Description logic (DL) formalisms are extensively used for ontological modelling, e.g., in biology [16] and chemistry [19]. One such example is the ChEBI ontology [20] - a reference terminology adopted by various biological databases for chemical annotation [23,8,10]. Despite their wide range of expressive features, DLs are severely limited in their ability to represent structures that are not tree-shaped. This explains, e.g., why ChEBI does not capture molecular structures in its ontology, thus excluding its main content from logical reasoning. In order to overcome this deficiency, numerous rulebased extensions of DLs have been proposed that provide certain kinds of graph-based modelling, such as description graphs [27]. However, in order to retain decidability of description graphs several constraints are imposed, such as role separation restrictions and a cumbersome acyclicity condition, that restrict the range of structures that can be modelled and thus hinder practical usability [26]. Moreover, when performing structurebased classification 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. Expressing completeness (closure) of finite structures in DLs is prohibitively inefficient, whereas nonmonotonic extensions of DLs remain largely unrealised in tools and applications [28]. This motivates the use of logical languages that draw upon rules enriched with nonmonotonic negation for the representation and classification of objects with a complex internal structure. Existential rules-function-free Horn rules with existential quantifiers in rule heads-have been proposed as a new expressive ontological language [6,2], 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 [5,26], stable model semantics [13], and well-founded semantics [17]. 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 stable models; this is highly desirable and easy to check, but too restrictive for many applications. Moreover, even without negation, satisfiability, fact entailment, ? This work was 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.
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 prevent 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 R-stratification. Our contributions can be summarised 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 (NP-complete 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 P P. 2 – 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 [22] to scale to the large number of rules in our experiments, and that DLV can then be used to discover missing relationships in ChEBI.
This paper is an adapted version of [24]; omitted proofs and additional details can be found in the companion report [25]. 2
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 ht1; : : : ; tni 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(e), Var(e), and Const(e) to denote the predicates, variables, and constants,
respectively, that occur in an expression e. Atoms, i.e., formulae without operators, are
written a, b , g. 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 := fnot a j a 2 Ag. A (nonmonotonic existential) rule is of the form
r :
8x:8z: B+ ^ not B
! 9y:H
(
The skolemisation sk(r) of a rule r as in (
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)) = 0/ ). 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 from HU(P).
An (Herbrand) interpretation M is a set of facts with ? 2= M. Satisfaction is defined as usual: M j= B+; not B holds if B+ M and B \ M = 0/ ; M j= (B+; B ; H) if M 6j= B+; not B or M j= H; and M j= P if M j= r for all r 2 P. The GelfondLifschitz reduct of P w.r.t. M is GL(P; M) := f(B+; 0/ ; H) j (B+; B ; H) 2 ground(P)and B \ M = 0/ g. M is a stable model of P, written M j=SM P, if M j= GL(P; M) and there is no smaller model M0 ( M with M0 j= GL(P; M). We consider cautious entailment: for a program P and a fact a, P j= a if a 2 M for all stable models M of P. Consequences of programs can be computed with the TP operator: Definition 1. For a program P, a set of facts F and r 2 P with sk(r) = (B+; B ; H), let r(F) := fHq j B+q F and B q \ F = 0/ g: Moreover, let TP(F) := F [ Sr2P r(F) and define TP0(F) := F, TPi+1(F) := TP(TPi (F)), and TP¥(F) := [ TPi (F): i 0
For a program P, a sequence of disjoint programs P = P1; : : : ; Pn is a stratification of P if P = Sin=1 Pi and, for all programs Pi; Pj 2 P, rules (B1+; B1 ; H1) 2 Pi and (B2+; B2 ; H2) 2 Pj, and every predicate R 2 Pred(H1), we have: (i) if R 2 Pred(B2+) then i j, and (ii) if R 2 Pred(B2 ) then i < j; P1; : : : ; Pn 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 [1]. Fact 1 Given a program P, a set of facts F, and a stable model M j=SM P [ F, we have M = TG¥L(P;M)(F). Also, if P = P1; : : : ; Pn is a stratification of P, then M := TP¥n (: : : TP¥1 (F) : : :) is the unique stable model of P if ? 2= M. 3
