The Semantic Web ontology language OWL 2 DL comes with a variety of language features that enable sophisticated and practically useful modeling. However, the use of these features has been severely restricted in order to retain decidability of the language. For example, OWL 2 DL does not allow a property to be both transitive and asymmetric, which would be desirable, e.g., for representing an ancestor relation. In this paper, we argue that the so-called \global restrictions" of OWL 2 DL preclude many useful forms of modeling, by providing a catalog of basic modeling patterns that would be available in OWL 2 DL if the global restrictions were discarded. We then report on the results of evaluating several state-of-the-art OWL 2 DL reasoners on problems that use combinations of features in a way that the global restrictions are violated. The systems turn out to rely heavily on the global restrictions and are thus largely incapable of coping with the modeling patterns. Based on our observations, we make suggestions for future lines of research on expressive description logic-style OWL reasoning.
The Semantic Web ontology language OWL 2 DL [
We start in Section 2 with a concise overview of OWL 2 DL and its global syntactic restrictions. In Section 3 we provide a catalog of basic modeling patterns that use complex properties in natural ways, but which are disallowed by the global restrictions. For all patterns, we give example use cases supporting their usefulness and practical relevance. For ontology authors, the catalog o ers a large set of new useful modeling options and demonstrates the extended modeling potential available, in principle at least, through the OWL 2 standard. In the same way, the catalog allows for a better understanding of the actual limitations of modeling in OWL 2 DL due to the global restrictions. To our knowledge, no comparable catalog of patterns exists.
In Section 4 we report on the results of an evaluation of several
state-of-theart OWL 2 DL reasoners using test problems that are based on the modeling
patterns. The evaluation results provide an understanding of what can be
expected from existing reasoners when they are applied to input data that violates
the global restrictions of OWL 2 DL. It has to be pointed out that such an
investigation is meaningful, as the OWL 2 standard does not require OWL 2
DL reasoners to reject input beyond OWL 2 DL, but only speci es how such
tools have to behave on legal OWL 2 DL input (see the de nition of tool
conformance for OWL 2 DL entailment checkers in [
A technical report is available [
OWL 2 DL
For space reasons, we refrain from repeating the structural speci cation of OWL 2
DL, and instead refer the reader to [
Recall that the basic modeling primitives in OWL are individuals, classes and properties, where the latter are interpreted by binary relations and strictly subdivided into data properties and object properties. Compared to its predecessor OWL 1, OWL 2 has been signi cantly extended by ways to describe characteristics and interdependencies on the object property level. In particular, SubObjectPropertyOf statements are allowed to take property chains as their rst argument, as, e.g., in
SubObjectPropertyOf( ObjectPropertyChain( :hasParent :hasBrother ) :hasUncle ) expressing that somebody's parent's brother is the uncle of that somebody. In database terms one could say that the uncle relation subsumes the join of the parent relation with the brother relation. Another novel property-centric modeling feature is property disjointness, as, e.g., in
DisjointObjectProperties( :hasParent :hasUncle ) expressing that somebody's parent cannot be that somebody's uncle. Furthermore, OWL 2 allows for characterizing properties as functional, inverse-functional, re exive, irre exive, symmetric, asymmetric, or transitive as, e.g., in TransitiveObjectProperty( :hasAncestor )
Recall further that the semantics of OWL 2 DL, called direct semantics [
In order to ensure decidability despite the high expressivity of the diverse
modeling features in OWL, the ways in which these features are allowed to interact
had to be restricted. This led to the so-called global restrictions that an OWL 2
DL ontology has to satisfy (see Chapter 11 of [
At the core of the restrictions is the notion of simple versus complex object properties. Roughly speaking, an object property is called complex, if it can be inferred from the join of two or more other object properties. For example, the above subproperty axiom renders the uncle property complex. The same holds for the ancestor property, since transitivity of a relation essentially means that the relation subsumes its own self-join.
The global restrictions severely constrain the ways in which complex properties can be used: according to the restriction on simple properties, complex properties are not allowed to occur in cardinality restrictions, self-restrictions, and property disjointness statements, nor is a complex property allowed to be characterized as functional, inverse-functional, irre exive, or asymmetric.
