Integrating OWL and Rules: A Syntax Proposal for Nominal Schemas

Abstract

This paper proposes an addition to OWL 2 syntax to incorporate nominal schemas, which is a new description-logic style extension of OWL 2 which was recently proposed, and which makes is possible to express \variable nominal classes" within axioms in an OWL 2 ontology. Nominal schemas make it possible to express DL-safe rules of arbitrary arity within the extended OWL paradigm, hence covering the well-known DL-safe SWRL language. To express this feature, we extend OWL 2 syntax to include necessary and minimal modi cations to both Functional and Manchester syntax grammars and mappings from these two syntaxes to Turtle/RDF. We also include several examples to clarify the proposal.

Introduction

Nominal schemas [ 4,5 ] are a new description-logic style extension of OWL 2 [ 10 ] which can be used like \variable nominal classes" within OWL 2 axioms. Although their semantics restricts them only to stand for named individuals, nominal schemas allow us to express arbitrarily shaped (Datalog) rules within the description logic (DL) paradigm, hence pushing the expressivity of OWL 2 DL and its fragments even further.

While the semantic intuition behind nominal schemas is the same as the one behind DL-safe variables presented in [ 8 ], the di erence lies in the fact that DL-safe variables are tied to rule languages, while nominal schemas integrate seamlessly with DL syntax. The proposed extension encompasses DL-safe variable SWRL [ 3,9,4 ] while staying within the DL/OWL language paradigm and without employing hybrid approaches.

Nominal schemas have been introduced as a new general constructor for DL, denoted by the letter V in the DL nomenclature. The addition of nominal schemas has been considered for several DLs such as SROIQ that underlies OWL 2 DL, and SROE L that underlies the OWL 2 EL pro le (de ne DL SROIQV and SROE LV , respectively, as extensions of the DLs SROIQ and SROE L). It has been shown in [ 5 ] that the worst-case complexity of SROIQV remains N2EXPTIME-complete, i.e., no worse than SROIQ. Furthermore, in the same paper, a tractable fragment of SROIQV has been identi ed. This fragment is called SROE LV n which is obtained by extending SROE L with nominal schemas in a slightly restricted form. Nevertheless, it still covers1 both OWL 2 EL and (DL-safe) OWL 2 RL.

We present an example of nominal schemas in the following. First, rules such as (1) are not expressible in the current OWL 2 DL standard. hasFather(x;y) ^ hasBrother(y;z) ^ hasTeacher(x;z) ! ChildTaughtByUncle(x) (1) Intuitively, this is due to the fact that the body of the above rules is not treeshaped. See [ 4 ] and [ 6 ] (which formally de nes not tree-shaped rules that can be expressed in SROIQ extended with role regularity) for further discussion. In contrast, using nominal schemas, rule (1) can be expressed as (2). 9hasTeacher:fzg u 9hasFather:9hasBrother:fzg v ChildTaughtByUncle: (2) The expression fzg is a nominal schema, which is to be read as a variable nominal that can only represent nominals (i.e., z binds to known individuals), where the binding is the same for all occurrences of the same nominal schema in an axiom. Variables x and y can still take arbitrary values and are hidden in the DL syntax, z needs to be restricted to be DL-safe to retain the conclusion. For a more detailed description of nominal schemas including their formal semantics see [ 5 ].

This document proposes representations for nominal schemas for the prominent variants of OWL syntax: Functional, Manchester, Turtle and RDF/XML. For an introduction of the OWL syntax, consult [ 10 ]. Mapping from Turtle triples to RDF/XML is a well de ned and automatized process so the RDF/XML based syntax will not be directly addressed in this document, it is assumed that it can be easily derived from the Turtle Syntax.

New reserved words are presented to mark the appearance of nominal schemas in the di erent syntax variants (Functional, Manchester and Turtle) as well as the necessary modi cations to their grammars (Functional and Manchester). The representation of nominal schemas in Turtle syntax is de ned by the mappings from Functional and Manchester.

Several approaches were considered for the representation and storage of nominal schemas, such as the use of entities with the ontology namespace, but this paper proposes the use of string literals. With this approach, we prevent the possible overlap that could be produced by giving the same name to two di erent nominal schemas. If these are declared as entities and, by error, two of them share the same name, they will end up pointing to the same node in an RDF graph when they most likely refer to di erent individuals.

