ geoOrigin(x)

It is easily checked that r1 - -→ r2 + -→ r3 + -→ r1, so {r1, r2, r3} is not R-stratified by Proposition 2. Although the program has a unique stable model for all sets of facts, there is no stratified order of rule applications that produces the stable model. In particular, the set of facts {inorganic(a), hA(a, b), c(b)} requires us to apply r3 before r1. This situation is undesired, since inorganic molecules usually do not contain carbon, and a refined notion of reliance should take this into account.

Definition 8 (Reliances under Constraints). Let r1 and r2 be rules, and let C be a set of constraints.

• r2 positively relies on r1 under C (written r1 + -→C r2) if there exists a set of facts F and a substitution θ that satisfy the conditions in Definition 2, and where F |= C.
• r2 negatively relies on r1 under C (written r1 - -→C r2) if there exists a set of facts F and a substitution θ that satisfy the conditions in Definition 4, and where F |= C.

The classes of programs that are R-acyclic under C and Rstratified under C are defined as in Definition 3 and 5, respectively, but using ± -→C instead of ± -→.

It should be noted that our earlier results treat constraints like any other rule of P. This is still possible here, e.g., if some constraints are not deemed to be relevant for showing stratification. Indeed, the fewer constraints are part of C, the fewer additional checks are needed to compute reliances.

Example 7. Consider the rules of Example 6 and the constraint c : inorganic(x) ∧ hA(x, y) ∧ c(y) → ⊥. With C := {c}, we find r3 ̸+ -→C r1, and indeed P1 := {r1}, P2 := {r2, r3} is an R-stratification under these constraints.

The consideration of constraints increases the complexity of checking positive reliances from NP to ΣP2 , i.e., the check can be performed in polynomial time by a nondeterministic Turing machine using an NP oracle. Yet, as before, the NP computations correspond to checking the applicability of a rule or constraint to a small set of facts, for which efficient implementations exist. A lower bound can be shown by reducing satisfiability of a quantified Boolean formula ∃p.∀q.ϕ to testing a positive reliance under a set of constraints.

Theorem 6. Given rules r1 and r2, and a set of constraints C, deciding whether r1 + -→C r2 is ΣP2 -complete. Checking whether a program P is R-acyclic under constraints is ΠP2 - complete.

As before, the relations + -→C and - -→C induce a graph of reliances under constraints. Analogously to Proposition 2, we can show that P is R-stratified under constraints if and only if this graph does not contain cycles that involve - -→C. This is the basis for deciding R-stratification under constraints, leading to the following result.

Theorem 7. Given rules r1 and r2, and a set of constraints C, the problem of deciding whether r1 - -→C r2 is in ∆P2 . Checking whether a program P is R-stratified under C is ΠP2 - complete.

Given an R-stratification of P under constraints C, we can again define a computation scheme to obtain unique stable models. C in this case is evaluated on all strata, though one can also defer constraint checking to the highest stratum.

Definition 9. For a set of facts F and a program P with Rstratification P = P1, . . . , Pn under constraints C, define S0P,C (F) := TC(F) and

The following result can be shown using the same overall proof structure as in Section 5. The main difference is that in all arguments that discuss potential reliances between rules, we also need to show satisfaction of the constraints. This is usually a consequence of the assumption that ⊥ is not derived.

Theorem 8. If ⊥ /∈ SnP,C (F), then SnP,C (F) is the unique stable model of F ∪ P ∪ C, or else F ∪ P ∪ C has no stable model.

Theorems 2 and 5 can be generalised to programs that are R-acyclic and R-stratified under constraints:

Theorem 9. For a set of facts F, a fact α, and a program P that is R-acyclic under a set of constraints C, deciding P∪F∪C |= α is coN2EXPTIME-complete (coNP-complete) w.r.t. program (data) complexity. If P is also R-stratified under C, deciding P ∪ F ∪ C |= α becomes 2EXPTIMEcomplete (P-complete) w.r.t. program (data) complexity.

7 Experimental Evaluation

In order to assess the practical utility of our solution, we conducted a case study with ChEBI. Our test datasets, software, and detailed results are published online [Magka et al., 2013].

