Seminar on Computational Intelligence

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Seminar on Computational Intelligence

• Topic
• Towards Fuzzy Description Logic for Semantic Web
• Presented by Martin Bitiboto

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Outline

• 1.Semantic Web
• Definition
• Historic
• WWW
• Intelligence Artificial in Semantic Web
• 2.Fuzzy description logic for Semantic Web
• 2.1. Description Logic
• 2.2. Classical SHOIN(D)
• 2.3. Preliminaries on Fuzzy Set Theories
• 2.4.Fuzzy SHOIN(D)
• Conclusion

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History of the Semantic Web

• Web was invented by Tim Berners-Lee (amongst
  others), when he was a physicist working at CERN
• This vision of the Web has become known as the
  Semantic Web

a plan for achieving a set of connected
applications for data on the Web in such a way as
to form a consistent logical web of data
an extension of the current web in which
information is given well-defined meaning, better
enabling computers and people to work in
cooperation
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Semantic Web

• Semantic Web is an extension of WWW, that enables
  machine to understand the contents of the Web and
  automatisation of process on the Web. So the
  contents of the Web will be understood of human
  and machine

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Semantic Web

• WWW World Wide Web contents only data that are
  human understandable

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Semantic Web

• Intelligence artificial may be use in SW to
  retrieve to classify data

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2. Fuzzy Description Logic for Semantic Web

• 2.1. Description Logic
• 2.2. Classical SHOIN(D)
• 2.3. Preliminaries on Fuzzy Set Theories
• 2.4.Fuzzy SHOIN(D)

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4. Fuzzy Description Logic for Semantic Web

• 4.1.Description Logics

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4.1.Description Logics (DL)

• Description Logics is the most recent name for
  a family of Knowledge Representation (KR)
  formalisms that represent the knowledge of an
  application domain (the world) by first
  defining the relevant concepts of the domain (its
  terminologies) and then using these concepts to
  specify properties of objects and individuals
  occurring in the domain (the world description).
• Description Logics are descended from so-called
  structured inheritance network Brachman,1977b
  1978, which were introduced to overcome the
  ambiguities in early semantic networks and frames
  and which were first realized in the system
  KL-One Brachman, Shmolze 1985

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4.1.Description Logics (DL)

• Architecture of Knowledge Representation based on
  Description Logics
  
• 
  Application Programs
  Rules

KR
TBox
Reasoning
Description language
ABox
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4.1.Description Logics (DL)

• KR system based on Description Logics provides
  facilities to set up knowledge bases, to reason
  about their contents and manipulate them.
• TBox introduced the terminologies i.e. the
  vocabulary of an application domain.
• ABox contains assertion about named individuals
  in term of this vocabulary.
• Reasoning reasoning allows to infer implicitly
  from knowledge that is explicitly contained in
  the knowledge base.
• TBox reasoning determine whether the a
  description is satisfiable (i.e.
  non-contradictory) or whether one description is
  more general than another one.

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4.1.Description Logics (DL)

• ABox reasoning determine whether its set of
  assertions is consistence or whether a particular
  individual is an instance of a given concept
  description.
• Description Language the language for building
  the description. It is a characteristic of each
  DL system.

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4. Fuzzy Description Logic for Semantic Web

• 4.2.Classical SHOIN(D)

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4.2.Classical SHOIN(D)

• Definition
• Syntax
• Semantic

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Definition of Classical SHOIN(D)

• SHOIN(D) is the corresponding Description Logic
  of OWL DL

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Syntax of classical SHOIN(D)

• SHOIN(D) allows to reason with concrete data
  types, such as strings and integers using
  so-called concrete domain
• Concrete domain a concrete domain D is a pair
  lt?D ,FDgt where ?D is an interpretation domain and
  FD is the set of concrete domain predicates d
  with a predefined arity n and interpretation
• dD ?nD
• Example for instance over integer20 may be an
  unary predicate denoting the set of integer
  greater or equal to 20.