The selected approach, the use of a xsd:string datatype, is also considered by the RIF XML format [ 11 ]. Note that the same nominal schema can never appear in two di erent statements of an ontology|more precisely, if the same variable occurs in di erent axioms, then they are considered distinct (i.e., local to the axiom), in a way similar to the use of variables in rules. So a speci c nominal schema is local to one single axiom. By using a string type, the occurrence of 1 without datatype-related features the nominal schema is exclusively bound to the axiom where it appears and the same string could be repeated in di erent axioms along the ontology safely. Even if two nominal schemas use the same string, they will be considered as di erent occurrences of a datatype and therefore, they can be understood as two separated nodes in an RDF graph.

Using the underscore to mark the appearance of a nominal schema, as it is done for Turtle blank nodes, was also considered. This approach was rejected because it could induce errors. Although in some cases both nominal schemas and blank nodes can represent individuals in an RDF graph they are completely di erent concepts. Using the underscore to mark both could be tricky and would make mappings from and to Turtle syntax di cult to de ne. With such a similar syntax the mapping may produce errors confusing nominal schemas with blank nodes and problems may arise when we want to move from the Turtle syntax to an RDF Graph.

The document is structured as follows. Section 2 contains the necessary modi cations that have to be made to the Manchester and Functional Syntax grammars in order to include nominal schemas. Section 3 refers to the mappings from these syntaxes to Turtle. Section 4 concludes. Appendix A contains two examples for the usage of nominal schemas in the di erent syntax variants that are discussed in the document. 2

Grammar Modi cations

We propose several changes to the grammars of the di erent OWL syntaxes in order to include nominal schemas. The presented changes are designed to be minimal and imply very small modi cations to the formal de nitions of these grammars.

Functional Syntax Grammar Modi cations

We de ne in this section the required modi cations we propose for the Functional Syntax grammar [ 1 ]. The reserved word ObjectVariable will be used to mark the appearance of a nominal schema. Nominal schemas will be in parentheses and will always be followed by the expression '^^xsd:string'. The changes are formally de ned in the next paragraphs.

Add the next production rule to the grammar: ObjectVariable := 'ObjectVariable (' quotedString'^^xsd:string)' Add the non-terminal symbol ObjectVariable, to the ClassExpression next production rule:

ClassExpression := Class j ObjectIntersectionOf j ObjectUnionOf j ObjectComplementOf j ObjectOneOf j ObjectSomeValuessFrom j ObjectAllValuesFrom j ObjectHasValue j ObjectHasSelf j ObjectMinCardinality j ObjectMaxCardinality j ObjectExactCardinality j DataSomeValuesFrom j DataAllValuesFrom j DataHasValue j DataMinCardinality j DataMaxCardinality j

DataExactCardinality j ObjectVariable

Although nominal schemas are not conceptually class expressions, their addition in this part of the grammar has been chosen in order to keep the modi cations as small as possible.

Manchester Syntax Grammar Modi cations

Again, the reserved word ObjectVariable will be used to mark the appearance of the nominal schemas in the Manchester Syntax [ 2 ]. As in the Functional Syntax, the nominal will be in parentheses and followed by '^^xsd:string'. The needed changes to this grammar are: Add the non-terminal symbol ObjectVariable to the atomic production rule: atomic := classIRI j 'f'individualList'g' j '('description')' j

ObjectVariable

Add the next production rule to the grammar:

ObjectVariable := 'ObjectVariable (' quotedString'^^xsd:string)' 3

Mapping FS and MS to Turtle

We de ne the syntax of nominal schemas in Turtle through the mapping from Functional and Manchester Syntaxes to the triple-notation. We assume that from this notation the process to move to RDF/XML is already formalized so, as said before, the XML syntax will not be directly addressed in this document.

Functional Syntax to and from Turtle

The W3C document containing the formal mapping from FS to Turtle can be found in [ 7 ]. To add nominal schemas syntax to the mappings, rst add the next row to the mapping from FS to Turtle:

Functional-Style Syntax S Triples Generated Main Node in an Invocation of T(S) of T(S) ObjectVariable("v1"^^xsd:string) :x rdf:type owl:ObjectVariable :x :x owl:variableId "v1" Then add the next row to the mapping from Turtle to FS:

RDF/XML Triples Functional Syntax :x rdf:type owl:ObjectVariable ObjectVariable("v1"^^xsd:string) :x owl:variableId "v1"

Manchester Syntax to and from Turtle

The mappings between Manchester Syntax and Turtle are de ned in a similar way as the one from the Functional Syntax. To include nominal schemas in this mapping, we rst need to add the next row to the table from MS to Turtle: Manchester-Style Syntax S Triples Generated Main Node in an Invocation of T(S) of T(S) Variable "v1"^^xsd:string :x rdf:type owl:ObjectVariable :x

:x owl:variableId "v1" Then add the next row to the mapping from Turtle to FS:

RDF/XML Triples Manchester Syntax :x rdf:type owl:ObjectVariable Variable "v1"^^xsd:string :x owl:variableId "v1"

4

Conclusions

In this document we propose ways for representing nominal schemas in the different syntaxes of the OWL language. Reserved words have been provided for Functional, Manchester, Turtle and RDF/XML syntaxes, along with the consistent modi cations to their grammars and mapping functions. Nominal schemas will be stored as string values in the OWL syntaxes to prevent overlapping errors. In the appendix of this document two examples are presented showing nominal schemas across the di erent covered syntaxes of OWL.

