Counting and Sets in OWL OWL augments restrictions with a full set-theoretic language, including intersections, unions, complements Using these, we combine  restrictions (e.g., the set of planets that go around the sun and have at least 1 moon)  the classes used to define restrictions (e.g., a Vegetarian is someone who eats food that is not Meat) OWL also includes restrictions that refer to cardinality, referring to the number of distinct values for a given property some individual has  We can then describe, e.g., “the set of planets that have at least 3 moons”

Reasoning with sets must cope with the following fundament aspects of the Semantic Web Open World Assumption (a consequence of the AAA slogan) There could always be something new that someone will say; we must assume that there is always more info that could be known Nonunique Naming Since the speakers on the Web won’t necessarily coordinate their naming efforts, the same entity could be known by more than one name (URI)

Unions and Intersections All set operations can be used on any class definition, including named classes and restrictions They use the collection (list) constructs of RDF U1 a owl:Class; owl:unionOf ( ns:A ns:B … ). I1 a owl:Class; owl:intersectionOf ( ns:A ns:B … ). The intersection/union of 2 or more classes it itself a class  Represented by either naming the class or by defining an anonymous class

An anonymous class can be used again if named using owl:equivalentClass bb:MajorLeagueBaseballPlayer owl:equivalentClass [ a owl:Class; owl:intersectionOf ( bb:MajorLeagueMember bb:Player bb:BaseballEmployee ) ]. Natural language descriptions often have a notion of intersection built in  “All planets orbiting the sun” is the intersection of all things that orbit the sun (a restriction) and all planets :SolarPlanet a owl:Class; owl:intersectionOf ( :Planet [ a owl:Restriction; owl:onProperty :orbits; owl:hasValue :TheSun] ).

 The set of major league baseball players is the intersection of all things that play on a major league team ( someValueFrom restriction) and baseball players :MajorLeagueBaseballPlayer a owl:Class; owl:intersectionOf ( :BaseballPlayer [ a owl:Restriction; owl:onProperty :playsFor; owl:someValuesFrom :MajorLeagueTeam] ).

Example: High-Priority Candidate Questions Earlier defined  a class of candidate questions based on dependencies on selected answers  priorities for the questions Now use set constructors to combine these to form a class of candidate questions of a given priority Repeat here the description of SelectedAnswer that classifies a dependent question as EnabledQuestion q:SelectedAnswer rdfs:subClassOf [ a owl:Restriction: owl:onProperty q:enablesCandidate; owl:allValuesFrom q:EnabledQuestion].

Define a class of questions we’re ready to ask based on 2 criteria  They’ve been enabled by the description above  They have high priority Do it with an intersectionOf constructor q:CandidateQuestion owl:equivalentClass [ a owl:Class; owl:intersectionOf ( q:EnabledQuestion q:HighPriorityQuestion )].

Or we could make a more relaxed description for candidate questions and include medium-priority questions q:CandidateQuestion owl:equivalentClass [ a owl:Class; owl:intersectionOf ( q:EnabledQuestion [ a owl:unionOf ( q:HighPriorityQuestion q:MediumPriorityQuestion )] )].

Closing the World Note the impact of the Open World Assumption on set operations and counting  Even the notion of set complement is subtle  Say, based on its absence, that something isn’t a member of a set But then find that it is But there are ways to assert in OWL that certain parts of the world are closed  Make inferences involving complements or counting much clearer  Example: James Dean made only 3 movies We know this; he’s been dead for 50 years

Enumerating Sets with owl:oneOf In the James Dean example, needn’t reject the Open World Assumption  Just need an enumeration of all the members of a given class (James Dean movies) Enumerate the members of a class with the owl:oneOf construct ss:SolarPlanet rdf:type owl:Class; owl:oneOf ( ss:Mercury ss:Venus ss:Earth ss:Mars ss:Jupiter ss:Saturn ss:Uranus ss:Neptune ).  This class contains 8 individuals and no others

owl:oneOf limits the AAA slogan  Nobody else can say there’s another distinct item  So use owl:oneOf carefully and only when the definition of the class isn’t likely to change (or at least not often) But owl:oneOf places no limitation on the Nonunique Naming Assumption  Makes no claim about whether, e.g., Mercury might be the same as Venus Combined with owl:someValuesFrom, owl:oneOf provides a generalization of owl:hasValue  owl:hasValue specified a single value a property can take  But owl:someValues from combined with owl:oneOf specifies a set of values a property can take

