Although the model-theoretic semantics of the languages used in the Semantic Web are crisps, the need arise to extend them to represent fuzzy data, in the same way fuzzy logic extend first-orderlogic. We will define a fuzzy counterpart of the RDF Model Theory for RDF (section 2) and RDF Schema (section 3). Last, we show how to implement the extended semantics in inference rules (section 4).
RDF has its own model-theoretic semantics, similar to that of rfist-order logic. To represent fuzzy data, we will denfie a syntactic and semantic extension of RDF, similar to the extension from first-order logic to fuzzy logic.
Even if fuzzy data can be simply seen as a juxtaposition of a triple and a number, the model-theoretic approach has well-known theoretical advantages.
We will try to be as plain as possible. Starting from RDF Syntax and RDF
Model Theory, we will make as few changes as possible. In the rest of the paper,
for the sake of brevity only the changes from RDF Semantics[
The RDF syntax must be extended to add to the triple hsubject, predicate,
objecti a value. Such a value can be taken as a real number in the interval [
This is not an extension from a 3-elements tuple to a 4-elements tuple as it may seem at a rfist glance. The added element has a syntactic nature different from the others: it is not an element of the domain of the discourse, but a property related to the formalism used by the language to represent uncertainty and vagueness.
The simple concrete syntax we define is as an extension of the EBNF of
N-Triples as given in [
N-Triples is a line-based, plain text format for encoding an RDF Graph, used for expressing RDF test cases. A statement has the form s p o., where s, p and o are respectively the subject, the predicate and the object of the statement. Our extended syntax add an optional prexfi n: to a statement in N-triple notation, where n is a decimal number representing the fuzzy truth-value of the triple. The use of decimal numbers instead of real numbers is only a limitation of the syntax and does not undermine the discussion.
The term triple, used in the EBNF for N-Triple, is replaced with the more generic term statement. Triple and statement are often used in semantic web literature as a synonym, but we prefer to use the latter to avoid confusion between a plain RDF statement (made actually of three parts) and a fuzzy RDF statement (that, although is still a triple semantically, is made up of four elements).
The fuzzy value is defined as optional. This way, the syntax is backwardcompatible; the intended semantics is that a statement with the form s p o. is equivalent to the statement 1: s p o.. With such a (syntactic only) default, we could take an inference engine implementing fuzzy RDF, let it parse plain RDF statements, and get the same results of a conventional RDF inference engine. Furthermore, as it would be clear in the description of fuzzy RDF inference rules (section 4), even the complexity of the computation would be of the same order.
We will not give an abstract syntax, nor a RDF/XML based syntax, as
they would not be useful. It can be shown that all “physical“ data (i.e., data
transmitted between host or processes) can be encoded using plain RDF reiefid
statements. The extended syntax will be used only in the paper to write down
the examples.
fuzzyNtripleDoc ::= line*
line ::= ws* ( comment | statement )? eoln
comment ::= ‘#’ ( character − ( cr | lf ) )*
statement ::= (value ws+)? subject ws+ predicate ws+ object ws* ‘.’ ws*
value ::= 1 | 0.[
encoding a language tag.
ws ::= space | tab
eoln ::= cr | lf | cr lf
space ::= #x20 /* US-ASCII space - decimal 32 */
cr ::= #xD /* US-ASCII carriage return - decimal 13 */
lf ::= #xA /* US-ASCII line feed - decimal 10 */
tab ::= #x9 /* US-ASCII horizontal tab - decimal 9 */
string ::= character* (with escapes as defined in section Strings of [
In this fuzzy counterpart, a couple hsubject, objecti has a membership degree to the extension of the predicate, given by the number prepended to the statement. The extension is not an ordinary set of couples anymore, but a fuzzy set of couples. In other words, a fuzzy RDF interpretation satisefis a statement n: s p o. if the membership degree of the couple, formed by the interpretation of the subject and the interpretation of the object, to the extension of the interpretation of the predicate, is greater or equal than n.
