1. FiRE Fuzzy Reasoning Engine Nikolaos Simou National Technical University of Athens

2. Outline Introduction to f-SHIN  Syntax  Semantics Fuzzy Reasoning Engine FiRE  Syntax  Reasoning Services FiRE Integration with Sesame  Benefits Demonstration

3. Uncertainty and knowledge representation Uncertainty is part of our lives… i.e. tall, fat, soon Description Logics (DLs) lacks the ability to represent imprecise or vague information This representation is necessary for many application domains (medical informatics, multimedia processing) DLs were extended using fuzzy set theory

4. Fuzzy Set Theory An object belongs to a fuzzy set to any degree between 0 and 1.  Tall(Tom) = 0.7 Set theoretic operations are performed by mathematical functions  1-x is used for negation  min(x,y) for intersection  max(x,y) for union  max(1-x,y) for implications

5. SHIN SHIN is an AL language AL-language is represented by a string of the form AL[U][E][N][C] U: union E: full existential quantification N:Number Restrictions C:Complement (negation) Negation is like union and full existential quantification  ALC = ALUE Similar for roles S : transitive roles H : role implication I : inverse roles

6. Syntax of Fuzzy SHIN f-SHIN concepts are formed in the same way as in SHIN C,D ::=⊤ | ⊥ | ¬C | C ⊓ D | C ⊔ D | ∃R.C | ∀R.C | ≥nR | ≤nR R,P::= R - | Trans(R) | P⊑R

7. Fuzzy knowledge bases A fuzzy knowledge base is a triple Σ= (T,R, A) where:  T is a finite set of fuzzy inclusion axioms: A ⊑ C or fuzzy equivalences: A ≡ C, called a fuzzy TBox  R is a finite set of fuzzy transitive role axioms: Trans(R) or fuzzy role inclusion axioms P ⊑ R, called a fuzzy RBox  A is a finite set of fuzzy assertions: 〈 a : C ⋈ n 〉 or 〈 (a, b) : R ⋈ n 〉, where ⋈ ∈ {≥,>,<, ≤ }, called a fuzzy ABox.

8. Semantics of f-SHIN A fuzzy interpretation is a pair I = (Δ I ×.I ) where Δ I is the domain of interpretation and.I is the interpretation function which maps An individual name α ∈ I to an element α I ∈ Δ I A concept name A to a membership function A I : Δ I →[ 0,1 ] A role name R to a membership function R I : Δ I × Δ I →[ 0,1 ]

9. Extending semantics to complex concepts

10. Interpreting Axioms

11. FiRE FiRE is a Fuzzy Reasoning Engine implemented using JAVA Currently supports f-SHIN DL Language Consist of an alphabet of concepts (C) roles (R) and individuals (I)

12. Declaring Alphabet Alphabet declaration is made using the keyword signature and it is divided into three smaller declarations the concepts declaration using keyword atomic-concepts, the roles declaration using keyword roles and the individuals declaration using keyword individuals.

13. Declaring Alphabet -An Example (signature :atomic-concepts (human person female male woman man parentC mother father sister brother parent- having-many-children tall clever thin organised parent- NOT-having-many-children parent-having-3-children beautifull fast strong) :roles ((has-gender :transitive t) (has-descendant :transitive t) (has-child :inverse has-descendant) (has-sibling) (has- degree) (has-sister) (has-brother) ( has-friend) ) :individuals (Alice Betty Charles Doris Eve Peter))

14. Declaring Alphabet Roles are declared using keyword roles. Each role is defined within brackets and it may have one or more of the following parameters KeywordDescriptionSyntax parentStates the parent Role:parent parentRole transitiveStates whether the role is transitive or not. (A role is not transitive by default) :transitive t :transitive f inverseStates the inverse role of a role. (All roles have an inverse role and if it is not stated directly then it is Inv-roleName e.g for role has-child inverse role by default is Inv-has-child) :inverse invRole