Challenge In a dialogue with Rimbaud, Rocky used the fact that  James Dean made only 3 movies and  a process of elimination to determine what movie Rimbaud saw How to represent this in OWL? Solution Define the class of James Dean movies as :JamesDeanMovie a owl:Class; owl:oneOf ( :Giant :EastOfEden :Rebel ). As usual, we define the meaning of owl:oneOf in terms of the inferences that can be drawn from it  There are several

First, can infer that each instance listed in owl:oneOf is indeed a member of the class  Given the above, can infer :Giant rdf:type :JamesDeanMovie. :EastOfEden rdf:type :JamesDeanMovie. :Rebel rdf:type :JamesDeanMovie. But owl:oneOf also asserts that these are the only members  In terms of inference, if we assert that some new thing is a member of the class, it must be owl:sameAs one of the members listed in the owl:oneOf list In our example, if someone introduces a new member of the class :RimbaudMovie rdf:type :JamesDeanMovie. can infer that :RimbaudMovie is owl:sameAs one of the movies mentioned

So far, we’ve expressed inferences in terms of new triples inferred But here the inference tells that some triple from a small set holds (don’t know which)  Can’t assert any new triple To do the elimination in OWL, must be able to say that some individual is not the same as another

Differentiating Individuals with owl:differentFrom Because of the Nonunique Naming Assumption, when we want different things, must name things that are explicitly different For this, OWL provides owl:differentFrom ss:Earth owl:differentFrom ss:Mars. owl:differentFrom supports several inferences when used with other constructs like owl:cardinality and owl:oneOf

Challenge 28 Use OWL to model the dialogue between Rocky and Rimbaud so OWL can infer that Rimbaud saw Rebel Solution At the beginning of the dialogue, Rocky knows that the movie Rimbaud saw was one of the 3 listed He guesses Giant. If correct, can assert :RimbaudMovie owl:sameAs :Giant.  But wrong, so assert :RimbaudMovie owl:differentFrom :Giant. Still must guess, this time East of Eden, and again wrong, so assert :RimbaudMovie owl:differentFrom :EastOfEden.

The semantics of owl:one of and owl:differentFrom now let us infer :RimbaudMovie owl:sameAs :Rebel. See Figure 1 for the assertions and inference

Differentiating Multiple Individuals To simplify specification of lists of items all different from one another, OWL provides 2 constructs used together: owl:AllDifferent, owl:distinctMembers Specify that an RDF list (i.e., collection) of items are mutually distinct by referring to the list in a triple using predicate owl:distinctMembers  The domain of owl:distinctMembers is owl:AllDifferent It’s customary for the subject of an owl:distinctMembers triple to be a bnode [ a owl:AllDifferent; owl:distinctMembers ( ss:Mercury ss:Venus … ss:Neptune )].

The following is typical :SolarPlanet a owl:Class; owl:intersectionOf ( owl:oneOf ( ss:Mercury, ss:Venus, ss:Earth, ss:Mars, ss:Jupiter, ss:Saturn, ss:Uranus, ss:Neptune ) [ a owl:AllDifferent; owl:distinctMembers (ss:Mercury, ss:Venus, ss:Earth, ss:Mars, ss:Jupiter, ss:Saturn, ss:Uranus, ss:Neptune ) ] ).

This is the same as asserting 28 owl:differentFrom triples, one for each pair of items in the list For the James Dean movies, [ a owl:AllDifferent; owl:distinctMembers ( :EastOfEden :Giant :Rebel )]. It’s common to use owl:oneOf and owl:AllDifferent as we’ve done, to say that a class is made up of an enumerated list of distinct elements

See Figure 2  The list is constructed with rdf:first, rdf:next, rdf:nil

Cardinality Cardinality restrictions let us express constraints on the number of individuals that can be related to a member of the restriction class  E.g., a baseball team has exactly 9 (distinct) players Cardinality restrictions also let us define sets of particular interest— the set of 1-act plays

