Annotation algebras for RDFS

Transcription

1 University of Edinburgh Semantic Web in Provenance Management, 2010

2 RDF definition U a set of RDF URI references (s, p, o) U U U an RDF triple s subject p predicate o object A finite set of triples an RDF graph

3 RDFS vocabulary RDF Schema a vocabulary of URI s with special semantics: rdfs:resource, rdfs:class, rdfs:literal, rdfs:subclassof, rdfs:subpropertyof, rdfs:domain, rdfs:range, rdf:, rdf:list,... ρdf [S. Muñoz, et al.] a fragment of RDFS vocabulary: sc, sp, dom, range,

4 RDFS vocabulary RDF Schema a vocabulary of URI s with special semantics: rdfs:resource, rdfs:class, rdfs:literal, rdfs:subclassof, rdfs:subpropertyof, rdfs:domain, rdfs:range, rdf:, rdf:list,... ρdf [S. Muñoz, et al.] a fragment of RDFS vocabulary: sc, sp, dom, range,

5 RDFS deductive system Sub-class: (A, sc, B) (B, sc, C) ; (A, sc, C) Sub-property: (A, sp, B) (B, sp, C) ; (A, sp, C) Typing: (X, A, Y) (A, dom, B) ; (X,, B) (X,, A) (A, sc, B). (X,, B) (X, A, Y) (A, sp, B). (X, B, Y) (X, A, Y) (A, range, B). (Y,, B)

6 Example of RDF graph {(Picasso, Paints, Guernica), (Picasso,, Cubist), (Paints, dom, Painter), (Cubist, sc, Painter) } (Picasso,, Painter)} Cubist sc Painter dom Picasso Paints Guernica

7 Example of RDF graph {(Picasso, Paints, Guernica), (Picasso,, Cubist), (Paints, dom, Painter), (Cubist, sc, Painter), } (Picasso,, Painter)} Cubist sc Painter dom Picasso Paints Guernica

8 Annotations for RDF Named graphs [J. Caroll, et al.] trust annotations, Temporal RDF [C. Gutierrez, et al.] temporal annotations, Colored RDF [G. Flouris, et al.] provenance annotations, Fuzzy RDF [U. Straccia] probabilistic annotations. In all of these works the fourth column introduced to store an annotation information about an RDF triple: (s, p, o) : v Last two of them introduce approaches how annotations pass through the RDFS inference rules.

9 Annotations for RDF Named graphs [J. Caroll, et al.] trust annotations, Temporal RDF [C. Gutierrez, et al.] temporal annotations, Colored RDF [G. Flouris, et al.] provenance annotations, Fuzzy RDF [U. Straccia] probabilistic annotations. In all of these works the fourth column introduced to store an annotation information about an RDF triple: (s, p, o) : v Last two of them introduce approaches how annotations pass through the RDFS inference rules.

10 Examples of annotated RDF graphs Cubist sc Painter dom Picasso Paints Guernica ({ } { }) ({1937} { 29, 000 Now}) = { , 1937} max( , 0.3 1) = 0.32

11 Examples of annotated RDF graphs Cubist sc : [ ] Painter : [ ] : [ , 1937] dom : [-29,000-Now] Picasso Paints : [1937] Guernica ({ } { }) ({1937} { 29, 000 Now}) = { , 1937} max( , 0.3 1) = 0.32

12 Examples of annotated RDF graphs sc Cubist Painter : 0.4 : 0.8 : 0.32 dom : 1 Picasso Paints : 0.3 Guernica ({ } { }) ({1937} { 29, 000 Now}) = { , 1937} max( , 0.3 1) = 0.32

13 Questions of this work What is similar in this examples? U. Straccia: Residuated Lattice What are the requirements for an annotation domain to be compatible with RDFS inference rules? Can we build the universal annotation domain such that any annotation can be obtained from it via a homomorphism?

14 Questions of this work What is similar in this examples? U. Straccia: Residuated Lattice What are the requirements for an annotation domain to be compatible with RDFS inference rules? Can we build the universal annotation domain such that any annotation can be obtained from it via a homomorphism?

15 Questions of this work What is similar in this examples? U. Straccia: Residuated Lattice What are the requirements for an annotation domain to be compatible with RDFS inference rules? Can we build the universal annotation domain such that any annotation can be obtained from it via a homomorphism?