The ChEBI database (release 97) contains about 20,000 molecular structures and taxonomic relations for about 8,000 chemical classes, while the DL-based ontology contains taxonomic information only. To obtain rules for reasoning, we considered a sample of 500 molecules, with sizes ranging from 2 to 138 atoms. The structure of each molecule (given in MDL Molfile format) was converted to rules of the form (2). Chemical classes, such as one-carbon molecule or organic hydroxy, do not have machine-readable descriptions in ChEBI. We selected 50 chemical classes and manually formalised their human-readable descriptions as rules, such as (3) and (6). In addition, we defined 30 molecule classes that are characterised by small substructures (functional groups of 2 to 8 atoms), e.g., organic hydroxy. We modelled each with two rules of the form (8) and (9), using distinct predicates r and n for each pair of rules. Finally, existential quantifiers were skolemised, and conjunctions in rule heads were decomposed into multiple rules. This led to a program P with 78,957 rules, the largest of which had 38 body atoms (8 negative). P was not stratified, but was R-stratified and R-acyclic. In addition, we generated a set F of 530 facts of the form C(aC), one for each molecule or functional group. This allowed us to compute subsumptions between chemical classes: C is subsumed by C ′ iff C ′ (aC) is in the unique stable model of P ∪ F .

We ran experiments on a desktop computer (2GHz quadcore CPU, 4GB RAM) running Linux. In a first experiment, we tried to compute a stable model of P ∪ F using DLV [Leone et al., 2006], but the system failed to compute this result within a time limit of 600 seconds. In a second experiment, we split P into R-strata and consecutively computed the stable model of each stratum. Of the five R-strata of P , the first stratum P1 contained 78,251 rules, while the 706 rules of the remaining four R-strata formed a stratified program P 5 2 . We thus used DLV to compute the stable model of P1 ∪ F, converted the result into a new set of facts S1P, and used DLV to compute the stable model of S1P ∪ P 5 2 . This took 17 seconds, with 13.5 seconds being used for actual reasoning in DLV.

We obtained 8,639 non-trivial subsumptions overall between chemical classes, which we compared to ChEBI's manually created taxonomy. This revealed several omissions in ChEBI, e.g., the fact that every organic hydroxy (ChEBI id 33822) is an organooxygen compound (ChEBI id 36963), illustrating the practical relevance of our approach.

8 RelatedWork

Nonmonotonic extensions for existential rules are considered by Calì et al. [2009] using stratified negation, and more recently by Gottlob et al. [2012] using well-founded semantics. Another approach to nonmonotonic ontological modelling are FDNC programs [Eiter and Simkus, 2010], which are related to DLs and inherit many of their limitations in modelling finite structures.

Local stratification generalises stratification by considering the (infinite) groundings of normal logic programs [Przymusinski, 1989]. This condition is undecidable [Cholak and Blair, 1994], but does not generalise R-stratification (see [Magka et al., 2013] for a counterexample). Further extensions along these lines led to weak stratification [Przymusinska and Przymunsinski, 1990], effective stratification [Bidoit and Froidevaux, 1991], modular stratification [Ross, 1994], and left-to-right dynamic stratification [Sagonas et al., 2001], all of which are known or suspected to be undecidable in the presence of function symbols.

Many other works study the problem of recognising negation-free programs with finite models, e.g., Fagin et al. [2005], Krötzsch and Rudolph [2011] and Cuenca Grau et al. [2012]. Deutsch et al. [2008] discuss how to generalise weak acyclicity by first partitioning the program into cyclic components; in the same way, reliances can be combined with any notion of acyclicity. Negation is rarely considered. Omegarestrictedness uses a kind of 'stratification' to ensure finiteness of stable models [Syrjänen, 2001]. Magka et al. [2012] define semantic acyclicity to ensure finite models in reasoning about structured objects but only consider stratified negation.

9 Conclusions

We showed that nonmonotonic existential rules can tackle complex real-world modelling problems and presented novel conditions to ensure efficient, deterministic reasoning. Our experiments indicate that our approach can dramatically increase the performance of existing reasoners, enabling them to address new practically interesting application areas.

For future work, it is thus very promising to integrate our approach into existing rule engines, which will also allow more extensive evaluations. Section 6 suggests that cyclic or non-stratified programs could be 'repaired' by adding suitable constraints, which could inspire new tools for rule modelling. Equality theories often lead to additional reliances, whereas datatypes and numeric constraints could be exploited to discard reliances-further work is needed to study these effects.

Acknowledgements This workwas supported by the Royal Society, the Seventh Framework Program (FP7) of the European Commission under Grant Agreement 318338, 'Optique', and the EPSRC projects ExODA, Score! and MaSI3.