OWL Datatypes: Design and Implementation Boris Motik and Ian Horrocks University of Oxford

2/22 Contents Introduction The Datatype System of OWL 2 The Datatypes of OWL 2 A Modular Datatype Checker Conclusion

3/22 Problems with Datatypes in OWL 1 Datatypes of OWL 1 are based on XML Schema (XSD) Problems with OWL 1 datatypes:  too few normative ones  no user-defined datatypes (e.g., intervals)  reasoning with some XSD datatypes is difficult  some XSD datatypes have an inappropriate semantics  there are datatype-less constants  certain semantic aspects are unclear  reasoning algorithms are unclear

4/22 Motivation OWL 2: a new version of OWL  considerably improves the datatype system of OWL Our results ensure that…  …the datatype system of OWL 2 is extensible  …certain language extensions are correctly defined  …OWL 2 supports datatypes that are practically feasible  …we know how to implement the datatypes of OWL 2  Make datatypes in OWL 2 better  Provide guidance for implementors

5/22 Contents Introduction The Datatype System of OWL 2 The Datatypes of OWL 2 A Modular Datatype Checker Conclusion

6/22 Datatype Map Each datatype d is described by:  a URI – gives the name of the datatype  a set of constants N C (d)  a set of facets pairs N F (d)  a value space (d) D  a data value (c) D 2 (d) D for each constant c  a facet value (f) D µ (d) D for each facet f Example: real  facets: x, · x, ¸ x, int Example: str  facets: h minLength n i, h maxLength n i, h length n i, h pattern “regExp” i

7/22 Data Ranges Facet expression: Boolean formula over facets  e.g., ¸ 5 Æ · 10 Datatype restriction: d[  ]  d is a datatype and  is a facet expression for d  e.g., real[ int Æ ¸ 5 Æ · 10 ]  OWL 2 Syntax: DatatypeRestriction( xsd:integer xsd:minInclusive “5”^^xsd:integer xsd:maxInclusive “10”^^xsd:integer ) Data range: > D, d[  ], { v 1, …, v n }, dr  will be extended in OWL 2 to all Boolean connectives

8/22 Using Data Ranges in Restrictions New datatype constructs:  qualified number restrictions  disjoint data properties Semantics is defined w.r.t. a datatype domain M D

9/22 Openness of the Datatype Domain M D is usually fixed in DL reasoning  datatype groups: M D is exactly the union of all value spaces Problem: adding new datatypes can change the meaning of certain axioms Example: > v 8 U.< 5 t 9 U.real  if real is the only datatype, then this axiom is a tautology  if we have both real and str, it is not a tautology  We do not fix M D in OWL 2  an ontology is satisfiable iff M D exists that at least contains the value spaces of all datatypes and for which all axioms are satisfied Proposition: consequences of OWL 2 ontologies are independent of the supported set of datatypes

10/22 Naming Data Ranges Teens ´ real[ int Æ > 12 Æ < 20 ]  semantics: (Teens) D = (real[ int Æ > 12 Æ < 20 ]) D  use Teens as a shortcut  e.g., Teenager ´ 9 hasAge.Teens Problem: we can write axioms about datatypes  A ´ real and A ´ > D  fixes M D to (real) D  prevents us from extending the set of datatypes  Make such axioms acyclic  each data range name can be defined only once and its definition cannot refer to itself  allows for simple unfolding of data range names

11/22 Datatype Reasoning Datatype checker decides satisfiability of conjunctions over assertions dr(t) and t 1 ¼ t 2  t (i) is a variable or a constant  example: { 5 }(x 1 ) Æ int[ > 4 Æ < 6 ](x 2 ) Æ x 1 ¼ x 2 Datatype checker can be integrated with a (hyper)tableau algorithm as usual Proposition: datatype checking is NP-hard  uses data property disjointness  seems like an innocuous feature!  even small additions to the language add complexity

12/22 Contents Introduction The Datatype System of OWL 2 The Datatypes of OWL 2 A Modular Datatype Checker Conclusion