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Syntax of classical SHOIN(D)

• RBox R consists of a finite set of transitivity
  axiom trans(R), and role inclusion axioms of the
  form R ? S and T ? U , where R and S are
  abstract role and U and T are concrete Role
• ABox A consists of a finite set of concept and
  role assertion axioms and individual (in)
  equality axioms
• The reflexivity Transitivity relationship is
  denoted with ?

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Syntax of classical SHOIN(D)

• Let C, Ra, Rc, Ia, Ic be non-empty finite and
  pair-wise disjoint sets of concepts name,
  abstract roles name, concrete roles name,
  abstract individual names and concrete individual
  names.
• Abstract role is abstract role name S or
  inverse S-1 of an abstract role name S (concrete
  role do not have inverse)

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Syntax of classical SHOIN(D)

• SHOIN (D) concepts is defined by the following
  syntactic rules
• C? ? ? A C1 ? C2 C1 ? C2C ? R.C
• ? R.C nS nSa1.an nTnT
• ?T1,....,Tn.D ?T1,,Tn.D
• D ? dc1,.,cn
• Where A is an atomic concept, R abstract
  role, S abstract simple role, Ti are concretes
  roles, d is a concrete domain predicate
• Example the concept
• Flower ? (?hasPetalWidht.(20mm ? 40mm)
  )??hasColor.Red)
• denotes the set of flowers having petal's
  dimension within 20mm and 40mm, whose color is
  red. 20mm and 40mm is concrete domain
  predicate.
• A SHOIN(D) knowledge base K ltT,R,Agt consists
  of a TBox T, a RBox R, and an ABox A

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Semantics of classical SHOIN(D)

• Interpretation I ( ?I, .I) an Interpretation
  I with respect to a concrete domain D is a pair
• I ( ?I, .I) , where ?I is non empty set
  called domain , disjoint from ?d and ( .I) is an
  interpretation function
• interpretation function .I assigns to each
• c ? C a subset of ?I , to each R ? Ra a
  subset of ?I x ?I , to each a ? Ia an element in
  ?I , to each c ? Ic an element in ?d , to each T
  ? Rc a subset ?I x ?D and to each n-ary concrete
  predicate d the interpretation dd ? ?nD

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Semantics of classical SHOIN(D)

• The mapping .I is extended to concepts and
  role as usual
• ?I ?I , ?I Ø ,
• (C1 ?C2)I C1I n C2I
• (C1 ?C2)I C1I U C2I
• (C)I ?I \CI
• (S-1) I ltx, ygt ltx, ygt ? S I
• (? R.C)I x ? ?I RI(x) ? CI
• (? R.C)I x ? ?I RI(x) n CI Ø
• (nS)I x ? ?I SI(x) n
• (nS )I x ? ?I SI(x) n
• a1.anI a1I.anI
• Similarity for other construct
• RI (x) y ltx, ygt ? RI
• In the following (1S) denoted (1S) ? (1S )

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Semantics of classical SHOIN(D)

• The satisfiability of an axiom E in an
  interpretation I ( ?I, .I) , denoted I ? E is
  defined as follows
• I ? C ? D iff CI ? DI ,
• I ? R ? S iff RI ? SI ,
• I ? T ? U iff TI ? UI ,
• I ? trans( R) iff RI is transitive,
• I ? aC iff aI ? CI ,
• I ? (a, b) R iff ltaI , bI gt ? RI ,
• I ? a b R iff aI bI ,
• I ? a? b R iff aI ? bI ,

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Semantics of classical SHOIN(D)