The syntax is similar to other restrictions—e.g., the class of things with exactly 5 players [ a owl:Restriction; owl:onProperty :hasPlayer; owl:cardinality 9] Can also specify a lower bound—e.g., those things with at least 10 players [ a owl:Restriction; owl:onProperty :hasPlayer; owl:minCardinality 10] and an upper bound—e.g., those things with at most 2 players [ a owl:Restriction; owl:onProperty :hasPlayer; owl:maxCardinality 2]

Cardinality refers to the number of distinct values a property has It interacts closely with the Nonunique Naming Assumption and owl:differentFrom Regarding inference, if we can show that an individual has exactly (resp., at least, at most) n distinct values for a property, then it’s a member of the corresponding owl:cardinality (resp., owl:minCardinality, owl:maxCardinality ) restriction Similarly, if we assert that something is a member of an owl:cardinality restriction, then it must have exactly n distinct values for the property Similar conclusions follow from restrictions on minimum and maximum cardinality

Challenge 29 Rocky and Rimbaud continue their conversation Rimbaud: Do you own any James Dean movies? Rocky: They’re the only ones I own. Rimbaud: Then you own no more than three movies. Model these facts in OWL so that Rimbaud’s conclusion follows from the OWL semantics Solution Need a property to indicate that someone owns a movie :ownsMovie a owl:ObjectProperty.

In OWL, make general statements about an individual by asserting that they are a member of a restriction class :JamesDeanExclusive owl:equivalentClass [ a owl:Restriction; owl:onProperty :ownsMovie; owl:allValuesFrom :JamesDeanMovie]. :Rocky a :JamesDeanExclusive. Define the class of things that own at most 3 movies :FewMovieOwner owl:equivalentClass [ a owl:Restriction; owl:onProperty :ownsMovie; owl:maxCardinality 3].

Rocky’s conclusion is the triple :Rocky a :FewMovieOwner. This triple can be inferred from the model  All values of property :ownsMovie for Rocky come from class :JamesDeanMovie  There are only 3, all distinct  So Rocky owns at most 3 movies Figure 3 (next slide) shows the inference

Challenge 30 Model the following in OWL Rimbaud: How many movies do you own? Rocky: Three. Rimbaud: Then you own the one I saw last night, Rebel without a Cause. Solution Assert that Rocky is a member of the owl:cardinality restriction class for “the set of things that own exactly 3 movies” :ThreeMovieOwner owl:equivalentClass [ a owl:Restriction; owl:onProperty :ownsMovie; owl:cardinality 3]. :Rocky a :ThreeMovieOwner.

Now  Rocky owns exactly 3 distinct movies  All his movies are members of :JamesDeanMovie  There are just 3 different :JamesDeanMovies Infer  He owns each of them In particular, infer :Rocky :ownsMovie :Rebel. Figure 4 (next slide) shows the assertions and inferences

Small Cardinality Limits Restrictions of cardinalities to the small numbers 0 and 1 have special modeling utility minCardinality 1 Those with some (at least 1) value for the property E.g., onProperty ownsMovie minCardinality 1 specifies the set of those owning at least 1 movie maxCardinality 1 Those that, if they have a value for the property, have a unique value E.g., onProperty ownsMovie maxCardinality 1 specifies the set of those limiting themselves to 1 movie

minCaridnality 0 Those for which a value for the property is optional  Superfluous in OWL semantics (properties are always optional)  But the explicit assertion can be useful for model readability E.g., onProperty ownsMovie minCardinality 0 explicitly specifies the set of those for which owning a movie is optional maxCardinality 0 Those for which no value for the property is allowed E.g., onProperty ownsMovie maxCardinality 0 specifies the set of those owning no movies

Set Complement The complement of a class is another class whose members are all things not in the complemented class A complement applies to a single class, so define it with 1 triple ex:ClassA owl:complementOf ex:ClassB. The complement of a class includes everything in the universe not in that class  E.g., the complement of bb:MajorLeaguePlayer includes not only minor league players but also movies, planets, …

Usually work with the relative complement (e.g., the class of players who aren’t major league players) by combining complement and intersection—e.g., bb:MinorLeaguePlayer owl:intersectionOf ( [ a owl:Class; owl:complementOf bb:MajorLeaguePlayer ] bb:Player ).  Using the complement class just here—no need for a name Specified anonymously in bnode notation

Challenge 31 Continue the dialogue with an addition participant, Paul Paul: I have all James Dean’s movies. Rimbaud: Do you have other movies too? Paul: I have some movies that aren’t James Dean movies. Rocky: You must have at least four movies then. Model this situation and conclusion in OWL

Solution Define an inverse of :ownsMovie :ownedBy owl:inverseOf :ownsMovie. Define the class of all movies Paul owns :PaulsMovie a owl:Class; owl:intersectionOf ( [ a owl:Restriction; owl:onProperty :ownedBy; owl:hasValue :Paul] :Movie ). Paul owns every James Dean movie: every :JamesDeanMovie is a :PaulsMovie (but possibly not vice versa) :JamesDeanMovie rdfs:subClassOf :PaulsMovie.