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 represents methanol (CH3OH), described by MCH3OH(y): methanol(x) ! 9y:mol(x) ^ MCH3OH(y) ^ Vi6=1hA(x; yi) 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 [25] for details. Using not we can express syntactic inequality and define, e.g., molecules with exactly one carbon atom:
Vi2=1(hA(x; yi) ^ c(yi)) ^ not y1 = y2 ! multiC(x)
mol(x) ^ hA(x; y) ^ c(y) ^ not multiC(x) ! oneC(x)
Rules (
hA(x; y) ^ o(y) ! hasO(x)
MCOH(y) ^ Vi3=1 hA(x; yi) ! orgHydroxy(x)
(
hA(b; gi(b))i3=1; MCOH(g1(b); g2(b); g3(b))g
So, organic hydroxy molecules contain oxygen, as expected. However, if we consider all
of the above rules and facts together, then rather than M1 [ M2 we obtain M1 [ M2 [
fhA(a; gi(a))i3=1; MCOH(g1(a); g2(a); g3(a)); multiC(a)g n foneC(a)g as the unique
stable model, since rule (
Nonmonotonic negation can be used to overcome this problem. We replace rules (
MCOH(y) ^ Vi3=1 hA(x; yi) ^ not n(yi) ! oH(x) ^ r(x)
oH(x) ^ not r(x) ! 9y:MCOH(y) ^ Vi3=1 hA(x; yi) ^ n(yi)
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 (
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 applying r1 can potentially enable applying r2 later. We next formalise this idea of positive reliance between rules and define R-acyclic programs, which have finite stable models. Definition 2 (Positive Reliance). Let r1 and r2 be rules such that sk(r1) = (B1+; B1 ; H1) and sk(r2) = (B2+; B2 ; H2); w.l.o.g. assume that Var(r1) \ Var(r2) = 0/ . 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 q such that:
Thus, r1 !+ r2 holds if there is a situation 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(
Similar notions have been considered in the past. The activation relation by Greco et
al. [18] is similar to Definition 2, but allows F to contain function terms to accommodate
arbitrary programs with functions. Our stronger restriction is needed to show r(07) !+6 r(
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(
We prove that checking positive reliance for two rules is NP-complete. Similar results are shown by Deutsch et al. [12] and by Baget et al. [3] 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 NPcomplete. 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 is shown by reducing the word problem of 2EXPTIME-bounded non-deterministic Turing machines to fact entailment, adapting constructions by Calì et al. [7] and Krötzsch and Rudolph [21]. Theorem 2. Let P be an R-acyclic program and let F [ fa g 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 j= a is coN2EXPTIME-complete w.r.t. program complexity and coNP-complete w.r.t. data complexity. 5
While positive reliances capture when one rule ‘triggers’ another, the use of nonmonotonic negation may cause the opposite interaction where one rule ‘inhibits’ another; 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) = (B1+; B1 ; H1) and sk(r2) = (B2+; B2 ; H2); w.l.o.g. assume that Var(r1) \ Var(r2) = 0/ . 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 q such that:
B1+q
F
Example 3 Consider rule r(
P is R-stratified if it has an R-stratification.
Example 4 For P consisting of rules r(
Example 4 and Proposition 1 above show that R-stratification properly contains 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.
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) from Definition 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 H2q 6 F in Definition 4, then r2 ! r1 would not hold, and the program would be R-stratified. 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 R-stratification P = P1; : : : ; Pn, define SP0(F) := F and
SPi+1(F) := TP¥i+1 (SPi(F)) for 0
For the remainder of this section, let P denote an R-stratified program with Rstratification P = P1; : : : ; Pn, let F denote a set of facts, and define SPi := SPi(F).
We first show that SPn is a (not necessarily unique) stable model of F [ P, provided that ? 2= SPn. The next two lemmas are key ingredients to this proof. Intuitively speaking, Lemma 1 asserts that, if the body of a rule r 2 Pi is satisfied at some point while computing SPi, 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 ` 0, a rule r 2 Pi with skolemisaj 1 tion sk(r) = (B+; B ; H), and a substitution q . Then TP`j (SP ) j= B+q ; not B q implies SPk j= B+q ; not B q .
Lemma 2 complements the previous result. Intuitively speaking, it states that a rule r 2 Pi, which is clearly satisfied after computing Si , will remain satisfied in all later P 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 Then S Pj j= sk(r)q implies SPk j= sk(r)q .
Using Lemmas 1 and 2, we can show the following result.
Proposition 3. If ? 2= SPn, then SPn j=SM F [ P. k n, a rule r 2 Pi, and a substitution q .
The main result of this section is that stable models of R-stratified programs are unique. We prove so by first showing that M j=SM P [ F implies SPn = M, which is established by showing inductively that for all k 2 f0; : : : ; ng, SPk = T ¥ (F). GL(Sik=1 Pi;M) Theorem 4. If ? 2= SPn, then SPn is the unique stable model of F [ P. Otherwise F [ P does not have a stable model.
