The DL System (see also latest changes)

fuzzyDL is a Description Logic Reasoner supporting Fuzzy Logic and fuzzy Rough Set reasoning. The fuzzyDL system includes a reasoner for fuzzy SHIF with concrete fuzzy concepts (ALC augmented with transitive roles, a role hierarchy, inverse, reflexive, symmetric roles, functional roles, and explicit definition of fuzzy sets). There is also a Protege plug-in that supports Fuzzy OWL2, to build fuzzy ontologies.

fuzzyDL's most interesting features are:

• it extends the classical Description Logic SHIF to the fuzzy case
• it allows the explicti definition of fuzzy concepts with left-shoulder, right-shoulder, triangular and trapezoidal membership functions
• it supports concept modifiers in terms of linear hedges
• it supports General Inclusion Axioms
• it supports "Zadeh semantics" and Lukasiewicz Logic
• it is backward compatible, i.e. it supports classical description logic reasoning
• full support of Mixed Integer Linear Programming optimization
• it supports fuzzy Rough Set reasoning within fuzzy DLs

fuzzyDL is a joint project with Fernando Bobillo.

Some applications:

• Matchmaking
• Knowledge-based Image Interpretation
• Multi Criteria Decision Making
• Clinical Decision Support
• Logic-based Fuzzy Control Systems

An example:

The following knowledge defines the classes of Minors and Young Persons.

1. (define-fuzzy-concept LessThan18 crisp(0,100,0,18))
2. (define-fuzzy-concept LowAge left-shoulder(0,100,10,30))
3. (define-concept Minor (and Person (some age LessThan18)))
4. (define-concept YoungPerson (and Person (some age LowAge)))
5. (functional age)
6. (define-modifier very linear-modifier(3))
7. (define-concept VeryYoungPerson (very YoungPerson))

• Axiom 5 asserts that age is functional, i.e. a person has one age only.
• Axiom 1 asserts that the concept LessThan18 is a fuzzy set with a crisp membership function crisp(k1,k2,a,b)) of the form below, where [k1,k2] is the domain. Essentially, it says that the domain is [0,100] and that membership is 1 for x in [0,18] .

• Axiom 2 asserts that the concept LowAge is a fuzzy set with a left-shoulder membership function left-shoulder(k1,k2,a,b) of the form below.

• Axiom 3 defines the concept of Minor as a Person whose age is LessThan18, i.e. less than 18.
• Axiom 4 defines the concept of YoungPerson as a Person whose age is LowAge.

As a consequence, under Zadeh semantics we get that a YoungPerson is a Minor to degree 0.4. Indeed,

(concept-subsumes? Minor YoungPerson)

returns 0.4.

On the other hand

(concept-subsumes? YoungPerson Minor)

returns 0.6, which is the degree of being a Minor a YoungPerson.

Other membership functions supported by fuzzyDL are shown below.

Right-shoulder function	Trapezoidal function	Triangular function

Linear Modifier

• Axiom 6 defines the modifier very using linear-modifier(c), where a = c/(c+1), b=1/(c+1)
• Axiom 6 defines the concept VeryYoungPerson as a very YoungPerson.

As a consequence.

(concept-subsumes? VeryYoungPerson Minor)

returns 0.2, which is the degree of being a Minor a VeryYoungPerson.

On the other hand

(concept-subsumes? Minor VeryYoungPerson)

returns 0.8, which is the degree of being a VeryYoungPerson a Minor.