Formalism (EARLY DRAFT)

THIS IS OUT OF DATE REGARDING THE CURRENT IMPLEMENTATION

A LinkML model M consists of:

  • slots MS = {s1, ...}
  • classes MC = {c1,...}
  • enums MEn = {En1, ...}
  • types MT = {t1, ...}
  • subsets MSu = {su1, ..}

Each slot s ∈ MS and each class c ∈ MC has an interpretation Δ:

  • Δ(s) ∈ P( {SubClassOf, ClassAssertion, SomeValuesFrom, AllValuesFrom, AnnotationAssertion, ObjectPropertyAssertion, AnnotationPropertyAssertion, UnionOf, DataOneOf, ObjectOneOf } )
  • Δ(c) ∈ P( {Class, Individual, ObjectProperty, DataProperty, AnnotationAssertion} )

(here P denotes the powerset, i.e. Δ maps to class or slot to a set of valid terms)

The interpretation dictates how elements that instantiate the model are translated to OWL-DL

Informative note:

  • any class c ∈ MC where Class ∈ Δ(c) is considered a metaclass
  • any class c ∈ MC where *Property ∈ Δ(c) is considered a metaproperty
  • in the linkml metamodel, the class ClassDefinition is a metaclass

A LinkML element e ∈ E is either:

  1. An instance of a class in MC
  2. A reference to an instance of a class in MC
  3. A literal taken from the universe of literals L
  4. A permissible value from the set of permissible values of a member of MEn

Note that the distinction between 1 and 2 only holds for tree representations such as YAML and JSON serializations.

Each e has a set of slot-value assignments A(e), where each assignment is a pair s, V

  • s is a slot in MS

  • V is a list of elements v1, ..., vn each of which can be:

    • if s.range ∈ MT: v_i must be a Literal
    • if s.range ∈ MC: v_i must be either:
      • another element in I OR
      • a reference to an element in I
      • (this distinction only holds for yaml/json tree representations)
    • if s.range ∈ MEn: v_i must be a member of the permissible value in ME

  • A(i) ⊆ { (s,V) ∈ MS x I ∪ Ref(I) ∪ ME ∪ L }

Any slot marked as identifier is not considered among the list of slots.

  • The identifier slot for a class c is marked c.id
  • The identifier value for an element e is marked e.id

Only elements that instantiate classes in MC can have identifiers. Not all instantiating elements have identifiers - instantiating elements without anonymous

Any linkml element can be translated using the rules below. The translation is recursive, i.e. mapping an element that has slot-value associations will invoke mappings on the values of these associations.

If the linkml element graph is a tree, then invoking mapping on the root will map the whole graph

Mapping elements

The function Tr(e) will translate an element e to an OWL entity, and as a side-effect will generate OWL axioms

Table 1: Translation of elements to OWL entities

condition returns
e.type ∈ MC and hasId(e) IRI(e.id)
e.type ∈ MC and noId(e) BlankNode()
e.type ∈ Ref(MC) IRI(e)
e.type ∈ MEn IRI(e.meaning)
e.type ∈ T Literal(e)

Generation of OWL axioms

Invoking Tr(e) will additionally generate further OWL axioms:

  • for all (s,V) in A(e), a mapping Tr(e,s,V) is applied

See mapping slot values below.

Generation of OWL declarations

TODO

Function Δ(e.type) generates cond
C(i) * Class(IRI(i.id)) i.id exists
OP(i) * ObjectProperty(IRI(i.id)) i.id exists
DP(i) * DataProperty(IRI(i.id)) i.id exists
AP(i) * AnnotationProperty(IRI(i.id)) i.id exists
*|*|BlankNode(i)|i.id` not exists

Mapping slot values: Tr(i,s,V)

interpretation of an element i with slot s with values V=v1..vn

Table 2: single-valued or conjunctive lists

If s is not multivalued OR {UnionOf, DataOneOf, ObjectOneOf} ∩ Δ(s) = {} then the following table generates an axiom for all v ∈ V

Δ(e.type) Δ(s) expression cond
* SubClassOf ∈ Δ C(e) ⊑ C(v)
* SubObjectPropertyOf ∈ Δ OP(e) ⊑ OP(v)
* ClassAssertion ∈ Δ I(e) : C(v)
Class SomeValuesFrom ∈ Δ C(e) ⊑ ∃ OP(s) C(v)
Class AllValuesFrom ∈ Δ C(e) ⊑ ∀ OP(s) C(v)
Individual SomeValuesFrom ∈ Δ I(e) : ∃ OP(s) C(v)
Individual AllValuesFrom ∈ Δ I(e) : ∀ OP(s) C(v)
* AnnotationAssertion ∈ Δ OR Δ = {} Ann(P(s) IRI(e) Tr(v))
* ObjectPropertyAssertion ∈ Δ OPA(OP(s) IRI(e) Tr(v))
* DataPropertyAssertion ∈ Δ DPA(DP(s) IRI(e) Tr(v))

See DL article on Wikipedia for explanation of symbols

multi-valued or disjunctive lists

If s is multivalued AND {UnionOf, DataOneOf, ObjectOneOf} ∩ Δ(s) ≠ {} then apply the following steps, and then apply table 2 rules

Δ(s) Tr(V)
UnionOf ∈ Δ UnionOf(C(v1), ..., C(vn))
ObjectOneOf ∈ Δ ObjectOneOf(I(v1), ..., I(vn))
DataOneOf ∈ Δ DataOneOf(D(v1), ..., D(vn))

pre-processing

For uncommitted slots that have a slot iri with a pre-existing interpretation, these are used:

  • If s.iri = rdf:type, and Δ(s) = {} then set Δ(s) = {ClassAssertion}
  • If s.iri = rdfs:subClassOf, and Δ(s) = {} then set Δ(s) = {SubClassOf}
  • etc TODO