Another severe restriction is on the co-occurrence of subproperty axioms, that is, the restriction on the property hierarchy. The rationale behind this rather technical restriction is to ensure that the set of property chains used to infer a complex property can be described as a regular language. Next to discarding certain subproperty axioms right away, this also prohibits the coexistence of certain such axioms.
Further restrictions apply to OWL 2 DL (referring to the use of blank nodes), but they are not of interest in this paper. 3
This section presents a catalog of modeling patterns based on usage of OWL 2 language features in a way that violates the global restrictions of OWL 2 DL. The catalog consists of twelve modeling patterns, most of them representing di erent combinations of axioms de ning complex properties, such as transitivity axioms, and language constructs that may only be used with simple properties, such as asymmetric property axioms; see Section 2 for a more detailed list of disallowed combinations of language constructs in OWL 2 DL. Each modeling pattern is described with a concrete example of a family relationship given in OWL 2 functional syntax, and is accompanied by an explanation of the con icts with the OWL 2 DL speci cation. Additional use cases from other application areas provide evidence for the generality, usefulness, and relevance of the pattern.
We have to point out that the patterns were not taken from any existing OWL ontologies. As the patterns are explicitly disallowed in OWL 2 DL, and as many of the involved language features were introduced only recently as part of OWL 2, one cannot expect to nd many of these patterns in real-world ontologies today. Rather, the goal is to demonstrate the drastic increase of modeling power in case the global restrictions of OWL 2 DL are discarded. Our motivations for choosing the modeling patterns were simplicity, plausibility, potential relevance, and generality.
We are aware that for some of the patterns it is possible to nd a semantically equivalent reformulation that is valid in OWL 2 DL. However, the purpose of our pattern catalog is not to present semantic scenarios that cannot be expressed in OWL 2 DL, but rather to o er to ontology authors a set of new modeling options that appear natural and simple using the features of OWL 2. We believe that for an ontology author, a complex or non-obvious reformulation of a pattern, in order to keep the ontology in OWL 2 DL, will often be unacceptable. Still, we consider work on such translations relevant as a means of \repairing" ontologies that use our modeling patterns, so that OWL 2 DL reasoners have a better chance of coping with such input (cf. the results in Section 4).
Strict partial orders are asymmetric transitive relations, such as the ancestor relationship between people:
TransitiveObjectProperty( :hasAncestor )
AsymmetricObjectProperty( :hasAncestor ) OWL 2 DL does not allow transitive properties to be asymmetric. Additional use cases include: comparison relations such as greater-than, part-whole relationships, and operational research tasks such as critical path analysis and supply chain management. 3.2
Property chain axioms allow composite relations to be built, such as the uncle relation in terms of the parent and brother relations. Naturally, the uncle relation should be speci ed to be asymmetric:
SubObjectPropertyOf(
ObjectPropertyChain( :hasParent :hasBrother ) :hasUncle )
AsymmetricObjectProperty( :hasUncle ) OWL 2 DL does not allow composite properties to be asymmetric. Another use case is an asymmetric nth-order predecessor relation for a xed number n, such as a grandparent de ned as a parent's parent (n = 2). 3.3
Relations are often de ned as pairs of complementary but mutually exclusive terms that are transitive, such as the ancestor and descendant relationships: TransitiveObjectProperty( :hasAncestor ) TransitiveObjectProperty( :hasDescendant )
DisjointObjectProperties( :hasDescendant :hasAncestor )
OWL 2 DL does not allow disjointness of transitive properties. Another use
case is disjoint pairs of comparison relations, such as greater-than and
smallerthan. Another disallowed example of two disjoint relations of which only one is
transitive is given by the SKOS semantic relations skos:broaderTransitive
and skos:related [
Relations composed using property chain axioms are often disjoint from one or more of the component relations. For example, when composing the uncle relation in terms of the parent and brother relations, then, realistically, all three relations are mutually disjoint:
SubObjectPropertyOf(
ObjectPropertyChain( :hasParent :hasBrother ) :hasUncle )