• Example
• let us consider the simple ontology TBox T
  about car with empty RBox (R Ø)
• Car ?(1maker) ?(1passager) ?(1speed)
• (1maker) ? Car T? ?maker.Maker
• (1passenger) ? Car ??passenger.IN
• (1speed) ? Car T? ?speed.km/h
• Roadster ? Cabriolet??passenger.2
• Cabriolet ? Car? ?topType.SoftTop
• SportsCar Car? ?speed. 240km/h
• in T the value for speed ranges over the
  concrete domain of kilometer per hour, while the
  value of the passenger ranges over the concrete
  domain of natural number, IN. the concrete
  predicate 240km/h is true if the value is
  greater than or equal to 240 km/h
• The ABox A
• mgb Roadster? (?maker. mg) ?
  ?(?speed.170 km/h)
• enzo Car? (?maker. ferrari) ? (?speed.
  gt340km/h)
• tt Car? (?maker. audi) ? (?speed.
  gt243km/h)

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4. Fuzzy Description Logic for Semantic Web

• 4.3. Preliminaries on Fuzzy Set Theories

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4.3. Preliminaries on Fuzzy Set Theories

• Fuzzy sets Fuzzy sets have been introduced by
  Zadeh as an approach to handle vagueness and
  uncertainty. While by classical sets an object
  belongs a set or not, Fuzzy sets have a gradation
  of belonging. Fuzzy sets theories allow an object
  to be partial member of the set.
• Membership function Fuzzy sets are defined by
  a membership function.

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4.3. Preliminaries on Fuzzy Sets Theories

• µA represents the degree of the membership of
  which x element of Universal set X, belongs to
  the fuzzy set A.
• µA is usually expressed as a member between 0 and
  1 µA (x) X-gt0,1
• If µA (x)1 then x is definitely belongs to A,
  while µA (x) 0.8 mean that x is close to be an
  element of A.

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4.3. Preliminaries on Fuzzy Set Theories

• Subset let A, B elements of F (X) . A is a
  subset of B iff A(x) B(x) ,for any element x of
  X
• Complement of fuzzy set A ,A is defined by
  membership function
• µA(x) n(µA(x) ), for any element x of X.

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4.3. Preliminaries on Fuzzy Set Theories

• negation function n n0,1-gt0,1 has to
  satisfy the following conditions and extends
  Boolean negation.
• -n(0)1 and n(1)0
• -?a, b, ? 0,1 ,n(a) n(b) implies n(b)
  n(a)
• -? a ? 0,1 n(n(a)) a
• Some examples of negation functions
• Lukasiewicz negation nL(a) 1-a (syntax
  L )
• Goedel negation nG(0)1 and nG(a)0 if
  agt0 (syntax G)

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4.3. Preliminaries on Fuzzy Set Theories

• Intersection the intersection of two fuzzy sets
  is given by
• µ(A ?B)(x) t(µA(x), µB(x)), where t is a
  triangular norm, or simply t-norm.
• t-norm is a function t0,10,1-gt0,1
  that has to satisfy the following
  conditions
• -? a ? 0,1, t(a,1)a
• -? a, b, c ? 0,1 b c implies t(a, b)
  t(a, c).
• -? a, b ? 0,1 t(a,b) t(b,a)
• -? a, b, c ? 0,1, t(a,t(b,c)) t( t(a,
  b), c)
• -? a ? 0,1, t(a,0) 0
• some examples of t-norm
• Lukasiewicz t-norm tL(a, b)
  max(ab-1,0) (syntax ?G)
• Gödel t-norm tG(a, b) min(a, b)
  (syntax ?G)
• Product t-norm tP(a, b) a.b (syntax
  ?P)

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4.3. Preliminaries on Fuzzy Set Theories

• Union union of two fuzzy set is given by
• µ(A v B)(x) s( µA(x), µB(x)), where s is a
  triangular co-norm, or simply s-norm.
• s-norm is a function s 0,1x0,1-gt0,1
  that has to satisfy the following conditions
• - ? a ? 0,1, s(a,0)a
• - ? a, b, c ? 0,1, bc implies
• s(a, b) s(a, c)
• - ? a, b ? 0,1 s(a, b) s(b, a)
• - ? a, b, c ? 0,1, s(a, s(b, c)) s(s(a,
  b), c)
• - ? a, b ? 0,1, s(a, b) n(t(n(a),
  n(b))) (Morgan Law)
• some examples of s-norm
• Lukasiewicz s-norm sL(a, b) min(ab,1)
  (syntax vL)
• Gödel s-norm sG(a, b) max(a, b) (syntax
  vG)
• Product t-norm tP(a, b) (ab-a.b)
  (syntax vp)