We have chosen not to make the mapping between vocabulary items and domain fuzzy. Instead, the membership of a resource to the domain is fuzzy. This is a step which poses some theoretical problems, in particular when we have to deal with properties in simple interpretations. In RDF interpretation, the property domain IP is a subset of the resource domain IR, so in fuzzy RDF interpretations would be enough to make IP a fuzzy subset of IR; in simple interpretations, instead, there is no formal relation between IP and IR, so when the mapping IS from URI references to (IR ∪ IP ) becomes fuzzy we need a further device. The chosen solution is to denfie a domain IDP for properties, so that IP is a fuzzy subset of IDP , and to modify the denfiition of IS to a mapping URI references ∈ V → (IR ∪ IDP ). RDF interpretations does not need IDP , as IP can be shown to be a fuzzy subset of IR.
ulary V is defined by:
A simple fuzzy interpretation I of a vocab1. A non empty set IR of resources, called the domain or universe of I 2. A non empty set IDP , called the property domain of I 3. A fuzzy subset IP of IDP , called the set of properties of I 4. A fuzzy mapping IEXT : IP → 2IR× IR, i.e. the fuzzy set of pairs hx, yi with x, y ∈ IR. 5. A mapping IS from URI references ∈ V → (IR ∪ IDP ) 6. A mapping IL from typed literals ∈ V → IR 7. A distinguished subset LV ⊆ IR, called the set of literal values, which contains all the plain literals of V
The belonging of an element to the properties domain is strictly related to the use of such element as a property in a statement. Therefore, we have defined a membership degree to the property domain, intuitively related to the truth value of the statements in which the resource is used as a property. 2.3
The next step is to denfie the semantic conditions an interpretation must satisfy in order to be a model for a graph. We state the semantic conditions that relate the membership degree of a couple hsubject, objecti to an extension and the truth of a fuzzy statement.
We will use the abbreviated Zadeh’s notation A(x) = n, instead of μ A(x) = n,
to state that the membership degree of the element x to the set A is equal to
n [
– if E is a plain literal aaa ∈ V , then I(E) = aaa – if E is a plain literal aaa@ttt ∈ V , then I(E) = haaa, ttti – if E is a typed literal ∈ V , then I(E) = IL(E) – if E is a URI reference ∈ V , then I(E) = IS(E) – if E is a ground triple n: s p o., then I(E) = true if s, p and o ∈ V ,
IP (I(p)) ≥ n and IEXT (I(p))(hI(s), I(o)i) ≥ n, otherwise I(E) = false. – if E is a ground RDF graph, than I(E) = false if I(E0) = false for some triple E0 ∈ E, otherwise I(E) = true
Only the condition of truth and falsity of a ground statement in the interpretation is affected. The given formulation of the condition has as a consequence that a graph where the same statement appears more than once, with different membership degrees, is equivalent to a graph where the statement appears only once, with a membership degree equal to the maximum of the membership degrees.
Note that whether a statement is a model for a graph or not is not a fuzzy concept; it is either true or false. However, it could be interesting to compute the minimum and maximum membership degree to an extensions a couple must have in an interpretation to be a model of a given graph. This minimum degree has a role similar to the degree of truth of a statement in a knowledge base. 2.4
The definition of simple interpretation is not affected. A set S of RDF graphs (simply) entails a graph E if every interpretation which satisefis every member of S also satisfies E.
Given a graph G and a triple hs, p, oi, it could be interesting to compute the minimum and maximum value of n such that G entails n: s p o.. Those bounds must be taken in account when we compute the deductive closure of the graph, as it is not unique.
Section 2 of RDF Semantics [
The Empty Graph Lemma can be shown using the same proof. The denfiition of an empty graph is the same as in plain RDF: an empty graph is a graph with no statements at all. It is important to note that an empty graph can not be defined as a graph with no not-zero-valued statements. Statements such as 0: s p o., although pretty useless, cannot be ignored, as the semantic requirement that s, p and o must belong to the graph’s vocabulary still applies.
Subgraph Lemma, Instance Lemma and Merging Lemma retain both their validity and their proofs with the new semantics.
Interpolation Lemma, Anonymity Lemma, Monotonicity Lemma and
Compactness Lemma make use in their proof of a way of constructing an
interpretation of a graph using lexical items in the graph itself, the so called Herbrand
interpretation [
The (simple) Herbrand fuzzy interpretation of G, written Herb(G), can be denfied as follows.
– LVHerb(G) is the set of all plain literals in G; – IRHerb(G) is the set of all names and blank nodes which occur in subject or object position of statements in G; – IDPHerb(G) is the set of URI references which occur in the property position of statements in G; – IPHerb(G)(p) is the maximum of n for all statements in which p occur in property position; – IEXTHerb(G)(hs, oi) is the maximum n for all the statements n: s p o. in
G – ISHerb(G) and ILHerb(G) are both identity mappings on the appropriate parts of the vocabulary of G.