13/22 Numeric Datatypes The following ontology is unsatisfiable:  > v 8 hasWeight.xsd:double  hasWeight(Paul, “76”^^xsd:integer)  in XSD, the integer 76 is not contained in xsd:double  no notion of typecasts in OWL XML Schema does not have real numbers  OWL 2 redefines XSD numeric datatypes  owl:realPlus = owl:real [ { -0, +inf, -inf, NaN }  owl:real is the set of all real numbers  all XSD numeric datatypes are subsets of owl:real  facets:  minExclusive, maxExclusive, minInclusive, maxInclusive

14/22 String Datatypes Plain RDF literals with a language tag do not belong to any XSD datatype  vs.  OWL 2 uses a new rdf:text datatype  value space contains pairs h string, languageTag i  will be used in RIF as well  xsd:string was retrofitted to rdf:text  value space contains pairs h string, “” i  The set of characters is assume to be infinite  E.g., ¸ n U.(str[ length 1])(a) is satisfiable iff n · m, where m is the number characters  m will change in future, which could change the meaning of this axiom

15/22 Other Datatypes Date/time:  many XSD date/time datatypes are difficult to reason with  e.g., xsd:gMonthDay represents a recurring point in time but recurrences are irregular due to leap seconds and years  XSD supports dates without time zones  OWL 2 supports only xsd:dateTime with required time zone  facets: minExclusive, maxExclusive, minInclusive, maxInclusive xsd:boolean xsd:hexBinary and xsd:base64Binary xsd:anyURI  disjoint with xsd:string

16/22 Contents Introduction The Datatype System of OWL 2 The Datatypes of OWL 2 A Modular Datatype Checker Conclusion

17/22 Modular Datatype Checking We assume that all datatypes are disjoint  xsd:integer is understood as a facet of owl:real  provides us with a natural modularization boundary Each datatype d needs a datatype handler:  minc d (d[  ], n)  true iff (d[  ]) D contains at least n elements  enu d (d[  ])  defined only if (d[  ]) D is finite  enumerates the extension of d[  ]  in d (c, d[  ])  true iff c D 2 (d[  ]) D  eq d (c 1, c 2 )  true iff c 1 D = c 2 D

18/22 The Algorithm Input: a conjunction  of assertions Output: true iff the conjunction is satisfiable 1.Normalize  such that each variable x in it occurs in exactly one assertion d[  ](x) 2.Simplify   delete from  assertions containing certain variables  in all remaining assertions of the form d[  ](x), the data range d[  ] is finite 3.Replace d[  ](x) with D(x) for D = enu d (d[  ]) 4.Guess values for all variables 5.Check whether the guess satisfies  Can be reduced to SAT

19/22 If  contains a variable x such that  x occurs in  in exactly one assertion d[  ](x),  x occurs in  in m assertions of the form x ¼ x’,  x occurs in  in n assertions of the form x ¼ c, and  minc d (d[  ], m+n+1) = true then delete in  all assertions containing x  If | (d[  ]) D | ¸ m+n+1, then we can satisfy x for any choice of values for x’  the constraints on x are irrelevant for the satisfiability of  Key to practical reasoning:  data ranges in practice are likely to be large (even infinite) The Simplification Step

20/22 Handling Numbers and Strings Numbers:  represent facets as intervals of the form dt(low, high)  facet expressions can be normalized using a suitable interval algebra Strings:  represent facets as regular languages  facet expressions can be normalized using standard results for Boolean operations with regular languages  caveat: the underlying alphabet is infinite  need to adapt Boolean operations on regular languages In both cases, datatype handlers are easily implemented for normalized expressions

21/22 Contents Introduction The Datatype System of OWL 2 The Datatypes of OWL 2 A Modular Datatype Checker Conclusion

22/22 Conclusion The algorithm has been implemented in the HermiT reasoner  a new OWL 2 reasoner based on hypertableau  No formal evaluation yet, but… Supporting datatypes did not noticeably change classification times  data ranges used in practice are often “large enough”