Acknowledgements: This work was partially supported by the National Science Foundation under award 1017225 \III: Small: TROn|Tractable Reasoning with Ontologies." The rst author acknowledges support from Programa de Intercambio de la Universidad Ponti cia de Salamanca 2010/11. The second author acknowledges support by a Fulbright Indonesia Presidential Scholarship PhD Grant 2010. A.1

Rule Syntax

Syntax Examples

Example 1

hasFather(x; y)^hasBrother(y; z)^hasTeacher(x; z)^ ! ChildTaughtByUncle(x)

Functional Syntax

SubClassOf( ) :ChildTaughtByUncle )

RDF/XML Syntax

9hasFather:(9hasBrother:fzg) u 9hasTeacher:fzg v ChildTaughtByUncle

ObjectIntersectionOf(

ObjectSomeValuesFrom( :hasFather

ObjectSomeValuesFrom( :hasBrother ObjectVariable("v1"^^xsd:string) )) ObjectSomeValuesFrom( :hasTeacher ObjectVariable("v1"^^xsd:string) ) :x1 rdfs:subClassOf :ChildTaughtByUncle :x1 rdf:type owl:Class :x1 owl:intersectionOf ( :x2 :x3) :x2 rdf:type owl:Restriction :x2 owl:onProperty :hasFather :x2 owl:someValuesFrom :x5 :x4 rdf:type owl:Restriction :x4 owl:onProperty :hasBrother :x4 owl:someValuesFrom :x6 :x5 rdf:type owl:ObjectVariable :x5 owl:variableId "v1" :x3 rdf:type owl:Restriction :x3 owl:onProperty :hasTeacher :x3 owl:someValuesFrom :x4

Example 2

Rule Syntax

hasReviewAssignment(v; x) ^ hasAuthor(x; y) ^ atVenue(x,z) ^ hasSubmittedPaper(v; u) ^ hasAuthor(u; y) ^ atVenue(u; z) ! ReviewerWithCon ictingAssignment(v)

DL Syntax

9hasReviewAssignment:(9hasAuthor:fag u 9atVenue:fbg) u 9hasSubmittedPaper:(9hasAuthor:fag u 9atVenue:fbg) v ReviewerWithCon ictingAssignment

Functional Syntax

SubClassOf(

ObjectIntersectionOf(

ObjectSomeValuesFrom ( :hasReviewAssign ObjectIntersectionOf ( ObjectSomeValuesFrom (:hasAuthor ObjectVariable("v1"^^xsd:string)) ObjectSomeValuesFrom (:atVenue ObjectVariable("v2"^^xsd:string)) ) ) ObjectSomeValuesFrom ( :hasSubmittedPaper ObjectIntersectionOf ( ObjectSomeValuesFrom (:hasAuthor ObjectVariable("v1"^^xsd:string)) ObjectSomeValuesFrom (:atVenue ObjectVariable("v2"^^xsd:string)) ) )

) :ReviewerWithCon ictingAssignment ) RDF/XML Syntax :x1 rdfs:subClassOf :ReviewerWithCon ictingAssignment :x1 rdf:type owl:Class :x1 owl:intersectionOf ( :x2 :x3) :x2 owl:onProperty :hasReviewAssign :x3 owl:onProperty :hasSubmittedPaper :x2 owl:intersectionOf ( :x4 :x5) :x3 owl:intersectionOf ( :x8 :x9) :x6 rdf:type owl:ObjectVariable :x6 owl:variableId "v1"

Manchester Syntax

Class: ReviewerWithCon ictingAssignment SubtClassOf: ( hasReviewAssign some ( (hasAuthor some (Variable "v1"^^xsd:string)) and (atVenue some (Variable "v2"^^xsd:string)) ) ) and ( :hasSubmittedPaper some ( (hasAuthor some (Variable "v1"^^xsd:string)) and (atVenue some (Variable "v2"^^xsd:string)) ) )

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