Using this definition of Herbrand interpretation instead of that in Appendix
A of [
RDF Semantic Conditions – IP (x) = IEXT (I(rdf : type))(hx, I(rdf : Property)) – If ”xxx”∧∧rdf : XMLLiteral ∈ V and xxx is a well-typed XML literal string, then • IL(”xxx” ∧ ∧rdf : XMLLiteral) is the XML value of xxx; • IL(”xxx” ∧ ∧rdf : XMLLiteral) ∈ LV ; • IEXT (I(rdf : type)) (hIL(”xxx” ∧ ∧rdf : XMLLiteral),
I(rdf : XMLLiteral)i) = 1 – If ”xxx”∧∧rdf : XMLLiteral ∈ V and xxx is an ill-typed XML literal string, then • IL(”xxx” ∧ ∧rdf : XMLLiteral) ∈/ LV ; • IEXT (I(rdf : type)) (hIL(”xxx” ∧ ∧rdf : XMLLiteral),
I(rdf : XMLLiteral)i) = 0
The rfist RDF semantic condition has the consequence that IP must be a subset of IR. Given such a fact, there is no more need of IDP , as it was for simple interpretation. IP can be directly denfied as a fuzzy subset of IR.
The second and third conditions equal to see the well-formedness of an XML Literal as crisp truth-valued. We could conceive an external machinery that can be considered completely trustworthy as it classify an XML literal as well-formed or not.
RDF axiomatic triples By definition, we give axiomatic triples a unit truth
value. Given the (syntactic) convention that a triple s p o. is equivalent to the
fuzzy statement 1: s p o., we can take the table of axiomatic triples of RDF in
section 3.1 of [
The path from RDF Schema to Fuzzy RDF Schema follows the same guidelines of the previous section.
The RDFS semantics is conveniently stated in terms of a new semantic
construct: the class [
In fuzzy RDF, extensions are fuzzy set of couples; in fuzzy RDFS, class extensions are fuzzy sets of domain’s elements. 3.1
A RDFS interpretation denfie the domains for resources ( IR), literals (IL) and literal values (LV ) in terms of classes. In fuzzy RDFS they are fuzzy subdomains of IR.
We will give RDFS semantic conditions and axiomatic triples, then we will try to explain the more problematic definitions: domains/ranges (section 3.2) and subproperties/subclasses (section 3.3).
RDFS semantic conditions – ICEXT (y)(x) = IEXT (I(rdf : type))(hx, yi) • IC = ICEXT (I(rdfs : Class)) • IR = ICEXT (I(rdfs : Resource)) • IL = ICEXT (I(rdfs : Literal)) – ICEXT (y)(u) ≥ min(IEXT (I(rdfs : domain))(hx, yi), IEXT (x)(hu, vi)) – ICEXT (y)(u) ≥ min(IEXT (I(rdfs : range))(hx, yi), IEXT (x)(hu, vi)) – IEXT (I(rdfs : subPropertyOf)) is transitive and reeflxive on IP – If IEXT (rdfs : subPropertyOf)(hx, yi) = n, then IP (x) ≥ n, IP (y) ≥ n, minha,bi{1 − IEXT (x)(ha, bi) + IEXT (y)(ha, bi)} ≥ n – IEXT (I(rdfs : subClassOf))(hx, I(rdfs : Resource)i) = IC(x) – If IEXT (rdfs : subClassOf)(hx, yi) = n, then IC(x) ≥ n, IC(y) ≥ n, mina{1 − IC(x)(a) + IC(y)(a)} ≥ n. – IEXT (I(rdfs:subClassOf)) is transitive and reeflxive on IC – IEXT (I(rdfs : subPropertyOf))(hx, I(rdfs : member)i) =
ICEXT (I(rdfs : ContainerMembershipProperty))(x) – ICEXT (I(rdfs : Datatype))(x) =
IEXT (I(rdfs : subClassOf))(hx, I(rdfs : Literal)i)
RDFS axiomatic triples As for RDF axiomatic triples, fuzzy RDFS axioms are
the same of plain RDFS, from section 4.2 of RDF Semantics [
To define the semantics of subClassOf and subPropertyOf, we need a relation of set inclusion between fuzzy sets that takes into account also the degree of the relation of inclusion itself. This relation must be transitive and reeflxive.
Zadeh’s denfiition of fuzzy subset [
Dubois and Prade [
A α B
⇐⇒ x ∈ (A ∪ B)α ∀x ∈ X , where α is a parameter and (· )α is the α -cut4. This relation is transitive only for α > 12 .
Other definitions of weak inclusion make use of inclusion grades. An inclusion grade I(A, B) is a scalar measure of the inclusion of the set A in the set B. In general, A ⊆ α B iff I(A, B) ≥ α , where ⊆ α denote a weak inclusion with inclusion grade α .