DisjointObjectProperties( :hasUncle :hasParent :hasBrother ) OWL 2 DL does not allow disjointness of composite properties. Another use case is an nth-order predecessor relation for a xed number n, such as a grandparent de ned as a parent's parent (n = 2), where the grandparent and parent relations are disjoint. 3.5
For some transitive relations it may be desirable to specify the minimum number of relationships per individual. For example, every person has at least two ancestors:
TransitiveObjectProperty( :hasAncestor ) SubClassOf( :Person
ObjectMinCardinality( 2 :hasAncestor :Person ) ) OWL 2 DL does not allow cardinality restrictions on transitive properties. Other use cases are comparison relations over unbounded domains, such as greater-than for numbers, where for any given number n there are always numbers m > n. 3.6
For some relations composed by property chain axioms it may be desirable to de ne them to be functional. For example, every person has at most one living maternal grandfather, being the father of the person's mother:
SubObjectPropertyOf(
ObjectPropertyChain( :hasMother :hasFather ) :hasMaternalGrandfather )
FunctionalObjectProperty( :hasMaternalGrandfather ) OWL 2 DL does not allow composite properties to be functional. An additional use case is a part-ownership relation, in scenarios where items can only have a single owner and where the owner of an item also owns all parts of the item. 3.7
Some relationships between two individuals may \propagate" to two other individuals, due to a speci c constellation of relationships that holds between all four individuals. For example, if Mary has mother Susan, and Bill has father John, where Susan and John are relatives, then Mary and Bill are also relatives. This can be expressed using property chain axioms:
SubObjectPropertyOf(
ObjectPropertyChain( :hasMother :hasRelative
ObjectInverseOf( :hasFather ) ) :hasRelative ) This representation violates the regularity conditions for the property hierarchy of OWL 2 DL, as in chains of size 3 or larger, an inner property of the chain (:hasRelative in position 2) must not also occur as the composite property. An additional use case would be to characterize identical composite items, such as computers, to have identical corresponding components, such as the computer's processors.
Although there is no general method in OWL 2 to fully de ne a composite relation, one can sometimes encode a close characterization by interlacing two property chain axioms. For example, one can de ne an uncle as a cousin's father, and a cousin as an uncle's child:
SubObjectPropertyOf(
ObjectPropertyChain( :hasCousin :hasFather ) :hasUncle ) SubObjectPropertyOf(
ObjectPropertyChain( :hasUncle ObjectInverseOf( :hasFather ) ) :hasCousin ) Circular dependencies on the property hierarchy violate the regularity conditions of OWL 2 DL. 3.9
Equivalence relations are transitive, symmetric and re exive, but for re exivity, a global scope is often not desirable, as it would entail that everything has the relationship. For example, being a relative to someone may be seen as an equivalence relation, provided that one accepts that everyone is a relative of himself. However, one would probably want to limit this relation to people, excluding, for instance, machines or ideas. OWL 2 supports this notion of a \locally-re exive" property by self-restrictions:
SymmetricObjectProperty( :hasRelativeOrSelf ) TransitiveObjectProperty( :hasRelativeOrSelf )
EquivalentClasses( :Person ObjectHasSelf( :hasRelativeOrSelf ) )
OWL 2 DL does not allow self-restriction of transitive properties. Other use
cases include the grouping of people according to some feature, such as
having the same profession or nationality. SKOS speci es the mapping property
skos:exactMatch as symmetric and transitive [
The re exive-transitive closure of a parent relation de ned over the class of people is the smallest super relation that is both transitive and re exive, where re exivity is scoped to the class of people, i.e., a person's ancestor or oneself. While it is not possible to represent the smallest such relation in OWL 2, a coarse approximation is possible using a self-restricted transitive super property: SubObjectPropertyOf( :hasParent :hasAncestorOrSelf ) TransitiveObjectProperty( :hasAncestorOrSelf )
EquivalentClasses( :Person ObjectHasSelf( :hasAncestorOrSelf ) ) OWL 2 DL does not allow self-restriction of transitive properties. Another example is skos:broaderTransitive, the transitive extension of the SKOS semantic property skos:broader [2, S22,S24], for which it would be plausible and consistent with the SKOS standard to additionally make it locally re exive to the class of SKOS concepts. 3.11