Paul also owns other movies Capture this in OWL First note that the following anonymous class includes everything that isn’t a James Dean Movie [ owl:complementOf :JamesDeanMovie] Building on this, the following is an anonymous class of all things having some value for :ownsMovie that isn’t a James Dean movie [ a owl:Restriction; owl:onProperty :ownsMovie; owl:someValuesFrom [ owl:complementOf :JamesDeanMovie]] Finally, we assert that Paul is such a thing :Paul a [ a owl:Restriction; owl:onProperty :ownsMovie; owl:someValuesFrom [ owl:complementOf :JamesDeanMovie]].

Want to conclude that Paul is a member of a class of people owning 4 or more movies Remains to define this class :ManyMovieOwner owl:equivalentClass [ a owl:Restriction; owl:onProperty :ownsMovie; owl:minCardinality 4]. Paul owns all 3 James Dean movies and at least 1 non-James Dean movie  At least 4 in all So infer :Paul rdf:type :ManyMovieOwner. Figure 5 shows the assertions and conclusion

Disjoint Sets Specify that 2 classes are non-overlapping, or disjoint, with owl:disjointFrom :Man owl:disjointFrom :Woman. For any members of disjoint classes, can infer that they are owl:differentFrom one another Given :Irene a :Woman. :Ralph a :Man. infer :Irene owl:differentFrom :Ralph.

Challenge 32 Continuing with the dialogue: Paul: Not only do I have all the James Dean movies, but I also have movies with Judy Garland, Tom Cruise, Judi Dench, and Antonio Banderas. Rocky: You must own at least seven movies. Paul: How did you know that? Rocky: Because none of those people played in movies together. Model this situation and conclusion in OWL

Solution Express that Paul owns, e.g., a Judy Garland movie by asserting that he’s a member of the class of things that own Judy Garland movies :Paul a [ a owl:Restriction; owl:onProperty :ownsMovie; owl:someValuesFrom :JudyGarlandMove]. :Paul a [ a owl:Restriction; owl:onProperty :ownsMovie; owl:someValuesFrom :JudiDenchMove]. :Paul a [ a owl:Restriction; owl:onProperty :ownsMovie; owl:someValuesFrom :TomCruiseMove]. :Paul a [ a owl:Restriction; owl:onProperty :ownsMovie; owl:someValuesFrom :AntonioBanderasMove].

Define the set of people who own 7 or more movies :SevenMovieOwner a owl:Restriction; own:onProperty ownsMovie; owl:minCardinality 7. We don’t know that Paul’s a member of this class until we know that all the sets of movies are disjoint :JamesDeanMovie owl:disjointWith :JudyGarlandMovie. :JamesDeanMovie owl:disjointWith :TomCruiseMovie. :JamesDeanMovie owl:disjointWith :JudiDenchMovie. :JamesDeanMovie owl:disjointWith :AntonioBanderasMovie. :JudyGarlandMovie owl:disjointWith :TomCuiseMovie. :JudyGarlandMovie owl:disjointWith :JudyDenchMovie. :JudyGarlandMovie owl:disjointWith :AntonioBanderasMovie. :TomCruiseMovie owl:disjointWith :JudyDenchMovie. :TomCruiseMovie owl:disjointWith :AntonioBanderasMovie. :JudiDenchMovie owl:disjointWith :AntonioBanderasMovie.