We have chosen to use the inclusion grade denfied as [
I(A, B) = infx∈X (A | − | where inf is the infimum and | − | is the bounded difference 5.
When A ⊆ B, I(A, B) = 1 [
This inclusion grade could also be written as I(A, B) = infx∈X (1− max(0, A(x)− B(x))) = infx∈X min(1, (1 − A + B)).
Furthermore, let’s suppose that there is at least an x such that A(x) > B(x). Then I(A, B) could be written as infx∈X (1 − A + B). The semantic condition requires such measure to be greater than or equal to n, where n is the truth value of the statement. In this case the semantic condition reduces to infx∈X (1 − A + B) ≥ n .
It could be interesting to ask how much this denfiition differs from the condition for classical fuzzy subsets, A(x) ≤ B(x).
If A ⊆ B, then I(A, B) = 1, so the semantic condition holds for any n ∈ [
Let’s call d(x) the difference d(x) = A(x)− B(x), so that 1− A+B = 1− d. We suppose that there is at least an x such that A(x) > B(x), so d(x) has at least a positive value. The semantic condition could then be written infx∈X (1 − d(x)) ≥ n. The maximum positive value of the difference d equal to 1 − n.
As n is the truth value of the statement that asserts the relation of subproperty or subclass, and 1 − n represent the lack of truth of the same statement, we can conclude that the maximum allowable positive difference between A(x) and B(x) is equal to the lack of truth on the subproperty or subclass relation. 4
RDF Model Theory’s entailment rules [
Given the way fuzzy RDF semantics is denfied, the corresponding inference rules for fuzzy RDF are analogous; only the fuzzy truth values of inferred statements must be computed. The simplest possible choice that respect the semantics is: – With rules as P ` Q, having only one antecedent, the truth value of the consequent Q is taken to be the same of the antecedent P . – With rules as P, Q ` R, the truth value of R is the minimum between the truth values of P and Q.
The inference rules for RDF/RDFS are shown in table 2. They were derived
from the rules used by the Sesame[
B)(x) = max(0, A(x) − B(x)) [
Sesame is a generic architecture for storing and querying RDF and RDF
Schema. It makes use of a forward-chaining inferencer to compute and store the
closure of its knowledge base whenever a transaction adds data to the
repository[
To obtain a fuzzy RDF storage and inference tool it is only a matter of modify Sesame RDF-MT inferencer, making it compute the correct truth values for inferred statements, and to extend the underlying storage to make room for a truth value (i.e., a number) for each statement.
This shows how a simple proof-of-concept fuzzy RDF inferencer is easy to implement. The starting point is the code base of an inference engine that implements the RDF model theory.
It can be shown that an inference engine implementing such rules is correct: all its rules are valid, in the sense that a graph entails any larger graph that is obtained by applying the rules to the original graph. There is no formal proof that it is also complete, but there is not such a proof for plain RDF Model Theory inference rules either. consequent iii: aaa rdf:type rdf:Property kkk: xxx rdf:type zzz where kkk = min(iii, jjj) kkk: xxx rdf:type zzz where kkk = min(iii, jjj) kkk: uuu rdf:type zzz where kkk = min(iii, jjj) kkk: uuu rdf:type zzz where kkk=min(iii, jjj) jjj: xxx rdf:type rdfs:Resource iii: uuu rdf:type rdfs:Resource kkk: aaa rdfs:subPropertyOf ccc where kkk=min(iii, jjj) kkk: aaa rdfs:subPropertyOf ccc where kkk=min(iii, jjj) iii: xxx rdfs:subPropertyOf xxx reflexivity of rdfs:subPropertyOf kkk: xxx bbb yyy where kkk=min(iii, jjj) kkk: xxx bbb yyy where kkk=min(iii, jjj) iii: xxx rdfs:subClassOf rdfs:Resource iii: xxx rdfs:subClassOf xxx reflexivity of rdfs:subClassOf kkk: xxx rdfs:subClassOf zzz where kkk=min(iii, jjj) kkk: xxx rdfs:subClassOf zzz where kkk=min(iii, jjj) kkk: aaa rdf:type yyy where kkk=min(iii, jjj) kkk: aaa rdf:type yyy where kkk=min(iii, jjj) iii: xxx rdfs:subPropertyOf rdfs:member jjj: xxx rdfs:subClassOf rdfs:Literal jjj: rdf:_* rdf:type rdfs:ContainerMembershipProperty This is an extra rule for list membership properties ( _1, _2, _3, ...). The RDF MT does not specify a production for this.