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4.3. Preliminaries on Fuzzy Set Theories

• Implication denoted ?that gives the truth value
  of A ? B , when the truth value of A and B is
  known. The fuzzy implication is given by a
  function i0,1x0,1-gt0,1 that has to satisfy
  the following conditions
• - ? a ? 0,1, i(0,a)1
• - ? a ? 0,1, i(a,1)1
• - i(1,0)1
• - ? a, b, c ? 0,1, a b implies
• i(a, c) i(b, c)
• - ? a, b, c ? 0,1, bc implies
• i(a, b) i(a, c)

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4.3. Preliminaries on Fuzzy Set Theories

• Some example of fuzzy implication
• Kleene-Dienes implication
• ? a, b, ? 0,1, ikd(a, b) max(1-a,b)
  (syntax ?kd)
• generalization of classical implication into
  fuzzy implication
• -for classical logic implication
• (a gt b )ltgt (a v b)
• in fuzzy implication ? a, b, ? 0,1,
  i(a, b) s(n(a), b)
• so for Lukasiewicz implication iL (a,
  b)sL(nL(a), b)
• and for Goedel implication iG (a,
  b)sG(nG(a), b)
• but for Product implication iP ?sp(nP(a), b)

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4.3. Preliminaries on Fuzzy Set Theories

• Some example of fuzzy implication
• If we turn (a v b) into (a ? b), then
• for Lukasiewicz implication
• iL (a, b) nL (tL ((a,nL(b)))
• but for Goedel implication
• iG (a, b) ?nG (tG ((a,nG(b))
• And for Product implication
• iP(a, b) ?nP (tP ((a, nP(b))

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4.3. Preliminaries on Fuzzy Set Theories

• Some example of fuzzy implication
• residuum based implication
• -in classical logic (a v b) can be
  rewritten as
• max c?0,1 a ?b b
• so in fuzzy implication
• ? a, b, ? 0,1, i(a, b) sup c ? 0,1
  t(a, c) b
• For residuum based implication
• - i(a, b) 1 if a b
• -if agt b then according to the chosen t-norm
  we have e.g.
• Lukasiewicz implication iL(a, b) 1-ab
  (syntax ?L)
• Goedel implication iG(a, b)b (syntax ?G)
  
• Product implication iP(a, b) a/b (syntax
  ?P)

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4.3. Preliminaries on Fuzzy Set Theories

• degree of subsumption define a degree of subset
  relationship between two fuzzy sets. Let fuzzy
  sets A and B elements of the crisp set X
• Degree of subsumption between A and B ,
  denoted A ?B, is given by
• inf x?X i(µA(x), µB(x)) where I is an
  implication function.

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4.3. Preliminaries on Fuzzy Set Theories

• A (binary) fuzzy relation R over two countable
  crisp sets X and Y is a function R X x Y-gt0,1.
• The inverse of R is R-1 Y x X-gt0,1 with
  membership function R-1(y,x)R (x,y), for every
• x ? X and y ?Y
• The composition of two fuzzy relation
• R1 X x Y -gt0,1 and R2 Y x Z -gt0,1 , the
  composition of R1 and R2 is defined as
• (R1oR2 )(x, z) sup y ?Y t(R1(x,y), R2(y,z)
  ), where t is t-norm.
• Transitivity of fuzzy relation
• fuzzy relation R X x Y-gt0,1 is said
  transitive iff
• R(x, y) (Ro R) (x, y)

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4.3. Preliminaries on Fuzzy Set Theories