Given  Paul has 3 James Dean movies and at least 1 movie with each of the other actors mentioned  None of these movies appears twice (all sets disjoint) an inference engine can confirm :Paul a :SevenMovieOwner. Figure 6 shows these assertions and inferences

Note how owl:someValuesFrom interacts with cardinality  Each owl:someValueFrom restriction guarantees the existence of one value for the specified property  Given the values are distinct, can count at least 1 per owl:someValuesFrom restriction Just as owl:AllDifferent lets us specify that several individuals are mutually distinct, could have something like owl:AllDisjoint to specify that a set of classes are mutually disjoint  The OWL standard didn’t include such a construct  But some proposals for OWL extensions include it

Prerequisites Revisited We explored how prerequisites can be modeled using owl:allValuesFrom But had a problem with the Open World Assumption: How can we tell that all prerequisites have been satisfied? Now show several ways to close the world Recall that we modeled the fact that something has all its prerequisites selected is a q:EnabledQuestion as q:hasPrerequisite a owl:ObjectProperty. [ a owl:Restriction; owl:onProperty q:hasPrerequisite; owl:allValuesFrom q:SelectedAnswer] rdfs:subClassOf q:EnableQuestion.

No Prerequisites To assert that an individual has no values for some property, use a cardinality restriction c:WhatProblem a [ a owl:Restriction; owl:onProperty q:hasPrerequisite; owl:cardinality 0]. So there are no triples of the following form, i.e., c:WhatProblem has no prerequisites c:WhatProblem q:hasPrerequisite ?x. So c:WhatProlem satisfies c:WhatProblem a [ a owl:Restriction; owl:onProperty q:hasPrerequisite; owl:allValuesFrom q:SelectedAnswer]. Hence c:WhatProblem a q:EnabledQuestion.

In summary, this holds since all of this problem’s prerequisites have been selected  Vacuously true since it has no prerequisites Note:  x  A(x)  B(x) is true if it’s true for every substitution instance x / a, where a denotes an individual  A(a)  B(a) is true if A(a) is false  If A(a) is false for every individual a, then A(a)  B(a) is true for every individual  I.e.,  x  A(x)  B(x) is true when nothing satisfies A(x)

Counting Prerequisites Also determine that something satisfies its prerequisites by counting how many there are (as with the James Dean movies) Suppose something has 1 prerequisite d:TvSymptom a [ a owl:Restriction; owl:onProperty hasPrerequisite; owl:cardinality 1]. And suppose we know 1 prerequisite and its type d:TvSymptom q:hasPrerequisite d:STV. d:STV a q:SelectedAnswer. One prerequisite is a member of q:SelectedAnswer and there aren’t any other prerequisites  So all prerequisites are members of q:SelectedAnswer d:TVSymptom a [ a owl:Restriction; owl:onProperty q:hasPrerequisite; owl:allValuesFrom q:SelectedAnswer].

Can make inferences with larger counts (as with the James Dean movies) if all entities are different Given d:TVTurnedOn a [ a owl:Restriction; owl:onProperty q:hasPrerequisite; owl:cardinality 2]. d:TVTurnedOn q:hasPrerequisite d:TVSnothing. d:TVTurnedOn q:hasPrerequisite d:STVSnosound. d:TVSnothing owl:differentFrom d:STVSnosound. d:TVSnothing a q:SelectedAnswer. d:STVSnosound a q:SelectedAnswer. infer (since only 2 prerequisites and we know which they are) d:TVTurnedOn a [ a owl:Restriction; owl:onProperty q:hasPrerequisite; owl:allValuesFrom q:SelectedAnswer].

Guarantees of Existence Recall: Membership of individual A in an owl:allValuesFrom restriction on property P doesn’t guarantee existence of a triple of the form A P ?x. If A is a member of a restriction owl:onProperty P owl:someValuesFrom class C, can infer there’s a pair of triples A P ?x. ?x rdf:type C.  So owl:someValuesFrom does guarantee the existence of some value for the given property owl:hasValue not only guarantees such a triple but even specifies it  I.e., if A is a member of the restriction owl:onProperty P owl:hasValue X, can infer A P X.

Contradictions Challenge 33 Model the following situation and conclusion in OWL Rocky: I have a couple of Judy Garland movies. Rimbaud: But you said you own only James Dean movies, and he and Judy Garland weren’t in any movie together.