Cyclic relationships constructed from one binary relation, such as the \loves" relation, may be of arbitrary size. For example, a person may love only himself or another person mutually, or there may be a cycle of unreturned love including several people. Each person in such a cycle can be represented as an instance of the class of \loved lovers" and, thus, instanceship in this class indicates that a person is part of such a cyclic relationship. Class instanceship can be expressed in terms of a self-restricted transitive super property of the loves property: SubObjectPropertyOf( :loves :z ) TransitiveObjectProperty( :z )
SubClassOf( ObjectHasSelf( :z ) :LovedLover ) OWL 2 DL does not allow self-restriction of transitive properties. Another use case is chemical ring molecules of arbitrary size, where all bonds are of the same sort, such as Cycloalkanes. 3.12
Certain cyclic relations or coincidence relations can be composed from a set of di erent basic relations, such as the concept of a legitimate child, that is, a person with a father and a mother who are married. Occurrence of such relationships in a knowledge base can be indicated by instanceship in a class of legitimate children, modeled using a property chain axiom and a self-restriction: SubObjectPropertyOf(
ObjectPropertyChain( :hasMother :hasSpouse
ObjectInverseOf( :hasFather ) ) :z )
SubClassOf( ObjectHasSelf( :z ) :LegitimateChild ) OWL 2 DL does not allow self-restriction of composite properties. Another use case is circular molecules of a xed size that are built from di erent sorts of bonds, such as Furan. 4
We now report on the results of evaluating several state-of-the-art OWL 2 DL
reasoners using test problems based on the modeling patterns of Section 3. The
focus was on nding out whether or not the reasoners can cope with the modeling
patterns. As mentioned in Section 1, compliant OWL 2 DL reasoners are not
required to reject input beyond the speci cation of OWL 2 DL, and experience
shows that existing systems often do reason upon such input. Hence it is a
legitimate question to ask how they behave on our modeling patterns. To give an
answer, we checked whether the reasoners are able to recognize certain \obvious
looking" logical conclusions from the modeling patterns according to the OWL 2
direct semantics. Reasoning performance was not considered an important aspect
of our evaluation.
Test Data. We created a test suite consisting of one test case per modeling
pattern. Each test case is built from two small ontologies, a premise and a conjecture.
The premise covers the main example of the corresponding modeling pattern
given in Section 3, typically extended by some additional assertions. The
conjecture is a small set of assertions that follow logically from the premise. Both
the premise and the conjecture ontology conform syntactically to the OWL 2
structural speci cation, and the conjecture is entailed from the premise
according to the OWL 2 direct semantics. The test cases were designed to be \not
too di cult to solve", so that the OWL 2 DL reasoners do not fail due to high
reasoning complexity. The complete test suite is described in full detail in the
appendix of the technical report [
University of Manchester, England, is a tableaux-based OWL 2 DL reasoner. { HermiT 1.3.6 (http://hermit-reasoner.com), created at the University of Oxford, England, is a tableaux-based OWL 2 DL reasoner. { Pellet 2.3.0 (http://clarkparsia.com/pellet), created by Clark & Parsia, USA, is a tableaux-based OWL 2 DL reasoner.
Testing Environment. All tests were conducted on a mobile computer \Lenovo
ThinkPad T410s" with an Intel R CoreTM i5 M520 CPU (4 cores) at 2.4 GHz
speed, 4 GB RAM, with Microsoft Windows 7 Professional (64-Bit) as the
operating system. The CPU timeout for applying a reasoner to a test case was
300 seconds. The possible outcomes of the test runs are as follows:
{ `+': termination with correct result
{ ` ': termination with wrong result
{ `?': processing error or timeout
Results. Table 1 shows the outcomes of the evaluation. The details of the results
can be found in the appendix of the technical report [
In this paper we have shown that many useful and relevant modeling options were available in OWL 2 DL if the global restrictions on complex properties would be relinquished. We have presented a catalog of twelve basic useful modeling patterns that are in the scope of the unrestricted structural speci cation and the direct semantics of OWL 2 but are beyond the scope of OWL 2 DL, including strict partial orders and di erent forms of circular relationships. Although the 5 For someone familiar with the methods used in state-of-the-art OWL reasoners this fact does not come as a big surprise. For instance, the restriction on the property hierarchy is a crucial prerequisite for preprocessing the ontology ahead of the actual core reasoning procedures. OWL 2 DL standard does not prevent compliant reasoners from processing such input, all the state-of-the-art OWL 2 DL reasoners that we tested were unable to cope with these modeling scenarios.
In our technical report [