• Fuzzy modifier fuzzy modifier applies to fuzzy
  set to change their membership. Well know
  examples are modifiers like very, more or less,
  slightly, etc.
• For example when I say it's very close to
  Zero, the world very modifies close to zero
  which is fuzzy set.
• a modifier m is a function m0,1-gt0,1
• the whole family of modifier is generated by
  function µp where µ is membership function and p
  ? 0,8.
• While p8 the modifier could be named
  exactly because it suppressed all membership
  function less than 1.0

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4. Fuzzy Description Logic for Semantic Web

• 4.4. Fuzzy SHOIN(D)

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4.4. Fuzzy SHOIN(D)

• The difficulty with classical SHOIN is to handle
  imprecise concept. So with fuzzy set and logic
  theories the classical SHOIN(D) is extended to
  handle this kind of concept.

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Syntax of fuzzy SHOIN(D)

• concrete Fuzzy domain is based on Fuzzy sets. And
  is a pair lt?D ,FDgt , where ?D is an
  interpretation domain ,FD is a set of concrete
  domain predicates with fuzzy a predefined arity n
  and an interpretation
• dd ?nD -gt0,1 with a n- ary fuzzy relation
  over ?D .
• a fuzzy RBox R is a finite set of of SHOIN(D)
  transitivity axioms trans(R) and fuzzy role
  inclusion axioms of the form
• lta n gt, lta ngt, lta gt ngt, and lta lt ngt where a
  is a SHOIN(D) role inclusion
• A fuzzy TBox T consists of fuzzy concept
  inclusion axioms of the form lta n gt, lta ngt, lta
  gt ngt, and lta lt ngt where a is a SHOIN(D) concept
  inclusion axiom (C ?D)
• A fuzzy ABox A consists of a set of fuzzy
  concept and fuzzy role assertion axiom of the
  form lta n gt, lta ngt, lta gt ngt, and lta lt ngt where
  a is a SHOIN(D) concept role assertion
• Fuzzy knowledge base KltT,R,Agt consists of a
  fuzzy TBox T, a fuzzy RBox R, and a fuzzy ABox A

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Semantics of fuzzy SHOIN(D)

• This semantics extends classical SHOIN(D)
  semantics. The main idea is that concepts and
  roles are interpreted as fuzzy subsets of an
  interpretation's domain. Therefore SHOIN(D)
  axioms, rather being satisfied (true) or
  unsatisfied (false) in an interpretation, become
  a degree of truth in 0,1.

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Semantics of fuzzy SHOIN(D)

• A fuzzy interpretation I w.r.t a concrete
  domain D is a pair I ( ?I, .I) consisting of a
  non empty set ?I called domain, disjoint from ?D
  and a fuzzy interpretation function I that
  assigns
• To each abstract concept c ? C a function CI ?I
  -gt 0,1
• To each role R ? Ra a function
• RI ?I x ?I -gt0,1
• To each a ? Ia an element in ?I
• To each c ? Ic an element in ?D
• To each concrete role T ? Rc a function
• RI ?I x ?D -gt0,1
• To each modifier m ? M a fixed function
• m 0,1-gt0,1
• To each n-ary concrete predicate d the
  relation
• dD ?D -gt0,1

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Semantics of fuzzy SHOIN(D)

• Example
• SportCar Car?? speed.very(High) ,where very
  is concept modifier and high is a fuzzy concrete
  predicate over the domain of speed expressed in
  kilometer per hour and may be defined as High
  (x) min (1, 0.oo4x)
• Concerning concepts and roles, the syntax is as
  for SHOIN(D), except that we allow modifiers in
  concept expressions. That is if M is a new
  alphabet for modifier symbols, m element of M is
  a modifier and C is a SHOIN(D) concept, then m(C)
  is fuzzy SHOIN(D) concept.

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Conclusion

• We have presented a classical SHOIN(D) and its
  fuzzy extension. Fuzzy SHOIN(D) allows modifiers,
  fuzzy concrete domain predicates and fuzzy axioms
  to appear in SHOIN(D). This allows the
  interpretation of vague concepts. And vague
  concepts are abundant in human knowledge, and
  thus appears likely in Web content.

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Thank you