This introduces a new aspect of inferencing in OWL In the simplest form of inferencing, we infer triples based on asserted ones With notions more advanced than RDFS-Plus,  some inferences themselves can’t be represented as triples  but can result in new triples when combined with other assertions But here no new triples are inferred at all  Rimbaud found a contradiction in what Rocky said

The contradiction arises because Rocky states the following :JamesDeanExclusive owl:equivalentClass [ a owl:Restriction; owl:onProperty :ownsMovie; owl:allValuesFrom :JamesDeanMovie]. :Rocky a :JamesDeanExclusive. :Rocky a [ a owl:Restriction; owl:onProperty :ownsMovie; owl:someValuesFrom :JudyGarlandMovie]. :JudyGarlandMovie owl:disjointWith :JamesDeanMovie. Figure 7 (next slide) shows the assertions (no inferences)

The OWL semantics can tell us that there’s a contradiction but not which assertion is wrong  The truth of an assertion relates, not to OWL semantics, but to the domain being modeled A model may fail to correspond to the actual state of affairs  If this is because it’s logically inconsistent (so couldn’t correspond to any state of affairs), the tools surrounding OWL can tell us so

Unsatisfiable Classes A contradiction involves a set of assertions that can’t all be true at the same time A similar situation arises when we define a class in an inconsistent way Illustrate this with a slight variation on the previous example  First define the class of people owning Judy Garland movies :JudyGarlandMovieOwner owl:equivalentClass [ a owl:Restriction; owl:onProperty :ownsMovie; owl:someValueFromm :JudyGarlandMovie].  Now, instead of claiming that Rocky is a member of this class and JamesDeanExclusive, define a class of such people :JDJG owl:intersectionOf ( :JudyGarlandMovieOwner :JamesDeanExclusive ).

We can define this class without asserting that Rocky (or anything else) is a member, hence without involving a contradiction  But the same argument showing that Rocky can’t be a member shows that nothing can be a member of this class Not only is the class empty, it must be empty—it (or the description of what it contains, which implicitly involves a variable) is unsatisfiable  Asserting that something is a member of an unsatisfiable class introducers a contradiction into the model

Figure 8 shows the assertions and conclusions JDJG, being the intersection of JudyGarlandMovieOwner and JamesDeanExclusive, is a subclass of both  It’s inferred to be a subclass of owl:Nothing, the necessarily empty class

Propagation of Unsatisfiable Classes An unsatisfiable class easily leads to other unsatisfiable class definitions subclass If a class can’t have any members, then no subclass of it can have a member (which would thereby be a member too of the class) someValuesFrom A restriction with owl:someValuesFrom an unsatisfiable class is itself unsatisfiable domain and range If a property has an unsatisfiable domain or range, it’s unusable  Asserting any triple using the property as predicate results in a contradiction

intersection of disjoints The intersection of 2 disjoint classes is unsatisfiable The intersection of any class with an unsatisfiable class is unsatisfiable Some operations don’t propagate unsatisfiability  The union of class C with an unsatisfiable class is just C  A restriction owl:allValuesFrom an unsatisfiable class can be satisfiable— vacuously so: no member can have the property in question The usefulness of these rules in modeling comes during analysis of the results of an inference engine  Many inference engines report on unsatisfiable classes  But, with several such classes, it’s hard to unravel the threads

Inferring Class Relationships Earlier focused on drawing inferences about individuals but have shifted to inferences about classes Some of the more common of the many patterns are the following  Use set notation for brevity (and clarity) Intersections and subclass A  B  A, A  B  B Union and subclass A  B  A, A  B  B Complement and subclass If A  B then  A   B

Subclass propagation through restriction owl:allValuesFrom  If A  B then { x |  y  P(x,y)  y  A }  { x |  y  P(x,y)  y  B } owl:someValuesFrom  If A  B then { x |  y  P(x,y)  y  A }  { x |  y  P(x,y)  y  B } hasValue, someValuesFrom, and subClassOf If a  A then { x | P(x,a) }  { x |  y  P(x,y)  y  A } Relative cardinalities For natural number c, d where d > c, { x | card({ y | P(x,y ) } ) = d }  { x | card({ y | P(x,y ) } )  c }

Examples Subclass propagation through restriction If :AllStarBaseballTeam is a subclass of :BaseballTeam, then the restriction (on any property, e.g., :playsFor ) owl:allValuesFrom :AllStarBaseballTeam is a subclass of the restriction (on the same property :playsFor ) owl:allValuesFrom :BaseballTeam hasValue, someValuesFrom, and subClassOf Suppose individual :TokyoGiants is a member of class :BaseballTeam The restriction on property :playsFor owl:hasValue :TokyoGiants is a subclass of the restriction on :playsFor owl:someValuesFrom :BaseballTeam