16 Questions of this work What is similar in this examples? U. Straccia: Residuated Lattice What are the requirements for an annotation domain to be compatible with RDFS inference rules? Can we build the universal annotation domain such that any annotation can be obtained from it via a homomorphism?

17 Connection to relational databases Similar questions for relational databases world: What are the requirements for an annotation domain to be compatible with positive relational algebra queries? What is the universal annotation domain such that any annotation can be obtained from it via a homomorphism? How-provenance [T. J. Green, G. Karvounarakis, V. Tannen]: annotation domain commutative semi-ring universal (free) structure semi-ring of positive polynomials over N

18 Connection to relational databases Similar questions for relational databases world: What are the requirements for an annotation domain to be compatible with positive relational algebra queries? What is the universal annotation domain such that any annotation can be obtained from it via a homomorphism? How-provenance [T. J. Green, G. Karvounarakis, V. Tannen]: annotation domain commutative semi-ring universal (free) structure semi-ring of positive polynomials over N

19 Annotated RDFS deductive system Sub-class: (A, sc, B) : v 1 (B, sc, C) : v 2 (A, sc, C) : v 1 v 2 ; (X,, A) : v 1 (A, sc, B) : v 2 (X,, B) : v 1 v 2. Sub-property: (A, sp, B) : v 1 (B, sp, C) : v 2 (A, sp, C) : v 1 v 2 ; (X, A, Y) : v 1 (A, sp, B) : v 2 (X, B, Y) : v 1 v 2. Typing: (X, A, Y) : v 1 (A, dom, B) : v 2 (X,, B) : v 1 v 2 ; (X, A, Y) : v 1(A, range, B) : v 2 (Y,, B) : v 1 v 2. Generalisation: (X, A, Y) : v 1 (X, A, Y) : v 2 (X, A, Y) : v 1 v 2.

20 Annotated RDFS deductive system Sub-class: (A, sc, B) : v 1 (B, sc, C) : v 2 (A, sc, C) : v 1 v 2 ; (X,, A) : v 1 (A, sc, B) : v 2 (X,, B) : v 1 v 2. Sub-property: (A, sp, B) : v 1 (B, sp, C) : v 2 (A, sp, C) : v 1 v 2 ; (X, A, Y) : v 1 (A, sp, B) : v 2 (X, B, Y) : v 1 v 2. Typing: (X, A, Y) : v 1 (A, dom, B) : v 2 (X,, B) : v 1 v 2 ; (X, A, Y) : v 1(A, range, B) : v 2 (Y,, B) : v 1 v 2. Generalisation: (X, A, Y) : v 1 (X, A, Y) : v 2 (X, A, Y) : v 1 v 2.

21 Annotated RDFS deductive system Sub-class: (A, sc, B) : v 1 (B, sc, C) : v 2 (A, sc, C) : v 1 v 2 ; (X,, A) : v 1 (A, sc, B) : v 2 (X,, B) : v 1 v 2. Sub-property: (A, sp, B) : v 1 (B, sp, C) : v 2 (A, sp, C) : v 1 v 2 ; (X, A, Y) : v 1 (A, sp, B) : v 2 (X, B, Y) : v 1 v 2. Typing: (X, A, Y) : v 1 (A, dom, B) : v 2 (X,, B) : v 1 v 2 ; (X, A, Y) : v 1(A, range, B) : v 2 (Y,, B) : v 1 v 2. Generalisation: (X, A, Y) : v 1 (X, A, Y) : v 2 (X, A, Y) : v 1 v 2.

22 Properties of annotation algebras (A, sc, B) : v 1 (B, sc, C) : v 2 (A, sc, C) : v 1 v 2 (Sub-class transitivity) = associativity of, (X, A, Y) : v 1 (X, A, Y) : v 2 (X, A, Y) : v 1 v 2 (Generalisation) = associativity and commutativity of,...

23 Dioids K = K,,,, is a dioid iff it is an idempotent semi-ring: semi-lattice by : (a b) c = a (b c), a b = b a, a = a, a a = a; monoid by : (a b) c = a (b c), a = a = a; distributivity over : a (b c) = (a b) (a c), (b c) a = (b a) (c a); -annihilation of : a = = a. A dioid is a -dioid iff: -annihilation of : a =.