The complexity of reasoning for programs that are both R-acyclic and R-stratified can thus be improved. Hardness is shown by using a reduction similar to Theorem 2 but with a deterministic Turing machine, that results in an R-stratified constructed program. Theorem 5. Let P be an R-acyclic R-stratified program, let F be a set of facts, and let a be a fact. Deciding P [ F j= a is 2EXPTIME-complete w.r.t. program complexity and P-complete w.r.t. data complexity. 6
To widen the classes of logic programs with unique stable models, it has been proposed to study stratification for a particular set of facts [4]. 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. In this section, we therefore propose a generalisation of R-acyclicity and R-stratification that considers constraints, that is, rules of the form B+ ! ? where B+ is a set of atoms. As Example 6 suggests, constraints restrict the possible inputs 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 : r2 : r3 : mol(x) ^ hA(x; y) ^ c(y) ! organic(x) mol(x) ^ not organic(x) ! inorganic(x)
inorganic(x) ! mol(x) ^ geoOrigin(x) It is easily checked that r1 ! r2 !+ r3 !+ r1, so fr1; r2; r3g 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 finorganic(a); hA(a; b); c(b)g 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.
of constraints.
– r2 positively relies on r1 under C (written r1 !+C r2) if there exists a set of facts F and a substitution q that satisfy the conditions in Definition 2, and where F j= C. – r2 negatively relies on r1 under C (written r1 !C r2) if there exists a set of facts F and a substitution q that satisfy the conditions in Definition 4, and where F j= C. The classes of programs that are R-acyclic under C and R-stratified 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 := fcg, we find r3 !+6C r1, and indeed P1 := fr1g, P2 := fr2; r3g is an R-stratification under these constraints.
Adding constraints increases the complexity of testing positive reliances from NP to S P, i.e. they can be checked in polynomial time by a nondeterministic Turing ma2 chine using an NP oracle. 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 is obtained by reducing satisfiability of a quantified Boolean formula 9p:8q:j to testing a positive reliance under a set of constraints. Theorem 6. Given rules r1 and r2, and a set of constraints C, deciding r1 !+C r2 is S2P-complete. Checking if a program P is R-acyclic under constraints is P2P-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 r1 !C r2 is in D2P. Checking if a program P is R-stratified under C is P2P-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 R-stratification P = P1; : : : ; Pn under constraints C, define SP0;C(F) := TC(F) and
SPi+;C1(F) := TP¥i+1[C(SPi;C(F)) for 0
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 ? 2= SPn;C(F), then SPn;C(F) is the unique stable model of F [ P [ C, or else F [ P [ C has no stable model.
Theorems 2 and 5 carry over for R-acyclic and R-stratified programs under constraints: Theorem 9. For a set of facts F, a fact a, and a program P that is R-acyclic under a set of constraints C, deciding P [ F [C j= a is coN2EXPTIME-complete (coNP-complete) w.r.t. program (data) complexity. If P is also R-stratified under C, deciding P [ F [C j= a becomes 2EXPTIME-complete (P-complete) w.r.t. program (data) complexity. 7
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 [25].
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 (
We ran experiments on a desktop computer (2GHz quad-core CPU, 4GB RAM) running Linux. In a first experiment, we tried to compute a stable model of P [ F using DLV [22], 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 P5. We thus used DLV to compute the stable model of P1 [ F, converted the result 2 into a new set of facts SP1, and used DLV to compute the stable model of SP1 [ P25. This took 17 seconds, with 13.5 seconds being used for actual reasoning in DLV.
We obtained 8,639 non-trivial subsumptions 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
Nonmonotonic extensions for existential rules are considered by Calì et al. [5] using stratified negation, and more recently by Gottlob et al. [17] using well-founded semantics. Another approach to nonmonotonic ontological modelling are F DN C programs [14], which are related to DLs and inherit many of their expressivity limitations.
Local stratification generalises stratification by considering the (infinite) groundings of normal logic programs [30]. This condition is undecidable [9], but does not generalise R-stratification (see [25] for a counterexample). Further extensions along these lines led to weak stratification [29], effective stratification [4], modular stratification [33], and left-to-right dynamic stratification [34], 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 [15,21,11]. Deutsch et al. [12] 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 [35]. Magka et al. [26] define semantic acyclicity to ensure finite models in reasoning about structured objects but only consider stratified negation. Finally, the use of constraints on sets of facts to enhance acyclicity and stratification resembles the use of dependencies (called extensional constraints) on ABoxes to optimise DL query rewriting [31,32]. 9
We showed that nonmonotonic existential rules under a stable model semantics can address complex real-world modelling problems, and presented novel conditions to ensure efficient, deterministic reasoning in these cases. Our experiments indicate that our approach can dramatically increase the performance of existing answer set programming engines, enabling them to address new, practically relevant 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.