Relative cardinalities Suppose a :ViableBaseballTeam must have at least 9 players on its roster and a :FullBaseballTeam has exactly 10 players  :FullBaseballTeam is a subclass of :ViableBaseBallTeam owl:someValuesFrom and owl:minCardinality Suppose :BaseballTeam has some pitcher, i.e., :BaseballTeam is a subclass of the restriction owl:onProperty :hasPlayer owl:someValuesFrom :Pitcher Can infer that it has at least 1 pitcher, i.e., :BaseballTeam is a subclass of the restriction owl:onProperty :hasPlayer owl:minCardinality 1 Any class relationship that can be proven based on the semantics of the OWL constructs will be inferred by the logic underlying OWL  We’ve seen only the more common propagation patterns

Inferring class relationships in OWL is in sharp contrast to OO modeling  In OO modeling, all instances are created as members of some class Their behavior is specified by the class structure  In OWL, the class structure can change as more info is learned about classes and individuals These aspects of OWL are direct consequences of the basic assumptions about the web AAA slogan, Open World assumption, Nonunique Naming assumption A strict data model (e.g., an OO model) is useful when there’s top-down governance of the system  Doesn’t work in an open, free system Our understanding of the structure of knowledge changes as we discover new things  OWL provides a consistent and systematic way to understand those changes

OWL’s ability to infer class relationships enables a style of modeling where subclass relationships are rarely asserted directly  Instead, relationships between classes are described in terms of unions, intersections, complements, restrictions  And the inference engine determines the class structure  If more info is learned about a given class or individual, more class structure can be inferred Illustrate this principle with the baseball model—Table 1

Here  indicates class equivalence,  indicates subclass

Figure 9 shows the class tree asserted

Some subclass relationships inferred from the definitions in Table 1 and our subclass inferencing patterns AllStarBaseballTeam  BaseballTeam  AllStarTeam, so  AllStarBaseballTeam  BaseballTeam  AllStarBaseballTeam  AllStarTeam Both AllStarBaseballPlayer and AllStarPlayer are someValuesFrom restrictions on property playsFor, referencing, resp., AllStarBaseballTeam AllStarTeam  Propagate AllStarBaseballTeam  AllStarTeam to infer AllStarBaseballPlayer  AllStarPlayer  Similar reasoning lets us infer AllStarBaseballPlayer  BaseballPlayer

JBallTeam  PacificLeagueTeam  CentralLeagueTeam, so  PacificLeagueTeam  JBallTeam Hiroshima Carp is a Central League Team so also a JBallTeam and, further, a BaseballTeam  A CarpPlayer is a hasValue restriction on the Carp, so CarpPlayer  BaseballPlayer

AllStarPlayer is equivalent to the someValuesFrom restriction onProperty playsFor  So any individual member of AllStarPlayer playsFor some team But domain( playsFor )  Player So that individual is also a Player  Just shown AllStarPlayer  Player AllStarTeam employs some AllStarPlayer and employs is the inverse of playsFor  So some person playsFor any AllStarTeam But range( playsFor ) is Team  So AllStarTeam  Team

Figure 10 shows the inferred class structure Every class is involved in some class inferencing pattern  So the inferred structure (unlike the asserted one) has considerable depth

Reasoning with Individuals and Classes From an RDF perspective, inferencing about individuals and inferencing about classes are very similar  New triples are added to the model based on asserted triples From a modeling perspective, the 2 kinds of reasoning are very different  One draws specific conclusions about individuals in a data stream (A-box reasoning)  The other draws general conclusions about classes of individuals (T-box reasoning)

The utility of reasoning about individuals in a Semantic Web context is clear  Info specified in one source is transformed according to a model for use in another context  Mappings from one context to the next are specified with such constructs as rdfs:subClassOf, rdfs:subPropertyOf, and various owl:Restrictions  Data are then transformed and processed according to these models and the inferences specified in the RDFS and OWL standards for each

The utility of reasoning about classes is more subtle  Determines how data are related in general, even before any data is present  Records relationships with r dfs:subClassOf, r dfs:subPropertyOf, rdfs:domain, rdfs:range  Once these general relationships are inferred, processing individual data is much easier