15. f-SHIN Constructors Syntax SyntaxDL NotationDescription *top* ⊤ Top concept *bottom* ⊥ Bottom concept (not C)¬CGeneral negation (and C 1...C n ) C 1 ⊓... ⊓ C n Conjuction (or C 1...C n ) C 1 ⊔… ⊔ C n Disjunction (some R C) ∃R.C Exist restriction (all R C) ∀ R.C Value restriction (at-most n R) ≤nR At most restriction (at-least n R) ≥nR At least restriction

16. Declaring Axioms There are two kinds of axioms inclusion and equivalence. Defined concept has to be an atomic concept and on the other hand property concept can be an atomic concept or a result of f-SHIN constructors. Syntax  (implies/equivalent definedConc propertyConc) Examples  (implies man (and person (some has-gender male)))  (equivalent father-having-only-sons (and man (all has-child man)))

17. Declaring Assertions Fuzzy Instances  Syntax Using the keyword instance followed by the individual, the concept the inequality symbol (>,>=,<,<=) and the degree of participation.  Example (instance Charles brother) (instance Charles (and tall thin) >= 0.7 )

18. Declaring Assertions Fuzzy Relations  Syntax Using the keyword related followed by the individuals names, the role that relates them, the inequality type and the degree of participation  Examples (related Peter Betty has-sister ) (related Peter Charles has-friend >= 0.8 )

19. FiRE Interface The FiRE user interface consists of an editor pane (upper left), inference services pane (upper right-Entailment, Subsumption,Glb), and output pane (the bottom-Tableaux Expansion, Output, Tableaux, Model, Classification).

20. Reasoning Services Consistency  Checks the knowledge for consistency.  The Tableaux expansion pane presents the tableaux expansion in detail showing all the operators applied.  The Output pane shows the statements declared, or differently the syntax mistakes  The Tableaux pane shows the results of tableaux expansion  The model pane shows the resulting model.

21. Reasoning Services Entailment  Entailment queries evaluate whether an individual participates in a concept with a specific degree. FiRE answers true or false to these queries  Syntax (instance individual concept ⋈ degree) or individual concept ⋈ degree  Example ( instance Doris (and human (some has-gender (or female male))) >= 1) or Doris (and human (some has-gender (or female male))) >= 1

22. Reasoning Services Subsumption  Subsumption queries evaluate whether a concept is subsumed by another concept. FiRE answers true or false to these queries  Syntax (implies C 1 C 2 ) or C 1 C 2  Example ( implies father man) or father man Greatest Lower Bound  Greatest lower bound queries evaluate the greatest possible degree of an individual participating in a concept  Syntax Similar to entailment queries

23. Reasoning Services Classification This action performs subsumption checks of all the possible combinations of concepts and it creates the taxonomy tree shown in Classification Pane.

24. What is sesame Sesame is an open source Java framework  Storing  Querying and  Reasoning with RDF and RDF Schema. (also supports N3, N-Triple, Turtle) It can be used  As a database for RDF and RDF Schema  As a Java library for applications that need to work with RDF internally

25. Sesame Architecture

26. Sesame Integration Converts knowledge to N-Triples and exports it to Sesame Evaluates greatest lower bound for all the individuals participating in all the concepts (Global GLB) and exports implicit knowledge to Sesame. Imports knowledge from sesame and evaluates Global GLB

27. Benefits Data Stored in a sesame repository  Data can then easily stored to a database Server-Client Architecture  Easy access to data Queries  Which individual is tall > 0.6, clever > 0.5 and also has a sister? Reasoning is off-line operation

28. Future Extensions Apply DL Optimizations. Fuzzy Queries Extend to other uncertainty formalisms. Extend the DL component’s expressiveness (SHOIN exist). Support data types. Support rules.

29. FiRE URL You can download FiRE from http://www.image.ece.ntua.gr/~nsimou/