24 Dioids K = K,,,, is a dioid iff it is an idempotent semi-ring: semi-lattice by : (a b) c = a (b c), a b = b a, a = a, a a = a; monoid by : (a b) c = a (b c), a = a = a; distributivity over : a (b c) = (a b) (a c), (b c) a = (b a) (c a); -annihilation of : a = = a. A dioid is a -dioid iff: -annihilation of : a =.

25 String dioids Σ an alphabet. Subsequence ordering u u on words in Σ : u = u 1 u 2... u n u = u 1 w 1 u 2 w 2... w n 1 u n. A finite set m Σ is an antichain if for all its elements are noncomparable. M[Σ] set of antichains. min(m) the set of minimal elements (w.r.t. ) of m. On M[Σ] are defined: m 1 + m 2 = min(m 1 m 2 ), m 1 m 2 = min({w 1 w 2 w 1 m 1, w 2 m 2 }). M[Σ] = M[Σ], +,,, {ɛ} string dioid over Σ.

26 Free -dioids Theorem Given a set of generators Σ the string dioid M[Σ] = M[Σ], +,,, {ɛ} is the free -dioid on Σ. That means, that for any -dioid K f = K,,,, and a valuation φ : Σ K there exists a unique homomorphism Eval φ : M[Σ] K such that for each a Σ holds Eval φ (a) = φ(a).

27 Application of string annotations Cubist sc Painter dom Picasso Paints Guernica

28 Application of string annotations Cubist sc : b Painter : a dom : d Picasso Paints : c Guernica Σ = {a, b, c, d}

29 Application of string annotations Cubist sc : b Painter : a : a b + c d dom : d Picasso Paints : c Guernica Σ = {a, b, c, d}

30 Application of string annotations Cubist sc : b : [ ] Painter : a : [ ] Picasso : a b + c d Paints : c : [1937] dom : d : [-29,000-Now] Guernica φ T : φ T (a) = [ ], φ T (b) = [ ], φ T (c) = [1937], φ T (d) = [ 29, 000 Now]

31 Application of string annotations Cubist sc : b : [ ] Painter : a : [ ] Picasso : a b + c d Paints : c : [1937] dom : d : [ , 1937] : [-29,000-Now] Guernica φ T : φ T (a) = [ ], φ T (b) = [ ], φ T (c) = [1937], φ T (d) = [ 29, 000 Now] Eval φt (a b + c d) = [ , 1937]

32 Application of string annotations sc Cubist : b : 0.4 Painter : a : 0.8 Picasso : a b + c d Paints : c : 0.3 dom : d : 1 Guernica φ P : φ P (a) = 0.8, φ P (b) = 0.4, φ P (c) = 0.3, φ P (d) = 1

33 Application of string annotations sc Cubist : b : 0.4 Painter : a : a b + c d : 0.8 : 0.32 Paints Picasso : c : 0.3 dom : d : 1 Guernica φ P : φ P (a) = 0.8, φ P (b) = 0.4, φ P (c) = 0.3, φ P (d) = 1 Eval φp (a b + c d) = 0.32

34 Discussion In both of the examples the instantiations of is commutative. In the work from relational databases world we are inspired of [T. J. Green, et al.] the product is also commutative. In our work the product in not commutative. Why? There is no evidence in the deductive system that it should be commutative. We do not say that it must be non-commutative, but may be non-commutative. In the paper we present an example of a form of default reasoning where we appear to need a non-commutative product.

35 Discussion In both of the examples the instantiations of is commutative. In the work from relational databases world we are inspired of [T. J. Green, et al.] the product is also commutative. In our work the product in not commutative. Why? There is no evidence in the deductive system that it should be commutative. We do not say that it must be non-commutative, but may be non-commutative. In the paper we present an example of a form of default reasoning where we appear to need a non-commutative product.

36 Thank You

37 Default RDFS... sc: Plantae sc: Linnaeus Pinophyta sc:... sc: Araucariaceae sc: sc: Noble Noble... Linnaeus... = = Noble Araucaria Wollemia sc: sc: A. araucana W. nobilis

38 Default RDFS... sc: Plantae sc: Linnaeus Pinophyta sc:... sc: Araucariaceae sc: sc: Noble Noble... Linnaeus... = = Noble Araucaria Wollemia sc: sc: A. araucana W. nobilis